J. S. Trefil Auth. Introduction to the Physics of Fluids and Solids 1975

Introduction to the Physics

of Fluids and Solids

J. S. Trefil Depar tment of Physics, University of Virginia

Pergamon Press Inc. New York • Toronto • Oxford • Sydney • Braunschweig

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Copyright © 1975, Pergamon Press Inc. Library of Congress Cataloging in Publication Data

Trefil, J S Introduction to the physics of fluids and solids.

Includes bibliographies. 1. Fluids. 2. Solids. I. Title.

QC145.2.T73 1975 531 74-2153 ISBN 0-08-018104-X

All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form, or by any means, electronic, mechanical, photocopying,

recording or otherwise, without prior permission of Pergamon Press Inc.

Printed in the United States of America

my sons Jim and Stefan

Preface

I t h a s b e c o m e i n c r e a s i n g l y c l e a r o v e r t h e p a s t f e w y e a r s t h a t a s i z a b l e

p e r c e n t a g e of t h e s t u d e n t s w h o l e a v e u n i v e r s i t i e s w i t h d e g r e e s in p h y s i c s

wi l l n o t e n d u p d o i n g r e s e a r c h in a r e a s n o r m a l l y iden t i f i ed w i t h c u r r e n t

r e s e a r c h . T h e i n c r e a s e d c o n c e r n w i t h t h e e n v i r o n m e n t a n d w i t h a p p l i e d

r e s e a r c h h a s m e a n t t h a t t h e s e s t u d e n t s o f t e n find t h e m s e l v e s w o r k i n g in

fields l ike o c e a n o g r a p h y o r a t m o s p h e r i c p h y s i c s . I n t h e l o n g - r a n g e

h i s t o r i c a l v i e w , t h i s i s n o t s t r a n g e , s i n c e t h e p h y s i c i s t h a s t r a d i t i o n a l l y

p l a y e d t h e r o l e of t h e g e n e r a l i s t in t h e p a s t . T h e q u e s t i o n a b o u t w h i c h I

h a v e b e c o m e i n c r e a s i n g l y c o n c e r n e d i s " A r e w e e q u i p p i n g o u r s t u d e n t s t o

b e t h e g e n e r a l i s t s of t h e f u t u r e ? "

T h e r e is a g r o w i n g b o d y of o p i n i o n in t h e p h y s i c s c o m m u n i t y t h a t is

c o m i n g t o t h e c o n c l u s i o n t h a t t h i s q u e s t i o n m u s t b e a n s w e r e d in t h e

n e g a t i v e . M y o w n t h e o r y a b o u t h o w t h i s s t a t e of af fa i rs c a m e a b o u t i s t h a t

w e h a v e , t o a l a r g e e x t e n t , s t o p p e d t e a c h i n g p h y s i c s s t u d e n t s a b o u t m a n y

a r e a s of c l a s s i c a l p h y s i c s . T h a t t h i s s h o u l d h a v e h a p p e n e d is n o t

s u r p r i s i n g , s i n c e m o d e r n p h y s i c s r e s e a r c h is c o n c e r n e d a l m o s t e x c l u -

s i v e l y w i t h q u a n t u m s y s t e m s , s u c h a s n u c l e i , e l e m e n t a r y p a r t i c l e s , o r

e l e c t r o n s in a so l id . T h u s , t h e r e is a c o n s i d e r a b l e a d v a n t a g e t o t h e s t u d e n t

g o i n g i n t o t h e s e fields t o b e i n t r o d u c e d t o q u a n t u m m e c h a n i c s a s s o o n a s

p o s s i b l e in h i s u n d e r g r a d u a t e c a r e e r . U n f o r t u n a t e l y , t h i s a d v a n t a g e h a s

b e e n g a i n e d a t t h e e x p e n s e of d r o p p i n g t h e s t u d y of m a n y a r e a s of

c l a s s i c a l p h y s i c s f r o m t h e c u r r i c u l u m . W e a r e n o w c o n f r o n t e d w i t h a

s i t u a t i o n in w h i c h p h y s i c s g r a d u a t e s m a y h a v e l i t t le o r n o a w a r e n e s s of

t h e g r e a t b o d y of k n o w l e d g e of fluid m e c h a n i c s a n d e l a s t i c i t y w h i c h w a s

g a i n e d b e f o r e t h e b e g i n n i n g of t h i s c e n t u r y .

xi

xii Preface

O r d i n a r i l y , t h i s w o u l d b e u n f o r t u n a t e f r o m a c u l t u r a l p o i n t of v i e w , b u t

w o u l d b e of l i t t le p r a c t i c a l i m p o r t a n c e . T h e e m p l o y m e n t s i t u a t i o n

m e n t i o n e d a b o v e , h o w e v e r , g i v e s t h e q u e s t i o n of e d u c a t i o n in t h e s e fields

s o m e u r g e n c y , s i n c e it is p r e c i s e l y in t h e s e a r e a s t h a t m o s t of t h e a p p l i e d

r e s e a r c h wil l b e d o n e . T h i s p o i n t w a s b r o u g h t h o m e t o m e m o s t f o r c e f u l l y

w h e n I b e c a m e i n v o l v e d in s o m e i n t e r d i s c i p l i n a r y r e s e a r c h p r o j e c t s in

m e d i c i n e , a n d d i s c o v e r e d t o m y c h a g r i n t h a t I d i d n o t p o s s e s s t h e

b a c k g r o u n d n e c e s s a r y t o m a k e m e a n i n g f u l c o n t r i b u t i o n s in m a n y a r e a s of

t h e r e s e a r c h .

A f t e r r e f l ec t ing o n t h e s e p r o b l e m s , I d e c i d e d t o t r y t o p u t t o g e t h e r a

c o u r s e of l e c t u r e s w h i c h w o u l d a t t e m p t , in o n e s e m e s t e r , t o a l l o w

g r a d u a t e s a n d a d v a n c e d u n d e r g r a d u a t e s in p h y s i c s t o l e a r n a b o u t t h e s e

fields. T h e r e s t r i c t i o n t o a o n e s e m e s t e r c o u r s e h a s t h e a d v a n t a g e t h a t it

d o e s n o t u n d u l y d i s t o r t t h e o r d i n a r y c o u r s e s c h e d u l e s w h i c h a s t u d e n t is

e x p e c t e d t o c a r r y , a n d t h e o b v i o u s d i s a d v a n t a g e a s s o c i a t e d w i t h t r y i n g t o

c o v e r a lo t of m a t e r i a l in a s h o r t t i m e . M y c o l l e a g u e s a t t h e U n i v e r s i t y of

V i r g i n i a r e s p o n d e d t o t h i s i d e a w i t h a g r e a t d e a l of e n t h u s i a s m a n d

s u p p o r t , f o r w h i c h I a m d e e p l y in t h e i r d e b t , a n d t h e c o u r s e w a s o f f e r e d

u n d e r t h e t i t le " T o p i c s in C l a s s i c a l P h y s i c s . " T h i s b o o k is a n o u t g r o w t h of

t h e c o u r s e , w h i c h h a s b e e n g i v e n f o r t h e p a s t t h r e e y e a r s .

T h e p u r p o s e of t h i s t e x t is t w o f o l d . F i r s t , a n a t t e m p t is m a d e t o s h o w

t h e s t u d e n t t h a t t h e r e is n o e s s e n t i a l n e w k n o w l e d g e w h i c h h e m u s t

m a s t e r t o l e a r n a b o u t c o n t i n u u m m e c h a n i c s . I n f a c t , t h e b a s i c e q u a t i o n s

a r e s i m p l y t h e a p p l i c a t i o n s of l a w s which he already knows t o n e w

s i t u a t i o n s . F o r e x a m p l e , t h e E u l e r e q u a t i o n is s i m p l y a d i s g u i s e d f o r m of

N e w t o n ' s s e c o n d l a w .

S e c o n d , it is s h o w n t h a t o n c e t h e s e f e w b a s i c p r i n c i p l e s a r e u n d e r s t o o d ,

t h e y c a n b e a p p l i e d t o a n a l m o s t u n b e l i e v a b l e n u m b e r of s y s t e m s w h i c h

a r e s e e n in n a t u r e . T h u s , o n c e t h e l a w s g o v e r n i n g t h e m o t i o n of

n o n v i s c o u s fluids a r e u n d e r s t o o d , w e c a n e q u a l l y w e l l d i s c u s t h e

s t r u c t u r e of t h e g a l a x y (a s in C h a p t e r 2) o r n u c l e a r fission ( a s in C h a p -

t e r 7 ) .

T o e m p h a s i z e t h e s e c o n d p o i n t , a l a r g e n u m b e r of e x a m p l e s f r o m m a n y

fields of p h y s i c s h a v e b e e n c o l l e c t e d in t h e t e x t . P a r t l y t h i s i s i n t e n d e d t o

g i v e t h e flavor of d e v e l o p m e n t s in t h e s e fields, a n d p a r t l y it is i n t e n d e d t o

c o l l e c t , in o n e c o n v e n i e n t l o c a t i o n a n d in o n e c o h e r e n t d e v e l o p m e n t ,

p r o b l e m s f r o m a s m a n y p h y s i c s - r e l a t e d fields a s p o s s i b l e . C l e a r l y , e a c h

r e a d e r wil l h a v e h i s o w n t a s t e a s t o w h i c h e x a m p l e s s h o u l d h a v e b e e n

i n c l u d e d a n d w h i c h o m i t t e d . S p a c e c o n s i d e r a t i o n s a l o n e w o u l d d e c r e e

t h a t s o m e i m p o r t a n t a r e a s of p h y s i c s w o u l d h a v e t o b e lef t o u t . T h u s , t h e

Preface xiii

d i s c u s s i o n of s t e l l a r s t r u c t u r e i g n o r e s m a g n e t i c a n d t h e r m a l e f f e c t s , t h e

d i s c u s s i o n of b l o o d flow i g n o r e s d i f fus ion p r o c e s s e s , e t c . A n i n s t r u c t o r

u s i n g t h i s b o o k a s a t e x t c a n , of c o u r s e , s u p p l y h i s o w n e x a m p l e s if h e s o

d e s i r e s .

T h e g e n e r a l p r o c e d u r e f o l l o w e d in t h e d e v e l o p m e n t is t o i n t r o d u c e a

p h y s i c a l p r i n c i p l e f irst , w i t h a n e m p h a s i s o n t h e n a t u r e of t h e p r i n c i p l e a n d

i t s c o n n e c t i o n t o t h i n g s a l r e a d y f a m i l i a r t o t h e s t u d e n t , a n d t h e n t o a p p l y

t h e p r i n c i p l e t o s o m e i n t e r e s t i n g s y s t e m . S o m e t i m e s t h i s i s d o n e in

s e p a r a t e c h a p t e r s (e .g . , C h a p t e r 4 d e a l s w i t h t h e f o r m a l i s m f o r d e a l i n g

w i t h fluids in m o t i o n , C h a p t e r s 5 , 6, a n d 7 w i t h a p p l i c a t i o n s ) , a n d

s o m e t i m e s in t h e s a m e c h a p t e r s (e .g . , C h a p t e r 11 i n t r o d u c e s t h e p r i n c i p l e s

of s t a t i c s in e l a s t i c s o l i d s a n d a p p l i e s t h e m t o g e o l o g i c a l s y s t e m s ) . T h e

m a t h e m a t i c a l d i s c u s s i o n is m o r e o r l e s s s e l f - c o n t a i n e d , b u t s o m e

a p p e n d i c e s o n m a t h e m a t i c s a r e i n c l u d e d a t t h e e n d f o r t h e s a k e of

c o m p l e t e n e s s .

T h e c o m p l e t i o n of a b o o k l ike t h i s is c l e a r l y n o t t h e w o r k of a s ing le

i n d i v i d u a l . M a n y t h a n k s a r e d u e b o t h t o m y c o l l e a g u e s a n d t o t h e s t u d e n t s

w h o a c t e d a s s u b j e c t s f o r t h i s e x p e r i m e n t in p h y s i c s t e a c h i n g . B o t h

g r o u p s m a d e m a n y v a l u a b l e s u g g e s t i o n s w h i c h I h a v e i n c o r p o r a t e d i n t o

t h e d e v e l o p m e n t of t h e s u b j e c t .

S p e c i a l m e n t i o n s h o u l d b e m a d e of m y c o l l e a g u e s in t h e w o r k o n t h e

u r i n a r y d r o p s p e c t r o m e t e r ( s e e C h a p t e r 14), R o g e r s R i t t e r a n d N o r m a n

Z i n n e r , M . D . , w h o first i n t r o d u c e d m e t o t h e f a s c i n a t i n g field of m e d i c a l

r e s e a r c h , a n d t o G . A i e l l o a n d P . L a f r a n e e , w h o h a v e b e e n w o r k i n g a n d

l e a r n i n g w i t h u s .

F i n a l l y , I w o u l d l i ke t o t h a n k M r s . M a r y G u t s c h f o r h e r i n v a l u a b l e

a s s i s t a n c e i n p u t t i n g t h e m a n u s c r i p t t o g e t h e r , a s w e l l a s f o r h e r r e f u s a l t o

b e i n t i m i d a t e d b y t h e a m o u n t of w o r k i n v o l v e d , a n d m y w i f e , J e a n n e

W a p l e s , f o r h e r h e l p in t h e final s t a g e s of t h e o r g a n i z a t i o n .

Charlottesville, Virginia J . S . TREFIL

T h e A u t h o r

J a m e s S . T r e f i l ( P h . D . , S t a n f o r d U n i v e r s i t y ) is a n A s s o c i a t e

P r o f e s s o r o f P h y s i c s a n d F e l l o w i n t h e C e n t e r f o r A d v a n c e d

S t u d i e s a t t h e U n i v e r s i t y o f V i r g i n i a . H e h a s p u b l i s h e d e x t e n s i v e l y

i n t h e a r e a o f t h e o r e t i c a l h i g h e n e r g y p h y s i c s , a n d h a s h e l d v i s i t i n g

p o s i t i o n s a t s e v e r a l m a j o r l a b o r a t o r i e s i n t h a t f i e l d . M o r e r e c e n t l y ,

h e h a s b e c o m e i n t e r e s t e d i n t h e a p p l i c a t i o n s o f p h y s i c s t o

m e d i c i n e , a n d h a s c o n t r i b u t e d t o r e s e a r c h i n t h e f i e l d s o f u r o l o g y ,

c a r d i o l o g y , a n d r a d i o b i o l o g y .

1

Introduction to the Principles of Fluid Mechanics

Little drops of water Little grains of sand Make the mighty ocean And the pleasant land.

R. L. STEVENSON

A Child's Garden of Verses

F l u i d s a p p e a r e v e r y w h e r e a r o u n d u s in n a t u r e . I n t h i s s e c t i o n of t h e b o o k ,

w e sha l l d i s c u s s s o m e of t h e b a s i c l a w s w h i c h g o v e r n t h e b e h a v i o r of

f lu ids , a n d l o o k a t t h e a p p l i c a t i o n s of t h e s e l a w s t o v a r i o u s p h y s i c a l

s y s t e m s . W e sha l l s e e t h a t g o o d u n d e r s t a n d i n g s of t h e w o r k i n g s of m a n y

d i f f e ren t t y p e s of p h y s i c a l s y s t e m s c a n b e d e r i v e d in t h i s w a y .

P e r h a p s t h e m o s t a m a z i n g i d e a t h a t wi l l b e d e v e l o p e d i s t h a t fluid

m e c h a n i c s i s n o t l i m i t e d in i t s a p p l i c a t i o n s t o d i s c u s s i n g t h i n g s l i ke t h e

f low of f luids in l a b o r a t o r i e s , o r t h e m o t i o n of t i d e s o n t h e e a r t h , b u t t h a t it

c a n s u c c e s s f u l l y b e a p p l i e d t o s y s t e m s a s d i f f e ren t a s t h e a t o m i c n u c l e u s

o n t h e o n e h a n d , a n d t h e g a l a x y o n t h e o t h e r . B e c a u s e in d e a l i n g w i t h a

fluid, w e a r e in r e a l i t y d e a l i n g w i t h a s y s t e m w h i c h h a s m a n y p a r t i c l e s

w h i c h i n t e r a c t w i t h e a c h o t h e r , a n d b e c a u s e t h e m a i n u t i l i t y of fluid

m e c h a n i c s is t h e ab i l i t y t o d e v e l o p a f o r m a l i s m w h i c h d e a l s s o l e l y w i t h a

f e w m a c r o s c o p i c q u a n t i t i e s l i ke p r e s s u r e , i g n o r i n g t h e d e t a i l s of t h e

p a r t i c l e i n t e r a c t i o n s , t h e t e c h n i q u e s of f luid m e c h a n i c s h a v e o f t e n b e e n

f o u n d u s e f u l in m a k i n g m o d e l s of s y s t e m s w i t h c o m p l i c a t e d s t r u c t u r e

w h e r e i n t e r a c t i o n s ( e i t h e r n o t k n o w n o r v e r y difficult t o s t u d y ) t a k e p l a c e

b e t w e e n t h e c o n s t i t u e n t s . T h u s , t h e first s u c c e s s f u l m o d e l of t h e f i s s ion of

h e a v y e l e m e n t s w a s t h e l i qu id d r o p m o d e l of t h e n u c l e u s , w h i c h t r e a t s t h e

n u c l e u s a s a f luid, a n d t h u s r e p l a c e s t h e p r o b l e m of c a l c u l a t i n g t h e

1

2 Introduction to the Principles of Fluid Mechanics

i n t e r a c t i o n s of all of t h e p r o t o n s a n d n e u t r o n s w i t h t h e m u c h s i m p l e r

p r o b l e m of c a l c u l a t i n g t h e p r e s s u r e s a n d s u r f a c e t e n s i o n s in a fluid. Of

c o u r s e , t h i s t r e a t m e n t g i v e s o n l y a v e r y r o u g h a p p r o x i m a t i o n t o r e a l i t y ,

b u t it is n o n e t h e l e s s a v e r y u s e f u l w a y of a p p r o a c h i n g t h e p r o b l e m .

A c l a s s i c a l fluid is u s u a l l y d e f i n e d a s a m e d i u m w h i c h is inf in i te ly

d i v i s i b l e . O u r m o d e r n k n o w l e d g e of a t o m i c p h y s i c s te l l u s , of c o u r s e , t h a t

r e a l f lu ids a r e m a d e u p of a t o m s a n d m o l e c u l e s , a n d t h a t if w e g o t o s m a l l

e n o u g h s c a l e , t h e s t r u c t u r e of a fluid wi l l n o t b e c o n t i n u o u s . N e v e r t h e l e s s ,

t h e c l a s s i c a l p i c t u r e wi l l b e a p p r o x i m a t e l y c o r r e c t p r o v i d e d t h a t w e d o n o t

l o o k a t t h e fluid in t o o fine a d e t a i l . T h i s m e a n s , fo r e x a m p l e , w h e n w e

i n t r o d u c e " i n f i n i t e s i m a l " v o l u m e e l e m e n t s of t h e fluid, w e d o n o t m e a n t o

i m p l y t h a t t h e v o l u m e r e a l l y t e n d s t o z e r o , b u t m e r e l y t h a t t h e v o l u m e

e l e m e n t is v e r y s m a l l c o m p a r e d t o t h e o v e r a l l d i m e n s i o n s of t h e fluid, b u t

v e r y l a r g e c o m p a r e d t o t h e d i m e n s i o n s of t h e c o n s t i t u e n t a t o m s o r

m o l e c u l e s . S o l o n g a s w e t a l k a b o u t c l a s s i c a l m a c r o s c o p i c fluids, t h e r e

s h o u l d b e n o diff iculty in m a k i n g t h i s s o r t of a p p r o x i m a t i o n . I n d e e d , w h a t

i s " i n f i n i t e s i m a l " is l a r g e l y a m a t t e r of t h e k i n d of p r o b l e m o n e is w o r k i n g

o n . I t is n o t a t all u n u s u a l f o r c o s m o l o g i s t s t o c o n s i d e r " i n f i n i t e s i m a l "

v o l u m e e l e m e n t s w h o s e s i d e s a r e m e a s u r e d in m e g a p a r s e c s !

A. THE CONVECTIVE DERIVATION

If w e a r e g o i n g t o d e s c r i b e t h e m o t i o n of fluids, w e wi l l h a v e t o k n o w

h o w t o w r i t e N e w t o n ' s s e c o n d l a w f o r a n e l e m e n t of t h e fluid. T h i s l a w

t a k e s t h e f o r m

w h e r e m i s t h e m a s s of t h e e l e m e n t . W e a r e l e d n a t u r a l l y , t h e n , t o

c o n s i d e r t o t a l t i m e d e r i v a t i v e s of q u a n t i t i e s w h i c h d e s c r i b e t h e fluid

e l e m e n t s . W h i l e t h i s m a y s e e m s t r a i g h t f o r w a r d , t h e f a c t t h a t t h e fluid

e l e m e n t is in m o t i o n m a k e s it s o m e w h a t m o r e c o m p l i c a t e d t h a n it w o u l d

s e e m a t first g l a n c e . T o s e e w h y t h i s is s o , l e t u s c o n s i d e r s o m e q u a n t i t y /

a s s o c i a t e d w i t h a fluid e l e m e n t ( for d e f i n i t e n e s s , w e c o u l d t h i n k of p r e s s u r e

o r e n t r o p y o r v e l o c i t y ) . T h e n , if t h e e l e m e n t i s a t a p o s i t i o n x a t a t i m e t, a t a

t i m e t + A* it wi l l b e a t a n e w p o s i t i o n . ( S e e F i g . 1.1.) N o w t h e de f in i t ion of a

t i m e d e r i v a t i v e is

( l . A . l )

(1 .A.2)

The Convective Derivation 3

Fig. 1.1. The movement of the volume element.

W e s e e t h a t t h e f a c t t h a t in g e n e r a l t h e f u n c t i o n / d e p e n d s o n x, w h i c h is

i t se l f a f u n c t i o n of t i m e , m e a n s t h a t s o m e c a r e m u s t b e e x e r c i s e d in t a k i n g

t h e d e r i v a t i v e .

F o r m a l l y , w e c a n u s e t h e c h a i n r u l e of d i f f e r e n t i a t i o n t o w r i t e

(1 .A.3)

w h e r e t h e i n d e x i i n d i c a t e s w h i c h c o m p o n e n t of t h e v e c t o r x i s b e i n g

d i f f e r e n t i a t e d . ( T h i s n o t a t i o n is a t r i v i a l e x a m p l e of t h e m e t h o d of

C a r t e s i a n t e n s o r s w h i c h is d i s c u s s e d in A p p e n d i x I.) If w e d i v i d e t h r o u g h

t h e a b o v e b y dt, w e find

(1 .A.4)

B u t , b y de f in i t ion ,

w h e r e vt i s t h e i t h c o m p o n e n t of t h e v e l o c i t y of t h e fluid e l e m e n t .

T h e r e f o r e , t h e t o t a l d e r i v a t i v e of t h e f u n c t i o n / w i t h r e s p e c t t o t i m e is j u s t

(1 .A.5)

w h e r e w e h a v e u s e d t h e de f in i t ion of t h e g r a d i e n t o p e r a t o r in t h e l a t t e r

e q u a l i t y . T h i s t o t a l d e r i v a t i v e o c c u r s f r e q u e n t l y in fluid m e c h a n i c s , a n d is

g i v e n a s p e c i a l n a m e . I t is c a l l e d t h e c o n v e c t i v e d e r i v a t i v e , a n d is u s u a l l y

w r i t t e n

(1 .A.6)


T o fix t h i s i d e a f i rmly in m i n d , c o n s i d e r t h e f o l l o w i n g e x a m p l e : S u p p o s e

w e h a v e a fluid m o v i n g a r o u n d in a c o n t a i n e r , w h e r e o n e w a l l of t h e

c o n t a i n e r is a m o v a b l e p i s t o n . N o w l e t t h e f u n c t i o n / b e t h e p r e s s u r e

e x p e r i e n c e d b y a p a r t i c u l a r fluid e l e m e n t . T h e n t h e p r e s s u r e a s s e e n b y a n

o b s e r v e r r i d i n g a r o u n d o n t h e e l e m e n t wi l l v a r y a s a f u n c t i o n of t i m e f o r

t w o r e a s o n s — ( i ) t h e r e wi l l b e s o m e v a r i a t i o n in p r e s s u r e d u e t o t h e

m o t i o n of t h e p i s t o n ( th i s c o r r e s p o n d s t o t h e first t e r m in t h e c o n v e c t i v e

d e r i v a t i v e ) , a n d (ii) t h e c h a n g e s in p r e s s u r e r e s u l t i n g f r o m t h e f a c t t h a t t h e

e l e m e n t m o v e s t o d i f f e r en t r e g i o n s of t h e fluid, w h e r e t h e p r e s s u r e m a y b e

d i f f e ren t (e .g . , it m a y b e r i s i n g t o t h e t o p of t h e fluid, w h e r e t h e p r e s s u r e

wi l l b e l e s s ) . T h i s c o r r e s p o n d s t o t h e v • V t e r m in t h e c o n v e c t i v e

d e r i v a t i v e .

B. THE EULER EQUATION

T h e first f u n d a m e n t a l e q u a t i o n of h y d r o d y n a m i c s c o m e s f r o m a n

a p p l i c a t i o n of N e w t o n ' s s e c o n d l a w ( F = ma) t o fluid e l e m e n t s . W e k n o w

a p r e s s u r e (de f ined a s a f o r c e p e r u n i t a r e a ) is e x e r t e d u n i f o r m l y

e v e r y w h e r e i n s i d e a fluid. If w e c o n s i d e r a fluid e l e m e n t of l e n g t h Ax a n d

a r e A ( s e e F i g . 1.2.), t h e n t h e n e t f o r c e o n t h e e l e m e n t is

F = _ [ ( p + A P ) A - PA] = - ( A P ) A , ( l . B . l )

w h e r e t h e m i n u s s ign d e n o t e s t h a t t h e f o r c e a c t s in s u c h a w a y a s t o c a u s e

a flow f r o m r e g i o n s of h i g h e r p r e s s u r e t o r e g i o n s of l o w e r p r e s s u r e . If w e

m u l t i p l y a n d d i v i d e t h e r i g h t - h a n d s i d e of t h e e q u a t i o n b y Ax, a n d n o t e

t h a t A x A = V 0 , w h e r e V 0 is t h e v o l u m e , t h e n N e w t o n ' s l a w a p p l i e d t o t h e

v o l u m e e l e m e n t r e a d s

K ^ —* | Fig. 1.2. Forces on a volume element.

The Equation of Continuity 5

o r , in t e r m s of t h e d e n s i t y p = m / V o ,

o r , in t h r e e - d i m e n s i o n a l f o r m

(1 .B.2)

T h e a c c e l e r a t i o n t e r m of t h e l e f t - h a n d s i d e i n v o l v e s a t o t a l d e r i v a t i v e

s o it s h o u l d r e a l l y b e u n d e r s t o o d a s a c o n v e c t i v e d e r i v a t i v e in t h e s e n s e oi

S e c t i o n l . A . W e s h o u l d a l s o n o t e t h a t if f o r c e s o t h e r t h a n p r e s s u r e (e .g

g r a v i t y ) w e r e a c t i n g o n t h e fluid e l e m e n t , t h e y w o u l d a p p e a r o n the

r i g h t - h a n d s i d e of t h e e q u a t i o n . T h u s , t h e final f o r m of N e w t o n ' s s e c o n c

l a w a p p l i e d t o a fluid e l e m e n t i s

(1 .B.3)

w h e r e F e x t is a n y e x t e r n a l f o r c e o n t h e fluid e l e m e n t , s u c h a s g r a v i t y . T h i s

first f u n d a m e n t a l e q u a t i o n of h y d r o d y n a m i c s is k n o w n a s t h e E u l e r

e q u a t i o n .

A n a l t e r n a t e f o r m of t h e e q u a t i o n c a n b e d e r i v e d if w e u s e t h e r e s u l t of

P r o b l e m 1.1 t h a t

(1.B.4;

w h i c h , w h e n s u b s t i t u t e d i n t o E q . (1 .B .3) g i v e s

(1 .B.5)

If w e t a k e t h e c u r l of b o t h s i d e s of t h i s e q u a t i o n , a n d r e c a l l t h a t t h e c u r l of

t h e g r a d i e n t v a n i s h e s , w e g e t

(1 .B.6)

T h e s e t w o a l t e r n a t e f o r m s of t h e E u l e r e q u a t i o n s wi l l o c c a s i o n a l l y b e

u s e f u l in d e a l i n g w i t h p a r t i c u l a r p h y s i c a l p r o b l e m s .

C. THE EQUATION OF CONTINUITY

O n e of t h e b a s i c p r e c e p t s of c l a s s i c a l p h y s i c s is t h a t m a t t e r c a n n e i t h e r

b e c r e a t e d n o r d e s t r o y e d . T h e a p p l i c a t i o n of t h i s p r i n c i p l e t o fluid s y s t e m s


wil l l e a d u s t o o u r s e c o n d e q u a t i o n of m o t i o n , w h i c h is u s u a l l y c a l l e d t h e

e q u a t i o n of c o n t i n u i t y .

S u p p o s e w e h a v e a fluid w h o s e d e n s i t y ( in g e n e r a l a f u n c t i o n of t h e

c o o r d i n a t e s a n d t h e t i m e ) is g i v e n b y p(x, y, z, f) a n d w h e r e t h e v e l o c i t y of

t h e fluid e l e m e n t s i s g i v e n b y \(x, y, z, t). C o n s i d e r a l a r g e v o l u m e of t h e

fluid V 0 ( s e e F i g . 1.3). T h e m a s s of fluid i n s i d e t h e v o l u m e is j u s t

N o w in g e n e r a l fluid wi l l b e flowing in a n d o u t a c r o s s t h e s u r f a c e S

w h i c h b o u n d s t h e v o l u m e V0. T o find o u t w h a t t h i s flow i s , c o n s i d e r a n

e l e m e n t of s u r f a c e dS. S u p p o s e t h e fluid n e x t t o t h e s u r f a c e e l e m e n t h a s a

v e l o c i t y vn n o r m a l t o t h e s u r f a c e . T h e n i n a t i m e Af, all o f t h e fluid i n a

c y l i n d e r of l e n g t h vn A t a n d a r e a dS wi l l c r o s s t h e s u r f a c e e l e m e n t in t i m e

At . T h e t o t a l m a s s of fluid in t h e c y l i n d e r i s ( s e e F i g . 1.3) m = p(vn A t ) dS

s o t h e t o t a l m a s s o u t f l o w p e r u n i t t i m e is j u s t

w h e r e in t h e s e c o n d f o r m of t h e i n t e g r a l , w e h a v e a d o p t e d t h e u s u a l

c o n v e n t i o n of w r i t i n g t h e s u r f a c e e l e m e n t a s a v e c t o r w h o s e l e n g t h is

e q u a l t o t h e a r e a of t h e e l e m e n t , a n d w h o s e d i r e c t i o n i s n o r m a l t o t h e

s u r f a c e e l e m e n t .

T h e c o n s e r v a t i o n of m a s s w h i c h w e d i s c u s s e d a b o v e r e q u i r e s t h a t t h e

t i m e r a t e of c h a n g e of t h e m a s s in t h e v o l u m e V 0 b e e q u a l t o t h e o u t f l o w

of m a s s . T h i s i s a r e q u i r e m e n t t h a t t h e r e b e n o s u c h t h i n g a s a s o u r c e o r

( l . C . l )

Fig. 1.3. Flow through a closed surface in a fluid.

The Equation of Continuity 7

s i n k of a c l a s s i c a l f luid. M a t h e m a t i c a l l y , w e w r i t e

^ f PdV= \ pvdS, (1 .C .2)

Ol Jv0 JS

b u t G a u s s ' l a w s a y s t h a t

f pvdS = I V ( p v ) d V , IV0

s o t h a t t h e c o n s e r v a t i o n of m a s s c a n b e w r i t t e n

Jv 0 [ f + v -H d y = o - (l-C3)

S i n c e t h i s m u s t b e t r u e f o r a n y v o l u m e i n s i d e a fluid, it f o l l o w s t h a t t h e

i n t e g r a n d i t se l f m u s t v a n i s h , s o t h a t w e h a v e

| £ + V - ( p v ) = 0 . (1 .C .4)

I n t h i s f o r m , t h e r e q u i r e m e n t of t h e c o n s e r v a t i o n of m a s s is c a l l e d t h e

equation of continuity. I t wi l l p l a y a n e x t r e m e l y i m p o r t a n t r o l e in o u r

d e v e l o p m e n t of fluid m e c h a n i c s a n d , t o g e t h e r w i t h t h e E u l e r e q u a t i o n

w h i c h w e d i s c u s s e d in a p r e v i o u s s e c t i o n , p l a y s t h e r o l e of o n e of t h e

b a s i c e q u a t i o n s of h y d r o d y n a m i c s .

I n o u r a p p l i c a t i o n s of t h i s e q u a t i o n , w e sha l l o f t e n d e a l w i t h

incompressible fluids. T h e s e a r e fluids f o r w h i c h t h e d e n s i t y c a n b e

c o n s i d e r e d a c o n s t a n t . I n t h i s c a s e , t h e e q u a t i o n of c o n t i n u i t y t a k e s a

p a r t i c u l a r l y s i m p l e f o r m

V - v = 0 (1 .C .5)

S u p p o s e w e de f ine a fluid c u r r e n t d e n s i t y b y

i = p v . (1 .C.6)

T h e n t h e e q u a t i o n of c o n t i n u i t y t a k e s t h e f o r m

f f + V - j = 0. (1 .C.7)

T h i s i s p r e c i s e l y t h e s a m e e q u a t i o n t h a t o n e e n c o u n t e r s in e l e c t r o m a g n e -

t i s m , w h e r e p is t h e c h a r g e d e n s i t y a n d j i s e l e c t r i c a l c u r r e n t . T h e r e a s o n

f o r t h e s i m i l a r i t y in t h e e q u a t i o n s , of c o u r s e , i s t h a t j u s t a s w e p o s t u l a t e d

t h a t fluid m a s s c a n n e i t h e r b e c r e a t e d n o r d e s t r o y e d , in e l e c t r o m a g n e t i s m

o n e a l w a y s p o s t u l a t e d t h a t e l e c t r i c a l c h a r g e is c o n s e r v e d . O u r s e c o n d


w h e r e t h e s e c o n d e q u a l i t y f o l l o w s f r o m G a u s s ' l a w . T h u s , t h e t i m e r a t e of

c h a n g e of t h e m o m e n t u m in t h e v o l u m e V0 is t h e i n t e g r a l o f nik dSk o v e r

t h e s u r f a c e . T h e r e f o r e , in a n a l o g y t o o u r d e r i v a t i o n of t h e c o n t i n u i t y

e q u a t i o n , I I I k m u s t b e t h e m o m e n t u m flux in t h e I t h d i r e c t i o n o v e r t h e kth

s u r f a c e e l e m e n t , a n d h e n c e r e p r e s e n t s a n e t o u t f l o w of m o m e n t u m .

W e sha l l u s e t h i s m o m e n t u m t e n s o r f o r m of t h e E u l e r e q u a t i o n w h e n

w e i n t r o d u c e t h e i d e a of v i s c o s i t y l a t e r .

e q u a t i o n of m o t i o n , t h e n , c a n b e t h o u g h t of a s a s p e c i a l c a s e of a m o r e

f u n d a m e n t a l p r i n c i p l e of p h y s i c s w h i c h a r i s e s w h e n e v e r c o n s e r v e d

q u a n t i t i e s o c c u r in n a t u r e .

I n t h e C a r t e s i a n t e n s o r n o t a t i o n of A p p e n d i x A , t h e E u l e r e q u a t i o n c a n

b e w r i t t e n

(1 .C .8)

S i n c e t h e e q u a t i o n of c o n t i n u i t y g i v e s

a n d

(1 .C.9)

t h i s c a n b e r e w r i t t e n in t h e f o r m

(1 .C .10)

w h e r e w e h a v e d e f i n e d t h e t w o i n d e x t e n s o r 7rik b y

I I * =P8ik+pvivk. ( l . C . 1 1 )

T h i s t e n s o r i s c a l l e d t h e momentum flux tensor. T h e r e a s o n f o r t h i s n a m e

is q u i t e s i m p l e . W e k n o w t h a t t h e m o m e n t u m of a v o l u m e e l e m e n t i s j u s t

(p V 0 ) v s o t h a t t h e l e f t - h a n d s i d e of t h e a b o v e e q u a t i o n is j u s t t h e t i m e r a t e

of c h a n g e of t h e I t h c o m p o n e n t of t h e m o m e n t u m of t h e fluid p e r u n i t

v o l u m e . If w e a d d t h i s u p o v e r all of t h e e l e m e n t s in a v o l u m e V0, w e g e t

(1 .C .12)

Tlik dSk,

A Simple Example: The Static Star 9

D. A SIMPLE EXAMPLE: THE STATIC STAR

T h e s i m p l e s t a p p l i c a t i o n s of t h e E u l e r e q u a t i o n , of c o u r s e , wi l l b e f o r

t h e c a s e w h e r e v = 0 , t h e s t a t i c c a s e . I n t h e n e x t c h a p t e r , w e wi l l l o o k a t

m a n y e x a m p l e s of s t a t i c s y s t e m s , b u t f o r t h e m o m e n t , l e t u s b e g i n b y

c o n s i d e r i n g a s impl i f ied m o d e l f o r a s t a r . W e sha l l s e e t h a t t h e t w o

e q u a t i o n s w h i c h w e h a v e d e r i v e d d o n o t t h e m s e l v e s c o m p l e t e l y s p e c i f y

t h e s y s t e m w i t h w h i c h w e a r e d e a l i n g , b u t a n o t h e r p i e c e of i n f o r m a t i o n

wi l l b e n e e d e d . T h e e x t r a i n f o r m a t i o n i s e s s e n t i a l l y a s t a t e m e n t a b o u t t h e

k i n d of fluid of w h i c h t h e s y s t e m is m a d e .

If w e t h i n k of a s t a t i c s t a r , t h e f o r c e s a c t i n g o n a fluid e l e m e n t wi l l b e (i)

t h e p r e s s u r e a n d (ii) t h e g r a v i t a t i o n a l a t t r a c t i o n of t h e r e s t of t h e s t a r . T h i s

s e c o n d f o r c e is a n e x a m p l e of w h a t w a s c a l l e d F e x t i n E q . (1 .B .3 ) . I n

g e n e r a l , w e k n o w t h a t f o r a g r a v i t a t i o n a l f o r c e , w e c a n w r i t e

F e x t = - p V a ( l . D . l )

w h e r e ft i s t h e g r a v i t a t i o n a l p o t e n t i a l . W e k n o w t h a t H is r e l a t e d t o t h e

d e n s i t y of m a t t e r b y Poisson's equation

V2n = 4iTGp. (1 .D.2)

N o w t h e E u l e r e q u a t i o n in t h e s t a t i c c a s e r e d u c e s t o

—VP = - Vf t , (1 .D.3) P

w h i c h is j u s t t h e o r d i n a r y b a l a n c e of f o r c e s e q u a t i o n f r o m N e w t o n i a n

m e c h a n i c s . If w e t a k e t h e d i v e r g e n c e of b o t h s i d e s of t h i s e q u a t i o n , w e

find

V " (p V P ) = " = " 4 7 r G p * ( L D * 4 )

T h i s is t h e e q u a t i o n w h i c h w o u l d h a v e t o b e sa t i s f ied if t h e s t a r w e r e t o

b e in a s t a t e of e q u i l i b r i u m . A s it s t a n d s , h o w e v e r , it c a n n o t b e s o l v e d ,

s i n c e it r e l a t e s t w o s e p a r a t e q u a n t i t i e s — t h e p r e s s u r e a n d t h e d e n s i t y .

W h a t is n e e d e d is a r e l a t i o n b e t w e e n t h e s e t w o . T h i s i s e s s e n t i a l l y

i n f o r m a t i o n a b o u t t h e k i n d of fluid in t h e s t a r , s i n c e d i f f e r en t k i n d s of

fluids wi l l e x e r t d i f f e r en t p r e s s u r e w h e n k e p t a t t h e s a m e d e n s i t y .

T h e r e l a t i o n b e t w e e n p r e s s u r e a n d d e n s i t y i s c a l l e d a n equation of state.

T h e r e a d e r is p r o b a b l y f a m i l i a r w i t h o n e s u c h e q u a t i o n a l r e a d y , t h e i d e a l

g a s l a w , w h i c h s a y s

P=RpT, (1 .D .5)

w h e r e R is a c o n s t a n t a n d T is t h e t e m p e r a t u r e .


E. ENERGY BALANCE IN A FLUID

F o r t h e s a k e of c o m p l e t e n e s s , w e wi l l d i s c u s s t h e e n e r g y a s s o c i a t e d

w i t h f lu ids , a l t h o u g h w e sha l l h a v e f e w o c c a s i o n s t o u s e t h i s c o n c e p t in

s u b s e q u e n t d i s c u s s i o n s . L e t u s c o n s i d e r a fluid in a n e x t e r n a l field, s u c h

a s g r a v i t y , s o t h a t t h e f o r c e is j u s t

F = -Pvn

a n d t h e E u l e r e q u a t i o n is

If w e n o t e t h a t t h e t o t a l k i n e t i c e n e r g y of all of t h e fluid e l e m e n t s i s j u s t

(1 .D.6)

Spec i f i c s o l u t i o n s of t h i s e q u a t i o n a r e lef t t o t h e p r o b l e m s .

( l . E . l )

If w e t a k e t h e i n n e r p r o d u c t of t h e v e c t o r v w i t h t h i s e q u a t i o n , w e find,

a f t e r s o m e m a n i p u l a t i o n , t h a t

(1 .E .2 )

If w e a s s u m e t h a t t h e p o t e n t i a l ft i s i n d e p e n d e n t of t h e t i m e , s o t h a t

t h e n t h e c o n v e c t i v e d e r i v a t i v e of H wi l l b e

(1 .E .3 )

(1 .E .4 )

s o t h a t

F o r a s t a r c o m p o s e d of a n i d e a l g a s a t c o n s t a n t t e m p e r a t u r e , t h e e q u a t i o n

of e q u i l i b r i u m r e d u c e s t o

a n d t h e t o t a l p o t e n t i a l e n e r g y is

t h e n i n t e g r a t i n g E q . (1 .E .4 ) o v e r t h e v o l u m e V 0 g i v e s

(1 .E .5 )

w h e r e t h e l e f t - h a n d s i d e r e p r e s e n t s t h e t o t a l t i m e r a t e of c h a n g e of t h e

k i n e t i c p l u s p o t e n t i a l e n e r g y of t h e fluid s y s t e m . T e r m s s u c h a s t h i s a r e

f a m i l i a r f r o m o t h e r b r a n c h e s of p h y s i c s . T h e r i g h t - h a n d s i d e of t h e

e q u a t i o n , h o w e v e r , r e q u i r e s f u r t h e r i n v e s t i g a t i o n . If w e i n t e g r a t e b y

p a r t s , w e h a v e

T h e s e c o n d ( v o l u m e ) i n t e g r a l o n t h e r i g h t v a n i s h e s f o r a n i n c o m p r e s s i b l e

fluid. T h u s , w e a r e lef t w i t h t h e e q u a t i o n

T h e q u a n t i t y in t h e i n t e g r a n d h a s a s i m p l e i n t e r p r e t a t i o n . P dS is j u s t

t h e f o r c e a c t i n g a c r o s s t h e s u r f a c e e l e m e n t dS ( t h i s f o l l o w s f r o m t h e

de f in i t i on of t h e p r e s s u r e a s a f o r c e p e r u n i t a r e a ) . T h i s f o r c e t i m e s t h e

v e l o c i t y is s i m p l y t h e r a t e a t w h i c h t h e p r e s s u r e is d o i n g w o r k o n t h e fluid

w h i c h is c r o s s i n g t h e s u r f a c e e l e m e n t . W e s e e , t h e n , t h a t t h e a b o v e

e q u a t i o n is s i m p l y t h e r e q u i r e m e n t t h a t e n e r g y b e c o n s e r v e d — t h a t t h e

r a t e of c h a n g e of t h e e n e r g y of a fluid s y s t e m m u s t e q u a l t h e r a t e a t w h i c h

w o r k is d o n e a c r o s s t h e b o u n d a r i e s .

Of c o u r s e , t h i s i s n o t a n e w r e s u l t in t h e s e n s e t h a t w e k n o w t h a t e n e r g y

m u s t b e c o n s e r v e d . N e v e r t h e l e s s , it is c o m f o r t i n g t o s e e a f a m i l i a r l a w

e m e r g e f r o m o u r f o r m a l i s m .

SUMMARY

I n t h i s c h a p t e r , w e h a v e i n t r o d u c e d t h e b a s i c l a w s of fluid m o t i o n .

T h e s e l a w s a r e s e e n t o f o l l o w f r o m s o m e v e r y s i m p l e p h y s i c a l p r i n c i p l e s .

T h e s e p r i n c i p l e s a r e (i) m a t t e r c a n n e i t h e r b e c r e a t e d n o r d e s t r o y e d a n d

(ii) N e w t o n ' s s e c o n d l a w of m o t i o n . T h e p r i n c i p l e s g i v e r i s e t o t h e

e q u a t i o n s of c o n t i n u i t y a n d t h e E u l e r e q u a t i o n s , r e s p e c t i v e l y .

f P v d S + f P ( V - v ) d V . (1 .E .6 )

(1 .E .7 )

Summary 11

pttdV,


W e s a w t h a t t h e s e t w o e q u a t i o n s b y t h e m s e l v e s d i d n o t c o m p l e t e l y

de f ine t h e p h y s i c s of t h e s i m p l e s t a t i c s t a r , b u t t h a t o n e m o r e p i e c e of

i n f o r m a t i o n w a s n e c e s s a r y . T h i s p i e c e of i n f o r m a t i o n , in t h e f o r m of t h e

e q u a t i o n of s t a t e , w a s in r e a l i t y a s p e c i f i c a t i o n of t h e k i n d of fluid t h a t

c o m p o s e d t h e s y s t e m . I n m u c h of w h a t f o l l o w s , w e wi l l s p e a k of a n

i n c o m p r e s s i b l e fluid—a fluid f o r w h i c h p = c o n s t . T h i s , t o o , i s a n e q u a t i o n

of s t a t e .

O n t h e b a s i s of t h e s e v e r y s i m p l e p h y s i c a l p r i n c i p l e s , a l a r g e n u m b e r of

p h y s i c a l p r o b l e m s c a n b e t r e a t e d , a n d it is t o s o m e of t h e s e e x a m p l e s t h a t

w e n o w t u r n .

PROBLEMS

1.1. Using the method of Car tes ian tensor notat ion, show that

1.2. Show that for a fluid of densi ty p at rest in a gravitational field where the accelerat ion due to gravity at each point in the fluid is - g, that

where z is the vertical coordinate and P 0 is the pressure at a height h, and that the pressure is cons tant along lines of cons tant z.

1.3. Show that for an ideal gas at cons tant t empera ture , the only solutions to the equat ion of equilibrium for a star are unphysical (i.e. that they require infinite densit ies at some point in the star) . Are there any values of y in the poly tropic equation of state P = Kpy for which physical solutions are possible?

1.4. Le t us consider vectors and tensors defined in the x-y p lane. A rotat ion in the jc-y plane through an angle 6 is represented by the matrix

GijlGlmn 5 j m 8jn &in8jm,

and p rove the following identities

P =gp(h-z) + Po,

r - ( c o s ^ sin 0 \ \ - s i n 0 c o s 0 /

(a) Verify by explicit geometrical const ruct ion that the vec tor

v = ai + bj

t ransforms according to Eq . (1.A.4).

• V t r = v x (V x v) + (v • V)v,

V • (A x B) = B • (V x A) - A • (V x B)

Problems 13

(b) Verify by explicit calculation and const ruct ion that the quant i ty Uik, which

was defined in Eq . ( l . C . l l ) , is indeed a tensor of second rank.

1.5. Consider a fluid where the densi ty varies only with the z -coordinate , so that

Po i s son ' s equat ion becomes

(c) If the densi ty is taken to be symmetr ic about the plane z = 0

where

1.6. The force on a moving charge, according to the theory of e lec t rodynamics , is

F = q E + ^ v x B ,

where q is the value of the charge, c is a cons tant (equal to the speed of light), and E and B are the values of the electrical and magnet ic fields which are present .

(a) Consider a fluid which has mass densi ty p and charge densi ty o\ Wri te down the Euler equat ion for the motion of such a fluid in the case where the fields E and B are fixed by some mechanism external to the fluid.

(b) Wha t is the equat ion of continuity for p ? for cr?

1.7. Carry out the energy balance problem of Section l .E for the fluid descr ibed in P rob lem 1.6. In terpre t the new te rms which appear in the analogue of E q . (1.E.7).

1.8. An important the rmodynamic proper ty of a material is the en t ropy per unit volume, 5. An adiabatic react ion is defined as a react ion for which the en t ropy of a

(Hint: T h e change of variables

p = p 0 A(£) and

and assume further that the fluid is at a cons tan t t empera tu re , so that the equat ion of s tate is

P = c2p. Then show that

(a) c is the velocity of sound in the fluid. (b) The equat ion for the densi ty is


Show that in this case , the number of part icles per unit volume, N(r), becomes infinite as r - » o o .

(c) Show that as r - » o o , p(r) approaches a cons tant which is nonzero . Both par ts (b) and (c) show that the solar wind mus t b e a hydrodynamic , as opposed to a hydrosta t ic phenomenon (as might be guessed from the name) .

1.10. Consider the a tmosphere as an isothermal gas which has an equat ion of s tate given by

p = a + bP.

Determine the pressure as a function of height in such a system, assuming that the ear th ' s surface is flat and does not ro ta te . Explain where the t e rm "exponent ia l a t m o s p h e r e " arises.

1.11. Consider a fluid of densi ty p moving with velocity v along the z-axis . Imagine a surface of area dA which is inclined at an angle 6 to the z-axis , but which is parallel to the x-axis . Calculate the amount of m o m e n t u m flow across this surface per unit t ime by simple mechanics and through the use of the momen tum flux tensor defined in Eq . ( l . C . l l ) . Show that the resul ts are the same.

1.12. A spherical ba thysphere of radius JR and mass M descends into the ocean. Assuming that the ocean is made up of incompressible fluid, how far will it sink? Work the same problem for a bal loon rising into the air.

sys tem does not change. Show that for an adiabatic react ion,

•(ps) + V - ( p s v ) = 0,

where ps v is called the entropy flux density.

1.9. One of the most interesting phenomena discovered in the last quar ter century is that of the solar wind. It was discovered that there are particles a round the ear th which come from the sun.

(a) Consider a model in which the wind is taken to be the low-density tail of the solar mass distribution. If we assume that the solar particles are static, and that their equat ion of state is that of an ideal gas, so that

P = 2NkT9

where N is the number of part icles per unit volume, show that the Euler equat ion requires that

where Ms is the mass of the sun and M the mass of a molecule. (b) It can be shown that the t empera ture as a function of radius should go

rouehlv as

References 15

1.13. Assuming tha t wate r is a fluid of cons tan t densi ty , calculate the force per unit area at the bo t tom of the Grand Coulee Dam. W h y is it th icker at the bo t tom than at the top?

1.14. Consider a je t of fluid of velocity v and mass M per unit length incident on a plate as shown in the figure. T h e jet leaves the plate at an angle 6 t o its original direct ion, but the plate is ar ranged in such a way that the magni tude of the fluid velocity does not change. Calculate the force acting on the plate . This is the principle of the turb ine .

There are a number of readable books in the field of hydrodynamics, many of which are standard, well-known texts. Some texts of this sort which might be valuable to the reader are

H. Lamb, Hydrodynamics, Dover Publications, New York, 1945. This book was written in the heyday of classical physics (1879) and revised by the author in 1932. It is an interesting text, mainly because of the large number of examples which are worked out. It is somewhat heavy going for the modern reader, however, because it does not use vector notation.

L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon Press, London, 1959. A complete modern exposition of hydrodynamics. The student learning the subject will probably find the mathematical development a little terse, but a large number of topics are covered.

A. S. Ramsey, A Treatise on Hydrodynamics, G. Bell and Son, London, 1954. A readable book with many examples worked out.

I. Prigogine and R. Herman, Kinetic Theory of Vehicular Traffic, American Elsevier, New York, 1971.

This text applies the ideas of hydrodynamics to traffic flow, and illustrates the remarks made in the Introduction concerning the wide applicability of hydrodynamics.

In addition to the above, many of the texts cited as references in later chapters contain sections dealing with the basic laws of hydrodynamics.

REFERENCES

2

Fluids in Astrophysics

There are more things in heaven and earth, Horatio, Than are dreamt of in your philosophy.

WILLIAM SHAKESPEARE

Hamlet, Act I, Scene V

SOME APPLICATIONS TO ASTROPHYSICS

A. EQUATIONS OF MOTION

O n t h e b a s i s of t h e b a s i c p h y s i c a l p r i n c i p l e s w h i c h w e i n v e s t i g a t e d in

t h e p r e v i o u s c h a p t e r , w e c a n n o w b e g i n t o l o o k a t s o m e i n t e r e s t i n g

e x a m p l e s of s y s t e m s in n a t u r e . W e wil l b e g i n b y c o n s i d e r i n g a u n i f o r m

fluid w h i c h is r o t a t i n g f r e e f r o m e x t e r n a l f o r c e s , b u t w h e r e t h e m u t u a l

g r a v i t a t i o n a l a t t r a c t i o n of t h e p a r t i c l e s of t h e fluid f o r o n e a n o t h e r i s t a k e n

i n t o a c c o u n t . T h i s s o u n d s v e r y m u c h l ike a s i m p l e m o d e l f o r a n o b j e c t l i ke

a s t a r , a n d , i n d e e d , t h e m a i n a p p l i c a t i o n s of w h a t w e wi l l d e v e l o p in t h i s

c h a p t e r h a v e b e e n in t h e field of a s t r o n o m y .

W e sha l l b e g i n b y i n v e s t i g a t i n g t h e p o s s i b l e e q u i l i b r i u m s h a p e s t h a t a

s t a r c a n h a v e , a n d t h e n d i s c u s s t h e q u e s t i o n of s t ab i l i t y . W e sha l l s e e t h a t

it is p o s s i b l e t o m a k e def in i t e s t a t e m e n t s a b o u t w h e t h e r a s t a r c o u l d h a v e

a c e r t a i n s h a p e , o r w h e t h e r a s t a r w i t h a c e r t a i n s h a p e c o u l d r o t a t e w i t h a

g i v e n f r e q u e n c y .

E x c e p t w h e r e o t h e r w i s e s t a t e d , w e sha l l c o n c e r n o u r s e l v e s in t h i s

c h a p t e r w i t h a fluid w h i c h h a s a c o n s t a n t d e n s i t y . T h i s is a n a p p r o x i m a -

t i o n , a n d , l ike all a p p r o x i m a t i o n s , i t i s g o o d f o r s o m e s y s t e m s a n d n o t s o

16

Equations of Motion 17

g o o d f o r o t h e r s . I t s h o u l d b e p o i n t e d o u t , h o w e v e r , in t h e sp i r i t of S e c t i o n

l . D , t h a t t h i s a s s u m p t i o n c o n s t i t u t e s a n e q u a t i o n of s t a t e f o r t h e s y s t e m ,

s o t h a t t h e E u l e r e q u a t i o n a n d t h e e q u a t i o n of c o n t i n u i t y wi l l c o m p l e t e l y

de f ine t h e fluid m o t i o n .

L e t u s c o n s i d e r a m a s s e l e m e n t in a fluid b o d y ( s e e F i g . 2 .1) . L e t t h e

b o d y b e r o t a t i n g w i t h a n g u l a r f r e q u e n c y co a b o u t t h e z - a x i s . L e t r b e t h e

v e c t o r w h i c h d e s c r i b e s t h e p o s i t i o n of t h e e l e m e n t r e l a t i v e t o t h e c e n t e r of

t h e b o d y , a n d le t co b e t h e p e r p e n d i c u l a r d i s t a n c e f r o m t h e e l e m e n t t o t h e

z - a x i s ( th i s s o m e w h a t c l u m s y n o t a t i o n is s t a n d a r d fo r t h i s p r o b l e m ) .

L e t u s n o w g o t o a s e t of a x e s w h i c h a r e r o t a t i n g w i t h f r e q u e n c y co, a n d

a r e t h e r e f o r e fixed in t h e b o d y ( t h e s e a r e c a l l e d b o d y a x e s in c l a s s i c a l

m e c h a n i c s ) . I n t h i s s y s t e m , t h e b o d y a p p e a r s t o b e a t r e s t , s o t h a t t h e

v e l o c i t y of t h e fluid is e v e r y w h e r e z e r o . T h e p r o b l e m of c a l c u l a t i n g t h e

m o t i o n of t h e fluid p a r t i c l e s is t h e n r e d u c e d t o t h e m u c h s i m p l e r p r o b l e m

of b a l a n c i n g f o r c e s , o r hydrostatics.

A n o b s e r v e r in t h i s s y s t e m wil l s e e t h e f o l l o w i n g f o r c e s p e r u n i t m a s s

a c t i n g o n a fluid e l e m e n t :

(1) t h e p r e s s u r e f o r c e , g i v e n b y - — V P , P

(2) t h e g r a v i t a t i o n a l f o r c e , g i v e n b y - V O , w h e r e fi is t h e g r a v i t a t i o n a l

p o t e n t i a l ,

(3) t h e c e n t r i f u g a l f o r c e , g i v e n b y co2co.

Fig. 2.1. Coordinates for volume elements in a rotating body.

18 Fluids in Astrophysics

P u t t i n g t h e s e t o g e t h e r , w e find f o r t h e E u l e r e q u a t i o n

- a > 2 < 5 = - ^ V P - V f t , (2 .A.1)

o r , in t e r m s of t h e x-y-z s y s t e m of c o o r d i n a t e s

(2 .A.2) o)2y

I n all of o u r a p p l i c a t i o n s , w e h a v e m a d e t h e s i m p l i f y i n g a s s u m p t i o n t h a t

t h e d e n s i t y is n o t a f u n c t i o n of t h e c o o r d i n a t e s . I n t h i s c a s e , t h e first

e q u a t i o n c a n b e i n t e g r a t e d t o g i v e

(2 .A.3)

w h e r e / ( y , z) is a n i n t e g r a t i o n " c o n s t a n t " a s f a r a s a n e q u a t i o n in x is

c o n c e r n e d . D i f f e r e n t i a t i n g E q . (2 .A.3) w i t h r e s p e c t t o x c a n c o n v i n c e t h e

r e a d e r t h a t t h e r e is n o w a y of e x c l u d i n g s u c h a n a d d i t i v e f u n c t i o n t o t h e

s o l u t i o n , j u s t a s in o r d i n a r y d i f fe ren t i a l e q u a t i o n s t h e r e is n o w a y of

e x c l u d i n g a n a d d i t i v e c o n s t a n t f r o m s o l u t i o n s e x c e p t b y a p p l y i n g

b o u n d a r y v a l u e s .

I n a s i m i l a r w a y , t h e r e m a i n i n g E u l e r e q u a t i o n s c a n b e i n t e g r a t e d t o

g i v e

T h e l e f t - h a n d s i d e of al l of t h e s e e q u a t i o n s is t h e s a m e q u a n t i t y , s o w e

c a n d e t e r m i n e s o m e t h i n g a b o u t t h e a r b i t r a r y f u n c t i o n s b y d e m a n d i n g t h a t

t h e r i g h t - h a n d s i d e of e a c h e q u a t i o n r e d u c e t o t h e s a m e f u n c t i o n of t h e

c o o r d i n a t e s . I n f a c t , o n e c a n r e a d i l y s e e t h a t o n l y t h e c h o i c e

The Rotating Sphere 19

w h e r e C is a c o n s t a n t wi l l d o t h i s . H e n c e w e find f o r t h e i n t e g r a t e d f o r m

of t h e E u l e r e q u a t i o n t h e r e s u l t

(2 .A.4)

B. THE ROTATING SPHERE

A s a first e x a m p l e of t h e a p p l i c a t i o n of S e c t i o n 2 . A w e sha l l c o n s i d e r a

s p h e r e of r a d i u s a r o t a t i n g w i t h a n g u l a r f r e q u e n c y co a b o u t a n a x i s ( s e e

F i g . 2 .2) .

W e b e g i n b y c a l c u l a t i n g H , t h e g r a v i t a t i o n a l p o t e n t i a l a t a p o i n t i n s i d e

t h e s p h e r e a d i s t a n c e r f r o m t h e c e n t e r . T h e t o t a l m a s s e n c l o s e d w i t h i n a

s p h e r e of r a d i u s r is j u s t

M ( r ) = 3*r r 3 p ,

s o t h a t t h e p o t e n t i a l is j u s t

(2 .B.1)

(2 .B.2)

(2 .B.3)

Fig. 2.2. A rotating sphere.

P u t t i n g t h i s i n t o E q . (2 .A .4 ) , w e find

I t t h e n f o l l o w s t h a t t h e s u r f a c e s of c o n s t a n t p r e s s u r e wi l l b e g i v e n b y t h e

e q u a t i o n

c o n s t . (2 .B.4)


i .e . if

co = 0.

T h u s , o u r i n v e s t i g a t i o n of t h e s i m p l e s t r o t a t i n g b o d y — a s p h e r e — s h o w s

t h a t it c a n b e in a s t a t e of e q u i l i b r i u m o n l y f o r t h e t r i v i a l c a s e of n o

r o t a t i o n . T h e p h y s i c a l r e a s o n fo r t h i s is q u i t e s i m p l e . T h e c e n t r i f u g a l

f o r c e t e n d s t o t h r o w o u t m a t e r i a l n e a r t h e e q u a t o r m o r e t h a n a t t h e p o l e s ,

s o m o s t r o t a t i n g b o d i e s c a n b e e x p e c t e d t o h a v e a s o m e w h a t " s q u a s h e d "

a p p e a r a n c e . T h i s m e a n s t h a t w e sha l l h a v e t o t u r n o u r a t t e n t i o n t o m o r e

c o m p l i c a t e d g e o m e t r i e s if w e w a n t t o l o o k a t m o r e r e a l i s t i c c a s e s .

A l t h o u g h in t h e c a s e of a s p h e r e t h e o n l y s o l u t i o n t o o u r e q u a t i o n is t h e

t r iv ia l o n e of co = 0 , t h e m e t h o d w e u s e d wi l l b e r e p e a t e d f o r m o r e

c o m p l i c a t e d g e o m e t r i e s , w h e r e it wil l b e l e s s e a s y t o f o l l o w . T o r e v i e w : t o

s e e if t h e r e is a n e q u i l i b r i u m p o s s i b l e fo r a r o t a t i n g fluid, w e m u s t

(1) C a l c u l a t e t h e g r a v i t a t i o n a l p o t e n t i a l i n s i d e t h e fluid.

(2) I n s e r t t h i s p o t e n t i a l i n t o E q . (2 .A.4) t o d e t e r m i n e t h e s u r f a c e s of

c o n s t a n t p r e s s u r e .

(3) A s c e r t a i n w h e t h e r o n e of t h e s e s u r f a c e s c o u l d c o i n c i d e w i t h t h e

a c t u a l s u r f a c e of t h e b o d y .

If t h e a n s w e r t o t h e l a s t s t e p is y e s , t h e n a n e q u i l i b r i u m i s p o s s i b l e — i . e .

t h e b o d y c a n b e r o t a t e d w i t h o u t c h a n g i n g i ts s h a p e .

C. ELLIPSOIDS

T h e s i m p l e s t p o s s i b l e e q u i l i b r i u m s h a p e f o r a r o t a t i n g g r a v i t a t i n g fluid

o n c e t h e s p h e r e h a s b e e n e l i m i n a t e d is t h a t of a n e l l i p so id . C o n s i d e r s u c h

a b o d y r o t a t i n g w i t h f r e q u e n c y co a b o u t i t s z - a x i s , w h i c h w e t a k e t o l ie

a l o n g o n e of t h e m a j o r a x e s of t h e e l l i p se .

(2 .B.5)

C l e a r l y , t h e s u r f a c e wil l c o i n c i d e w i t h a s u r f a c e of c o n s t a n t p r e s s u r e o n l y

if

N o w in o r d e r t o h a v e a s t a b l e r o t a t i o n , it is n e c e s s a r y t h a t t h e s u r f a c e

of t h e b o d y b e a s u r f a c e of c o n s t a n t p r e s s u r e . O t h e r w i s e t h e r e wi l l b e a

p r e s s u r e g r a d i e n t b e t w e e n t w o p o i n t s o n t h e s u r f a c e a n d t h e r e wil l n o t b e

a n e q u i l i b r i u m . T h e e q u a t i o n fo r t h e s u r f a c e is g i v e n b y

x2 + y2 + z 2 = a \

Ellipsoids 21

z

Fig. 2.3. The rotating ellipsoid.

I n A p p e n d i x B , w e s h o w t h a t t h e g r a v i t a t i o n a l p o t e n t i a l a n a l o g o u s t o

E q . (2.B.2) f o r t h e s p h e r e is j u s t

(2.C.1)

(2.C.2) A = R ( a 2 + A ) 0 ? 2 + A ) ( c 2 + A ) l , / 2 ,

w h e r e

w h i c h f o r t h e s a k e of c o n v e n i e n c e w e c a n w r i t e

a = irpG(a0x2 + p0y

2 + joz2 - Xo), (2.C.3)

w h e r e

(2.C.4)

a n d w h e r e j 3 0 , 7 0 , a n d ^ 0 a r e s i m i l a r l y d e f i n e d .

W e c a n n o w p r o c e e d a s w e d id in t h e c a s e of t h e s p h e r e , p u t t i n g t h e

a b o v e e x p r e s s i o n f o r t h e p o t e n t i a l e n e r g y i n t o E q . (2 .A.4), t h e i n t e g r a t e d

E u l e r e q u a t i o n s , a n d d e m a n d i n g t h a t a s u r f a c e of c o n s t a n t p r e s s u r e

c o i n c i d e w i t h t h e s u r f a c e of t h e e l l i p s e , w h i c h in t h i s c a s e is g i v e n b y t h e

e x p r e s s i o n

I n s u c h a p r o c e d u r e , t h e i n f o r m a t i o n a b o u t t h e s h a p e of t h e e l l i p s o i d is

c o n t a i n e d in t h e c o n s t a n t s a0, j 3 0 , 7 0 , a n d xo-


T h e i n t e g r a t e d E u l e r e q u a t i o n , w i t h t h e p o t e n t i a l f o r t h e e l l i p so id ,

b e c o m e s

(2 .C.5)

T h e s u r f a c e s of c o n s t a n t p r e s s u r e c a n b e o b t a i n e d f r o m t h i s b y s e t t i n g t h e

r i g h t - h a n d s i d e of E q . (2 .C.5) e q u a l t o a c o n s t a n t . I n o r d e r f o r o n e of t h e s e

s u r f a c e s t o c o i n c i d e w i t h t h e s u r f a c e of t h e e l l i p s o i d , it is n e c e s s a r y t h a t

( u p t o a c o m m o n m u l t i p l i c a t i v e c o n s t a n t ) ,

(2 .C.6)

A c a s e of p a r t i c u l a r s i m p l i c i t y is t h a t of t h e e l l i p s o i d of r e v o l u t i o n ,

w h e r e w e h a v e

(2 .C.7)

T h i s c o r r e s p o n d s t o a b o d y in w h i c h t h e c r o s s s e c t i o n p e r p e n d i c u l a r t o t h e

a x i s of r o t a t i o n a r e c i r c l e s , a n d r e p r e s e n t s t h e n e x t s t e p in g e o m e t r i c a l

c o m p l i c a t i o n a f t e r t h e s p h e r e . I t i s c a l l e d t h e M a c l a u r i n e l l i p so id .

I t s h o u l d b e n o t e d t h a t w e a r e a l r e a d y a n t i c i p a t i n g a r e s u l t w h i c h w e

sha l l d e r i v e l a t e r w h e n w e w r i t e t h e r e l a t i o n b e t w e e n a a n d c a s w e d o in

E q . (2 .C.7) b e c a u s e n o m a t t e r w h a t v a l u e of £ w e p i c k , c wi l l a l w a y s b e

l e s s t h a n o r e q u a l t o a a n d b. T h u s w e a r e c o n s i d e r i n g o n l y o b l a t e

s p h e r o i d s . T h e p r o l a t e s p h e r o i d is lef t t o P r o b l e m 2.3 a t t h e e n d of t h e

c h a p t e r .

W e c a n n o w w r i t e d o w n t h e s t r u c t u r e c o n s t a n t s d i r e c t l y

(2 .C .8)

w h e r e w e h a v e u s e d t h e c h a n g e of v a r i a b l e s

c 2 + A = ( a 2 - c 2 ) a 2 ,

Ellipsoids 23

t o c a r r y o u t t h e i n t e g r a l s . S i m i l a r l y ,

y 0 = 2 ( £ 2 + ! ) ( ! - £ a r c c o t f ) . (2 .C.9)

T h e s t r u c t u r e c o n s t a n t \o c o u l d b e c o m p u t e d a s w e l l , b u t s i n c e E q . (2 .A.4)

c o n t a i n s a n a r b i t r a r y c o n s t a n t a n y w a y , w e c a n s i m p l y i n c o r p o r a t e x° i n t o

i t .

F o r t h i s s impl i f ied g e o m e t r y , t h e c o n d i t i o n t h a t t h e s u r f a c e of t h e

e l l i p so id c o r r e s p o n d s t o a s u r f a c e of c o n s t a n t p r e s s u r e r e d u c e s t o

w h e n t h e v a l u e s of a 0 a n d 7 0 c o m p u t e d e a r l i e r a r e s u b s t i t u t e d .

T h e r e a r e t w o i m p o r t a n t p o i n t s w h i c h c a n b e m a d e a b o u t t h i s

e q u i l i b r i u m c o n d i t i o n . F i r s t , w e s e e t h a t t h e q u e s t i o n of w h e t h e r o r n o t

e q u i l i b r i u m c a n b e e s t a b l i s h e d d e p e n d s o n l y o n £, w h i c h is r e l a t e d t o a

ratio of l e n g t h s of m a j o r a n d m i n o r a x e s of t h e e l l i p s e . T h u s , t h e s i z e o f

t h e e l l i p s e d o e s n o t m a t t e r a t all p r o v i d e d t h a t t h e p r o p o r t i o n s of t h e a x e s

a r e s u c h t h a t E q . (2 .C .11) c a n b e sa t i s f ied . T h u s , a p l a n e t o r a g a l a x y w i t h

a g i v e n £ ( i .e . a g i v e n r a t i o b e t w e e n m a j o r a n d m i n o r a x e s ) wi l l h a v e t h e

s a m e r a t i o of f r e q u e n c y of r o t a t i o n t o 27rGp a t e q u i l i b r i u m ( b u t s i n c e

27rGp d e p e n d s o n t h e d e n s i t y , t h e y n e e d n o t h a v e t h e s a m e f r e q u e n c y of

r o t a t i o n ) .

T o find o u t w h e t h e r s u c h a s o l u t i o n e x i s t s ( i .e . w h e t h e r a n e l l i p s o i d in

u n i f o r m r o t a t i o n c a n b e in e q u i l b r i u m ) , w e c a n l o o k a t a g r a p h of t h e r i g h t -

a n d l e f t - h a n d s i d e s of t h e e q u a t i o n a s a f u n c t i o n of £. If t h e l i ne w h i c h

r e p r e s e n t s t h e l e f t - h a n d s i d e i n t e r s e c t s t h e c u r v e w h i c h r e p r e s e n t s t h e

r i g h t - h a n d s i d e , t h e n E q . (2 .C .11) wil l h a v e a s o l u t i o n , a n d t h e b o d y wil l b e

in a s t a t e of e q u i l i b r i u m f o r t h a t v a l u e of £

T h e s h a p e of t h e r i g h t - h a n d s i d e c a n b e g u e s s e d w i t h o u t a c t u a l l y

c a l c u l a t i n g it b y n o t i n g t h a t a s £ - > ° o ?

w h i c h r e d u c e s t o

= f c o t " 1 ^ ^ l ) - 3 f 2 , (2 .C .11)

cor 1 f

s o t h a t t h e r i g h t - h a n d s i d e a p p r o a c h e s z e r o f r o m t h e p o s i t i v e s i d e .

S i m i l a r l y , a s £ - » 0 , c o t 1 £ - » TT/2 SO t h a t t h e r i g h t - h a n d s i d e g o e s t o z e r o

a s ( 7 r / 2 ) £ T h i s m e a n s t h a t t h e r i g h t - h a n d s i d e s t a r t s f r o m z e r o , g o e s

(2 .C .10)


p o s i t i v e , a n d r e t u r n s t o z e r o , s o t h a t t h e r e m u s t b e a m a x i m u m s o m e w h e r e

in b e t w e e n .

T h e s i t u a t i o n is s k e t c h e d in F i g . 2 .4. I n g e n e r a l , t h e l e f t - h a n d s i d e n e e d

n o t d e p e n d o n £ a t all ( a l t h o u g h f o r m o s t c a s e s of p h y s i c a l i n t e r e s t , it

w i l l — s e e b e l o w ) , s o it wil l a p p e a r o n t h e f igure a s a s t r a i g h t l i ne . T h e r e a r e

s e v e r a l d i s t i n c t c a s e s . I n t h e c a s e c o r r e s p o n d i n g t o t h e l i ne l a b e l e d " 1 " , it

is p o s s i b l e f o r t h e r i g h t - a n d l e f t - h a n d s i d e s of E q . (2 .C .11) t o b e e q u a l ,

a n d h e n c e f o r a s o l u t i o n t o e x i s t f o r w h i c h a n e l l i p s o i d c a n r o t a t e in

e q u i l i b r i u m . F o r t h e l i ne l a b e l e d " 3 " , t h i s is n o t t h e c a s e , a n d n o s o l u t i o n

t o o u r p r o b l e m wil l e x i s t . T h u s , if co2l2irpG is l a r g e e n o u g h , it wi l l b e

i m p o s s i b l e f o r t h e e l l i p s o i d t o r o t a t e in e q u i l i b r i u m . T h e c a s e s e p a r a t i n g

t h e s e t w o r e g i m e s i s t h e l i ne l a b e l e d " 2 " , w h e r e co2\2iTpG is j u s t e q u a l t o

t h e m a x i m u m v a l u e of t h e r i g h t - h a n d s i d e of E q . (2 .C .11) .

B y e x p l i c i t c a l c u l a t i o n s , it t u r n s o u t t h a t t h e v a l u e of t h e r i g h t - h a n d s i d e

a t i t s m a x i m u m is 0 .224 , s o t h a t t h e c r i t i c a l c a s e o c c u r s w h e n

I n o t h e r w o r d s , t h e m a x i m u m f r e q u e n c y a t w h i c h a M a c l a u r i n e l l i p s o i d

c a n r o t a t e is of t h e o r d e r of V27rpG. T h i s is a s p e c i a l c a s e of a m o r e

g e n e r a l r e s u l t w h i c h w e p r o v e in A p p e n d i x C , w h i c h s a y s t h a t it is

i m p o s s i b l e f o r a n y b o d y t o b e in e q u i l i b r i u m if it is r o t a t i n g f a s t e r t h a n a

c r i t i ca l f r e q u e n c y coc, w h e r e coc i s d e f i n e d b y

P h y s i c a l l y , w e c a n t h i n k of t h i s r e s u l t in t h e f o l l o w i n g w a y : W h e n a

m a s s is r o t a t i n g s l o w l y , it is p o s s i b l e f o r t h e g r a v i t a t i o n a l a t t r a c t i o n t o

o v e r c o m e t h e c e n t r i f u g a l f o r c e a n d h o l d t h e fluid t o g e t h e r . A s co i s

i n c r e a s e d , h o w e v e r , t h e c e n t r i f u g a l f o r c e wi l l b e c o m e t o o g r e a t , a n d t h e

fluid wi l l fly a p a r t .

co 2 — 2irpG. (2 .C.12)

© the function

value of

Fig. 2.4. Plot of the right-hand side of Eq. (2.C. 11) as a function of £

Ellipsoids 25

T h i s r e s u l t , w h i l e it is v a l i d in t h e g e n e r a l c a s e , d o e s n o t s h e d m u c h l ight

o n t h e p r o b l e m of c l a s s i c a l s t e l l a r s t r u c t u r e . T o u n d e r s t a n d w h y , w e n e e d

t o r e a l i z e t h a t w h e n w e d i s c u s s a m a s s of fluid r o t a t i n g in a v a c u u m , t h e r e

a r e t w o i m p o r t a n t q u a n t i t i e s w h i c h m u s t b e c o n s e r v e d . T h e s e a r e t h e

m a s s a n d t h e a n g u l a r m o m e n t u m . S i n c e w e a r e d e a l i n g w i t h a n

i n c o m p r e s s i b l e f luid, t h e c o n s e r v a t i o n of m a s s r e q u i r e s t h a t t h e v o l u m e

b e fixed a s w e l l .

T h e v o l u m e of a n e l l i p s o i d of r e v o l u t i o n is j u s t

(2 .C .13)

w h i l e t h e m o m e n t of i n e r t i a a b o u t t h e z - a x i s i s

(2 .C.14)

s o t h a t t h e a n g u l a r m o m e n t u m is

(2 .C.15)

w h e r e w e h a v e w r i t t e n e v e r y t h i n g in t e r m s of t h e c o n s e r v e d q u a n t i t i e s M ,

V, a n d L a n d t h e p a r a m e t e r £.

S o l v i n g E q . (2 .C .15) f o r co a n d s u b s t i t u t i n g i n t o E q . (2 .C .11 ) , w e find

(2 .C .16)

= ^ c o t ^ ( 3 ^ + l ) - 3 ^ 2 ,

w h i c h c a n b e w r i t t e n

(2 .C .17)

w h e r e w e h a v e de f ined

(2 .C.18)

T h e p o i n t of t h i s d i s c u s s i o n is t h a t w h e r e a s in g e n e r a l t h e f r e q u e n c y of

r o t a t i o n a n d t h e p a r a m e t e r £ c a n b e r e g a r d e d a s i n d e p e n d e n t , w h e n w e

r e q u i r e t h a t m a s s a n d a n g u l a r m o m e n t u m b e c o n s e r v e d , t h i s is n o l o n g e r


t h e c a s e , a n d co b e c o m e s a f u n c t i o n of £, a s in E q . (2 .C .16) . T h i s is t r u e

w h e t h e r t h e e q u i l i b r i u m e q u a t i o n (2 .C .11) is sa t i s f ied o r n o t .

W e c a n p r o c e e d a s b e f o r e , g r a p h i n g t h e r igh t - a n d l e f t - h a n d s i d e s of E q .

(2 .C .16) , a s s h o w n in F i g . 2.5 w h e r e t h e s t r a i g h t l ine r e p r e s e n t s t h e

q u a n t i t y L 2 / L c

2 ( w h i c h is n o w t r u l y i n d e p e n d e n t of £, s i n c e it d e p e n d s

o n l y o n t h e in i t ia l c o n d i t i o n s ) . T h e p o i n t of i n t e r s e c t i o n is t h e s o l u t i o n

w h i c h w e s e e k , a n d r e p r e s e n t s t h e c o n f i g u r a t i o n a t w h i c h a g i v e n m a s s

e l l i p s o i d w i t h a f ixed a n g u l a r m o m e n t u m wil l r o t a t e in e q u i l i b r i u m . W e s e e

t h a t fo r e a c h L , t h e r e is o n e a n d o n l y o n e e q u i l i b r i u m c o n f i g u r a t i o n f o r t h e

e l l i p so id .

T h e r e s u l t in t h e figure i s p h y s i c a l l y r e a s o n a b l e , s i n c e a s L is i n c r e a s e d ,

£ - * 0 . F r o m E q . (2 .C .7 ) , £ - » 0 c o r r e s p o n d s t o a f l a t t e n e d o u t " p a n c a k e , "

s o t h a t t h i s a g r e e s w i t h o u r i n t u i t i o n , w h i c h t e l l s u s t h a t a s w e s p i n a b o d y

f a s t e r a n d f a s t e r , i t wil l t e n d t o f l a t t en o u t . S i m i l a r l y , a s L is d e c r e a s e d ,

w h i c h c o r r e s p o n d s t o t h e e l l i p s o i d a p p r o a c h i n g a s p h e r e .

Of c o u r s e , it m u s t b e k e p t f i rmly in m i n d t h a t a l t h o u g h it a p p e a r s t h a t

t h e r e wil l b e a s o l u t i o n t o E q . (2 .C .16) fo r a n y L , t h e c o n s t r a i n t t h a t co

m u s t b e l e s s t h a n coc c o n t i n u e s t o r e s t r i c t t h e p o s s i b l e v a l u e s of L w h i c h

m a y b e a c h i e v e d fo r a g i v e n m a s s .

F o r t h e c a s e of t h e e a r t h , w h i c h h a s m e a n d e n s i t y 5.52 g / c m 3 , t h i s

c r i t i ca l f r e q u e n c y is

coc = 1.5 x 10~ 3 s e c " 1 ,

w h i c h c o r r e s p o n d s t o a p e r i o d of

right-hand side

Fig. 2.5. Plot of the right- and left-hand sides of Eq. (2.C.16).

= 4 . 2 5 x 10 3 s e c = 1.8 h r ,

w h i l e f o r t h e s u n , w h i c h h a s m e a n d e n s i t y 1.41 g / c m 3 , it is

coc = 7 . 7 x l ( T 4 s e c ~ \

Jacobi Ellipsoids 27

w h i c h is a p e r i o d of

Tc = 8 . 1 5 x 10 4 s e c = 22 .5 h r .

T h u s , b o t h of t h e s e b o d i e s r o t a t e a t f r e q u e n c i e s w e l l b e l o w t h e c r i t i c a l

f r e q u e n c y g i v e n a b o v e .

D. THE EARTH AS A FLUID

In l a t e r a p p l i c a t i o n s , w e sha l l o f t e n w i s h t o t r e a t t h e e a r t h i t se l f a s a

fluid m a s s . S u p p o s e w e w a n t t o k n o w h o w r e a l i s t i c s u c h a n a p p r o x i m a t i o n

c o u l d b e . O n e m e a s u r e of s u c h a n a p p r o x i m a t i o n w o u l d b e t o c a l c u l a t e i t s

r o t a t i o n a l f r e q u e n c y f r o m E q . (2 .C .11) , a n d t o c o m p a r e it w i t h t h e a c t u a l

f r e q u e n c y of r o t a t i o n . F o r t h e e a r t h , w e h a v e

a n d

s o t h a t

w h i c h g i v e s

a = b = 6.378 x 10 6 m

c = 6.357 x 10 6 m ,

f = 12.16,

= 0 .059 . (2 .D.1) predicted

W e c a n c o m p a r e t h i s t o t h e o b s e r v e d f r e q u e n c y ( t a k i n g <ac f r o m E q .

(2 .C .12) )

= 0 .048 . (2 .D.2) observed

T h e s e t w o a g r e e t o a b o u t 2 0 % , s o t h a t if w e c a n b e sa t i s f ied w i t h t h a t

s o r t of a c c u r a c y , w e c a n i n d e e d t r e a t t h e e a r t h a s a fluid m a s s ( e v e n

t h o u g h w e k n o w it t o b e so l id ) . W e sha l l u s e t h i s r e s u l t l a t e r w h e n w e

c a l c u l a t e t h e f r e e v i b r a t i o n s of t h e e a r t h .

E. JACOBI ELLIPSOIDS

A n e l l i p so id of r e v o l u t i o n in w h i c h all t h r e e a x e s a r e n o t e q u a l is c a l l e d

a Jacobi ellipsoid. F o r s u c h a c o n f i g u r a t i o n , t h e e q u i l i b r i u m c o n d i t i o n s


c a n b e c a s t in t h e f o r m :

(2 .E .1 )

(2 .E .2 )

W e c o u l d a t t h i s p o i n t p r o c e e d j u s t a s w e d i d in t h e c a s e of t h e

M a c l a u r i n e l l i p so id , b u t r e c a l l i n g t h e r e s u l t t h a t fo r e a c h e q u i l i b r i u m

c o n f i g u r a t i o n t h e r e is j u s t o n e f r e q u e n c y of r o t a t i o n w h i c h wi l l j u s t

b a l a n c e t h e f o r c e s a t t h e s u r f a c e , w e s u b t r a c t t h e a b o v e e q u a t i o n s t o g e t

(2 .E .3 )

S i m i l a r l y , m u l t i p l y i n g E q . (2 .E .1 ) b y b2 a n d E q . (2 .E .2 ) b y a2 a n d t h e n

s u b t r a c t i n g g i v e s

( a 0 - j 8 0 ) a 2 b 2 + y 0 c 2 ( a 2 - b2) = 0 . (2 .E .4 )

T h e s e c o n d of t h e s e e q u a t i o n s is i n d e p e n d e n t of t h e f r e q u e n c y . T h u s , if

w e c a n find a s e t of v a l u e s fo r a, b , a n d c w h i c h sa t i s fy i t , w e wi l l h a v e t h e

e q u i l i b r i u m c o n f i g u r a t i o n . W e c a n t h e n p u t t h e s e v a l u e s i n t o E q . (2 .E .3 )

a n d e v a l u a t e t h e f r e q u e n c y w h i c h c o r r e s p o n d s t o t h i s c o n f i g u r a t i o n .

P u t t i n g t h e i n t e g r a l f o r m s f o r t h e s t r u c t u r e c o n s t a n t s in E q . ( 2 . E . 4 ) , w e

find

(2 .E .5 )

If a = b ( t h e c a s e f o r t h e M a c l a u r i n e l l i p s o i d ) , t h e n t h i s c o n d i t i o n is

a u t o m a t i c a l l y sa t i s f ied , a n d w e h a v e f o u n d t h e e q u i l i b r i u m c o n d i t i o n s

f r o m S e c t i o n 2 . C . If a ^ b, h o w e v e r , w e c a n a s k t h e q u e s t i o n of w h e t h e r it

i s e v e r p o s s i b l e t o sa t i s fy t h e c o n d i t i o n in E q . ( 2 . E . 5 ) .

I n s t e a d of s o l v i n g t h e p r o b l e m e x p l i c i t l y , w e wi l l s h o w t h a t a s o l u t i o n

m u s t e x i s t . T o s e e t h i s , w e wil l c o n s i d e r t h e v a l u e of t h e i n t e g r a l f o r t w o

d i f f e ren t c a s e s .

C a s e (i) c = 0 .

I n t h i s c a s e , t h e s e c o n d t e r m in t h e i n t e g r a n d v a n i s h e s , a n d , s i n c e A is

a l w a y s p o s i t i v e , t h e i n t e g r a l m u s t b e p o s i t i v e a s w e l l .

C a s e (ii)

Rotation of the Galaxy 29

I n t h i s c a s e , t h e first t e r m in t h e i n t e g r a n d b e c o m e s a2b2l(a2b2 +

( a 2 + b 2 ) A + A 2 ) w h i l e t h e s e c o n d b e c o m e s a2b2l(a2b2 + (a2 + b2)k).

C l e a r l y , t h e s e c o n d wil l a l w a y s b e g r e a t e r t h a n t h e first, s o t h e i n t e g r a l in

t h i s c a s e m u s t b e n e g a t i v e .

T h u s , w e h a v e a s i t u a t i o n in w h i c h t h e i n t e g r a l p r o c e e d s f r o m a p o s i t i v e

v a l u e a t c = 0 t o a n e g a t i v e o n e a t c = a2b2l(a2 + b2). A t s o m e i n t e r -

m e d i a t e p o i n t , it m u s t p a s s t h r o u g h z e r o , a n d t h e v a l u e s of a, b, a n d c

a t t h i s p o i n t wil l g i v e t h e e q u i l i b r i u m v a l u e s . F o r s o m e n u m e r i c a l r e s u l t s ,

t h e r e a d e r is r e f e r r e d t o L a m b ( C h a p t e r X I I ) . F r o m t h e s e v a l u e s , t h e

e q u i l i b r i u m r o t a t i o n a l f r e q u e n c y c a n b e c a l c u l a t e d u s i n g E q . ( 2 . E . 3 ) .

O n e f u r t h e r p o i n t s h o u l d b e m a d e . W e c a n d i v i d e E q . ( 2 .E .5 ) b y

(a2b2c2) a n d o b t a i n a f o r m of t h e e q u i l i b r i u m c o n d i t i o n w h i c h d e p e n d s

o n l y o n t h e r a t i o s bja a n d cja. T h i s is t h e s c a l i n g r e s u l t w h i c h w e s a w

e a r l i e r f o r t h e M a c l a u r i n e l l i p so id . T h e e q u i l i b r i u m d e p e n d s o n l y o n

r e l a t i v e s i z e s , a n d n o t o n t h e a c t u a l m a g n i t u d e of t h e d i m e n s i o n s of t h e

r o t a t i n g b o d y .

F. ROTATION OF THE GALAXY

A n i n t e r e s t i n g a p p l i c a t i o n of w h a t h a s b e e n d o n e s o f a r is t o l o o k a t t h e

g r o s s s t r u c t u r e of t h e g a l a x y . O n e p r o b l e m of s o m e c u r r e n t i n t e r e s t

c e n t e r s a r o u n d t h e g a l a c t i c r o t a t i o n c u r v e s . T h e s e c u r v e s a r e e s s e n t i a l l y a

p l o t of t h e v e l o c i t y of a p a r t i c l e in t h e g a l a x y a s a f u n c t i o n of i t s d i s t a n c e

f r o m t h e c e n t e r of r o t a t i o n . T h e r e a r e s e v e r a l d i f f e r en t k i n d s of r o t a t i o n

c u r v e s t h a t o n e c a n i m a g i n e :

(i) " S o l i d b o d y " r o t a t i o n , in w h i c h e v e r y p a r t i c l e in t h e g a l a x y h a s t h e

s a m e a n g u l a r f r e q u e n c y a s t h e g a l a x y a s a w h o l e , s o t h a t v ( r ) oc r .

(ii) " C o n s t a n t v e l o c i t y " r o t a t i o n , in w h i c h e v e r y p a r t i c l e in t h e g a l a x y

h a s t h e s a m e s p e e d ( a n d h e n c e d i f f e ren t a n g u l a r f r e q u e n c i e s ) . I n

t h i s c a s e , v(r) = v0.

(iii) " K e p l e r i a n " r o t a t i o n , in w h i c h p a r t i c l e s f a r f r o m t h e c e n t e r s e e a

g r a v i t a t i o n a l f o r c e = Gm lr2 w h i c h j u s t b a l a n c e s t h e c e n t r i f u g a l

f o r c e , a n d g i v e s v{r) oc l/Vr.

I n f a c t , all t h r e e t y p e s of r o t a t i o n a r e s e e n in n a t u r e . A " t y p i c a l "

r o t a t i o n c u r v e ( s u c h a s t h a t f o r o u r o w n g a l a x y ) is s h o w n in F i g . 2 .6 . W e

s e e t h a t a t v e r y l a r g e d i s t a n c e s ( w h e r e t h e p a r t i c l e s s e e t h e r e s t of t h e

g a l a x y a s a p o i n t ) w e g e t t h e e x p e c t e d K e p l e r i a n r e v o l u t i o n , w h i l e f o r

s o m e r e g i o n of r ( w h i c h v a r i e s f r o m o n e g a l a x y t o t h e n e x t ) t h e r e i s

c o n s t a n t v e l o c i t y r o t a t i o n . A t v e r y s m a l l r, t h e r o t a t i o n b e c o m e s so l id


V(r)

r

Fig. 2.6. A typical galactic rotation curve.

b o d y . I t s h o u l d b e n o t e d t h a t t h e r e a r e g a l a x i e s in n a t u r e w h i c h a r e

p r e d o m i n a n t l y so l id b o d y a s o p p o s e d t o t h e o n e s h o w n a b o v e , w h i c h is

p r e d o m i n a n t l y c o n s t a n t v e l o c i t y .

N o w t h e g a l a x y is o b v i o u s l y a b o d y w h i c h is r o t a t i n g f r e e l y u n d e r i t s

o w n g r a v i t a t i o n a l a t t r a c t i o n , s o t h a t t h e m e t h o d s w e h a v e d e v e l o p e d f o r

t r e a t i n g s u c h b o d i e s a r e a p p r o p r i a t e h e r e . H o w e v e r , w e sha l l s e e t h a t t h e

m a i n i n f o r m a t i o n w h i c h c a n b e g a i n e d f r o m s t u d y i n g g a l a c t i c r o t a t i o n

c u r v e s h a s t o d o w i t h t h e d i s t r i b u t i o n of m a t t e r in a g a l a x y , s o w e wi l l

w a n t t o d r o p , f o r t h e m o m e n t , t h e r e q u i r e m e n t t h a t t h e d e n s i t y of t h e fluid

b e c o n s t a n t .

T h e g e n e r a l s t r u c t u r e of o u r g a l a x y is p i c t u r e d in F i g . 2.7 (all d i s t a n c e s

in l ight y e a r s ) . M o s t of t h e m a s s is c o n c e n t r a t e d in a c e n t r a l c o r e , b u t t h e

g a l a x y is m u c h w i d e r t h a n it is h i g h . T h i s l e a d s u s t o s u p p o s e t h a t w e c a n

r e p l a c e t h e a c t u a l p r o b l e m of c a l c u l a t i n g t h e s u r f a c e c o n d i t i o n s f o r t h e

r a t h e r c o m p l i c a t e d g e o m e t r y of t h e r e a l g a l a x y b y t h e m u c h s i m p l e r

p r o b l e m of c a l c u l a t i n g fo r a t w o - d i m e n s i o n a l d i s k r o t a t i n g a b o u t a n a x i s

p e r p e n d i c u l a r t o t h e p l a n e of t h e d i s k .

W e c a n s t a t e t h i s s u p p o s i t i o n w i t h s o m e w h a t m o r e r i g o r b y n o t i n g t h a t

t h e q u a n t i t i e s l i ke p r e s s u r e a n d g r a v i t a t i o n a l p o t e n t i a l c a n b e e x p e c t e d t o

v a r y q u i t e r a p i d l y in t h e z - d i r e c t i o n in t h e g a l a x y , b u t s h o u l d v a r y m u c h

m o r e s l o w l y in t h e x — y p l a n e . T h u s , w e c a n n e g l e c t d e r i v a t i v e s of t h e s e

q u a n t i t i e s w i t h r e s p e c t t o x a n d y. T h e E u l e r e q u a t i o n t h e n b e e n e s

80,000 »

1,000

Fig. 2.7. A side view of a typical galaxy.

(2 .F .1 )

Rotation of the Galaxy 31

Fig. 2.8. Coordinates for a volume element in a rotating galaxy.

w h e r e r is t h e v e c t o r in t h e x - y p l a n e . T h e P o i s s o n e q u a t i o n is

(2 .F .2 )

I n P r o b l e m 1.5 it w a s s h o w n t h a t e q u a t i o n s of t h i s t y p e l e a d t o a d e n s i t y

d i s t r i b u t i o n t h a t fa l l s off, a t l a r g e z, a s

T h u s , m o s t of t h e m a t t e r in t h e g a l a x y is l o c a t e d n e a r t h e p l a n e z = 0 , a n d

o u r a p p r o x i m a t i o n ( r e p l a c i n g t h e g a l a x y b y a d i s k ) wil l b e a g o o d o n e .

N o w c o n s i d e r s u c h a d i s k . T h e E u l e r e q u a t i o n fo r a p a r t i c l e a d i s t a n c e r

f r o m t h e c e n t e r ( n e g l e c t i n g d e r i v a t i v e s of t h e p r e s s u r e w i t h r e s p e c t t o r)

is j u s t

(2 .F .3 )

T h u s , t o find t h e r o t a t i o n a l f r e q u e n c y ( a n d h e n c e t h e v e l o c i t y ) of t h e

p o i n t a t r, w e n e e d t o c a l c u l a t e t h e g r a v i t a t i o n a l p o t e n t i a l a t r d u e t o t h e

o t h e r m a s s e l e m e n t s in t h e d i s k . W e d o t h i s b y c a l c u l a t i n g t h e p o t e n t i a l a t

r d u e t o a p o i n t a t r ' , a n d t h e n a d d i n g u p o v e r all r \ ( S e e F i g . 2.8.)

(2 .F .4 )

w h e r e M ( r ' ) i s t h e m a s s p e r u n i t a r e a a t t h e p o i n t r ' , a n d t h e q u a n t i t y t h a t

a p p e a r s in t h e d e n o m i n a t o r of t h e i n t e g r a n d is j u s t t h e d i s t a n c e |r - r'|.

R e c a l l i n g t h a t t h e v e l o c i t y of a p o i n t in t h e d i s k d e p e n d s o n d f t / d r , w e

s e e t h a t t h e f o r m of r o t a t i o n c u r v e t h a t a g i v e n g a l a x y wi l l h a v e d e p e n d s

v e r y s t r o n g l y o n M ( r ) , t h e d i s t r i b u t i o n of m a s s in t h e g a l a x y . I n P r o b l e m

2.2 , f o r e x a m p l e , w e s h o w t h a t a d i s k w i t h a u n i f o r m m a s s d i s t r i b u t i o n

l e a d s , a t l e a s t a t s m a l l r, t o so l id b o d y r o t a t i o n . L e t u s e x a m i n e s o m e o t h e r


* K 2

V 1 ^

a — ^

** * y

A

Fig. 2.9. The mass distribution derived from a Maclaurin ellipsoid.

s i m p l e e x a m p l e s t o s e e w h a t c o n c l u s i o n s w e c a n d r a w a b o u t t h e r e l a t i o n

b e t w e e n t h e m a s s d i s t r i b u t i o n in a ga l axy , a n d i t s r o t a t i o n c u r v e .

L e t u s b e g i n b y a s k i n g h o w o n e w o u l d g o a b o u t r e p l a c i n g o n e of o u r

e q u i l i b r i u m s h a p e s — s a y a M a c l a u r i n e l l i p s o i d — b y a flat d i s k . If w e t a k e

a n e l l i p s o i d a n d i m a g i n e it b r o k e n u p i n t o c o l u m n s ( s e e F i g . 2.9) a n d t h e n

i m a g i n e e a c h c o l u m n c o l l a p s e d i n t o t h e p l a n e z = 0 , b u t in s u c h a w a y t h a t

t h e m a s s in e a c h c o l u m n w o u l d b e c o n s e r v e d , t h e m a s s e n c l o s e d in e a c h

c o l u m n w o u l d b e

w h e r e A i s t h e a r e a of t h e c o l u m n . T h u s , t h e m a s s p e r u n i t a r e a in t h e d i s k

is j u s t

(2 .F .5 )

w h e r e w e h a v e w r i t t e n 2 c p = M 0 . N o w w e c o u l d g o a h e a d a n d p u t t h i s

m a s s d i s t r i b u t i o n i n t o t h e p o t e n t i a l i n t e g r a l in E q . (2 .F .3 ) a n d w o r k it o u t .

H o w e v e r , w e a l r e a d y h a v e a n e x p r e s s i o n f o r t h e p o t e n t i a l of a M a c l a u r i n

e l l i p so id ,

H = 7 r p ( < * 0 r 2 + y 0 z 2 - x o ) .

S i n c e w e a r e d e a l i n g w i t h a d i s k , w e c a n s e t z = 0 in t h e a b o v e , s o t h a t t h e

f o r c e b a l a n c e e q u a t i o n b e c o m e s

w h i c h is t h e c u r v e f o r p u r e so l id b o d y r o t a t i o n .

I n o t h e r w o r d s , if w e i m a g i n e t h e g a l a x y s t a r t i n g o u t a s a f l a t t e n e d

M a c l a u r i n e l l i p s o i d , w e w o u l d g e t p u r e so l id b o d y r o t a t i o n , u n l i k e t h a t

w h i c h is s e e n f o r a l a r g e n u m b e r of g a l a x i e s , i n c l u d i n g o u r o w n . H o w c a n

w e u n d e r s t a n d t h i s ?

The Ringsof Saturn 33

O n e w a y is t o n o t e t h a t t h e m a s s d i s t r i b u t i o n M ( r ) in E q . (2 .F .5 ) is

a c t u a l l y p r e t t y u n i f o r m o v e r l a r g e d i s t a n c e s in t h e g a l a x y . O n t h e o t h e r

h a n d , w e k n o w t h a t o u r g a l a x y h a s a c o r e , w i t h a n a p p r e c i a b l e p e r c e n t a g e

of i t s m a s s l y ing a t r e l a t i v e l y s m a l l d i s t a n c e s f r o m t h e g a l a c t i c c e n t e r . S u c h

a d i s t r i b u t i o n wi l l , of c o u r s e , b e p o o r l y r e p r e s e n t e d b y a s m o o t h d i s t r i b u -

t i o n of t h e t y p e g i v e n in E q . ( 2 .F .5 ) . S u p p o s e w e t r i e d a d i s t r i b u t i o n l ike

i n s t e a d . T h i s d i s t r i b u t i o n , a l t h o u g h s i n g u l a r a n d t h e r e f o r e n o t c o m p l e t e l y

r e a s o n a b l e , a t l e a s t d o e s h a v e t h e p r o p e r t y of m a k i n g t h e g a l a x y m o r e

m a s s i v e n e a r i t s c e n t e r . W e c a n p u t t h i s d i s t r i b u t i o n i n t o E q . ( 2 . F . 3 ) , a n d ,

p r o c e e d i n g j u s t a s b e f o r e , find t h a t

T h i s , of c o u r s e , is t h e c o n s t a n t v e l o c i t y r o t a t i o n w h i c h w a s d i s c u s s e d

a b o v e .

W e s e e , t h e n , t h a t d i f f e ren t m a s s d i s t r i b u t i o n s l e a d t o d i f f e ren t r o t a t i o n

l a w s , a n d t h a t m a s s d i s t r i b u t i o n s w h i c h p l a c e m o s t of t h e m a s s n e a r t h e

c e n t e r of t h e g a l a x y t e n d t o f a v o r c o n s t a n t v e l o c i t y r o t a t i o n , w h i l e t h o s e

w h i c h a r e m o r e u n i f o r m t e n d t o f a v o r so l id b o d y r o t a t i o n .

T h e q u e s t i o n of w h y a g a l a x y s h o u l d a s s u m e o n e m a s s d i s t r i b u t i o n

i n s t e a d of a n o t h e r is o n e w h i c h c a n n o t b e t r e a t e d w i t h t h e s i m p l e m e t h o d s

w e h a v e a t o u r d i s p o s a l a t t h i s p o i n t , b u t is a n i n t e r e s t i n g p r o b l e m in i tself .

G. THE RINGS OF SATURN

A s t r o n o m e r s h a v e p u z z l e d o v e r t h e r i n g s of S a t u r n e v e r s i n c e t h e y

w e r e d i s c o v e r e d . W h e n t h e s c i e n c e of fluid m e c h a n i c s w a s first

d e v e l o p e d , it w a s n a t u r a l t h a t t h e q u e s t i o n of w h e t h e r t h e y c o u l d b e

c o m p o s e d of a fluid in e q u i l i b r i u m s h o u l d h a v e c o m e u p . T h e p r o b l e m c a n

b e s t a t e d a s f o l l o w s : I m a g i n e a c e n t r a l b o d y of m a s s M s u r r o u n d e d b y a n

a n n u l u s of e l l ip t i ca l c r o s s s e c t i o n r o t a t i n g w i t h f r e q u e n c y co a b o u t t h e

M ( r ) = 0 (r>R),

w h i c h m e a n s t h a t

(rco)2 = v2 = 2irGy. (2 .F .6 )


— • = — E Z

H D *

Fig. 2.10. Side view of the rings of Saturn.

w h e r e t h e first t e r m r e p r e s e n t s t h e p o t e n t i a l a t a p o i n t in t h e a n n u l u s d u e

t o t h e a t t r a c t i o n of t h e c e n t r a l b o d y , w h i l e t h e s e c o n d ( w h i c h w e h a v e y e t

t o c a l c u l a t e ) r e p r e s e n t s t h e p o t e n t i a l d u e t o t h e r e s t of t h e m a t e r i a l in t h e

a n n u l u s .

W e c o u l d , of c o u r s e , c a l c u l a t e £lR d i r e c t l y , a s w e d i d t h e p o t e n t i a l f o r

t h e e l l i p so id , b u t w e c a n g e t it m u c h m o r e e a s i l y if w e n o t e t h a t u n d e r t h e

c o n d i t i o n s in E q . (2.G.1), w e c a n t r e a t t h e a n n u l u s (a t l e a s t f o r t h e p u r p o s e

of c a l c u l a t i n g flR) a s a n inf ini te c y l i n d e r of e l l ip t i ca l c r o s s s e c t i o n s , w h o s e

s u r f a c e is g i v e n b y t h e e q u a t i o n

I n t h i s c a s e ,

w h e r e

a n d

(2.G.2)

a = irpG(a0x2 + y0z

2), (2.G.3)

b o d y a s s h o w n in F i g . 2.10. L e t u s f u r t h e r a s s u m e t h a t

T h e p o t e n t i a l O w h i c h m u s t b e i n s e r t e d i n t o t h e E u l e r e q u a t i o n c a n b e

w r i t t e n

(2.G.1)

The Ringsof Saturn 35

T h e i n t e g r a t e d E u l e r e q u a t i o n is t h e n

(2 .G.4)

w h e r e w e h a v e d r o p p e d t e r m s h i g h e r t h a n s e c o n d o r d e r in x ID a n d z / D .

W e s e e i m m e d i a t e l y t h a t u n l e s s t h e coef f ic ien t of t h e t e r m l i n e a r in x

v a n i s h e s , t h e s u r f a c e s of c o n s t a n t p r e s s u r e wi l l n e v e r c o i n c i d e w i t h t h e

s u r f a c e of t h e a n n u l u s . T h i s m e a n s t h a t

i .e . t h a t t h e r e i s o n l y o n e f r e q u e n c y a t w h i c h t h e a n n u l u s c a n r o t a t e ,

r e g a r d l e s s of i t s s h a p e . T h i s i s a d e p a r t u r e f r o m o u r p r e v i o u s r e s u l t s , in

w h i c h a n e q u i l i b r i u m w a s p o s s i b l e a t a n y f r e q u e n c y u p t o coc. H e r e t h e

f r e q u e n c y is c o m p l e t e l y fixed b y t h e c e n t r a l b o d y .

W e n o t e in p a s s i n g t h a t t h i s f r e q u e n c y i s p r e c i s e l y t h a t w h i c h a s a t e l l i t e

in o r b i t a r o u n d t h e c e n t r a l m a s s w o u l d h a v e .

I n o r d e r f o r t h e s u r f a c e s of c o n s t a n t p r e s s u r e t o c o i n c i d e w i t h t h e

s u r f a c e of t h e r i n g , w e m u s t h a v e

T h u s , p r o v i d e d t h a t t h e r a t i o a/c c a n b e a d j u s t e d t o s a t i s f y t h i s

c o n d i t i o n ( w h e r e co i s n o l o n g e r f r e e , b u t d e t e r m i n e d b y E q . (2 .G .5 ) ) , t h e

r o t a t i n g r i n g wi l l b e in e q u i l i b r i u m .

H a v e w e , t h e n , f o u n d t h e s o l u t i o n t o t h e p r o b l e m of t h e c o m p o s i t i o n of

t h e r i n g s of S a t u r n ? U n f o r t u n a t e l y , t h e a n s w e r t o t h i s q u e s t i o n is n o . U p

t o t h i s p o i n t in t h e t e x t , w e h a v e c o n s i d e r e d o n l y t h e q u e s t i o n of w h e t h e r

o r n o t a fluid m a s s c o u l d b e in e q u i l i b r i u m . B u t t h e r e a r e b o t h u n s t a b l e

a n d s t a b l e e q u i l i b r i a , a n d it t u r n s o u t t h a t t h e o n e t r e a t e d in t h i s s e c t i o n i s

w h i c h g i v e s t h e e q u i l i b r i u m c o n d i t i o n a s

(2 .G.6)

(2 .G.5)


of t h e f o r m e r v a r i e t y . I n P r o b l e m 3.2, it is s h o w n t h a t a s m a l l p e r t u r b a t i o n

of t h e c e n t e r of t h e r i n g wil l l o w e r t h e e n e r g y of t h e r i n g s y s t e m , s o t h a t a

fluid r i n g of t h e t y p e w e h a v e d i s c u s s e d w o u l d n o t s u r v i v e l o n g in n a t u r e .

T h e c o n c e p t of s t ab i l i t y i s , h o w e v e r , a v e r y i m p o r t a n t o n e in fluid

m e c h a n i c s , a n d w e will n o w t u r n t o a full d i s c u s s i o n of it .

SUMMARY

W e h a v e s e e n t h a t b y g o i n g t o a f r a m e r o t a t i n g w i t h a fluid m a s s , t h e

d y n a m i c a l p r o b l e m of c a l c u l a t i n g t h e m o t i o n of s u c h fluids c a n b e

r e p l a c e d b y t h e s t a t i c p r o b l e m of b a l a n c i n g p r e s s u r e , c e n t r i f u g a l f o r c e ,

a n d g r a v i t a t i o n . T h e m e t h o d of c a l c u l a t i n g e q u i l i b r i u m s h a p e s fo r s u c h

b o d i e s is q u i t e s i m p l e in p r i n c i p l e ( a l t h o u g h s o m e t i m e s c o m p l i c a t e d

m a t h e m a t i c a l l y ) . W e s i m p l y c a l c u l a t e t h e g r a v i t a t i o n a l p o t e n t i a l f o r t h e

b o d y , i n s e r t t h i s i n t o t h e E u l e r e q u a t i o n , a n d d e m a n d t h a t a s u r f a c e of

c o n s t a n t p r e s s u r e c o i n c i d e w i t h t h e s u r f a c e of t h e b o d y . I n t h i s w a y ,

v a r i o u s p h y s i c a l s y s t e m s w e r e e x a m i n e d , i n c l u d i n g e l l i p s o i d s ( s u c h a s t h e

e a r t h ) , d i s k s ( s u c h a s t h e g a l a x y ) , a n d r i n g s ( s u c h a s t h o s e a r o u n d S a t u r n ) ,

a n d it w a s f o u n d t o b e p o s s i b l e t o find e q u i l i b r i u m c o n f i g u r a t i o n s f o r e a c h

s h a p e .

PROBLEMS

2 . 1 . Show that , unlike the ear th , the approximat ion of treating the sun as a rotating ideal fluid does not give good agreement be tween theory and observat ion for ctf/coc. The fact that the outer surface of the sun rota tes slowly has caused many problems in as t rophysics .

2.2. Consider a galaxy which has a mass distribution given by

Show that this leads to an expression for angular f requency given by

Hence , show that in the limit r/R0-+0, this distribution gives a solid body rotat ion just like the Maclaurin ellipsoid. (Hint: You might want to consult L . Mestel , R.A.S. Monthly Notices 126, 553 (1963).)

M(r) = Mo

M ( r ) = 0

(r < Ro),

(r>R0).

(Hint: You may find the following change of coordinates useful

r' sin 6' = s sin ip,

r — r' cos 6' = s cos ip.)

Problems 37

2.3. Show that it is impossible for a prolate spheroid to be in equilibrium. This cor responds to our intuition, which tells us that centrifugal force will tend to pull a rotat ing body out at the equator , thus leading to oblate shapes .

2.4. P rove that a rotat ing body in equilibrium must be symmetrical about a plane through its center and perpendicular to the axis of rotat ion. (Hint: Show that if this were not t rue , the pressure at the points on the surface at the tips of a column through the fluid perpendicular to the plane could not be equal.)

2.5. A very serious problem in as t ronomy is determining how much mat ter there is in the universe , since not all mat ter is luminous and therefore visible. Fo r example , we know that there is a lot of dust in the galaxy which can be detected only by looking at light which has come through it. Suppose in a distant galaxy we observed a densi ty of luminous mat ter

where R is the radius of the galaxy. Suppose we also observed a rotat ion curve

Find an express ion for TJ , the rat io of luminous to nonluminous mat ter from these

experimental ly determined numbers .

2.6. Calculate the angular momen tum of the sun and of the entire solar sys tem. Which bodies carry most of the angular m o m e n t u m ?

2.7. The theories of the sun ' s formation which are now accepted suggest that the sun condensed out of a gas which was initially rotating. As an example of this p rocess , consider a sphere of gas of mass M and angular momen tum L. Suppose that this sphere collapses by some process which we do not follow to a Maclaurin ellipsoid whose major axis is of length a.

(a) If we conserve M and L, write an express ion for the densi ty of the gas in the final s tate as a function of the paramete r £

(b) H e n c e wri te one (complicated) equat ion for f itself. (c) For a body like the sun, which is nearly spherical , solve for f and hence co,

the f requency of rotat ion. (d) If the original cloud was the size of the solar sys tem, how much did the sun

speed up when contract ing?

2.8. Using the methods of Appendix B, find the electrostat ic potential at the points inside an ellipsoid which has a charge a per unit volume.

2.9. Consider an ellipsoid which has a charge densi ty <x per unit volume and a mat ter densi ty 0 per unit volume.

(a) Derive the express ion corresponding to Eq. (2.A.4) for such a sys tem. (b) H e n c e find the surfaces of cons tant pressure , and wri te down the condit ion

which tells whe ther the ellipsoid can be in equilibrium.

given by

V = cr.


(c) Define a new critical f requency for the charged ellipsoid. Can it ever be zero? Give a physical interpretat ion of this result .

2.10. It has somet imes been suggested that the galaxies are moving away from each other because of a small electrostat ic charge on each galaxy. H o w would Eq. (2.F.3) be changed if this were so? Unde r what condit ions would the galactic rotat ion curve for a charged and uncharged galaxy be the same?

2.11. Calculate the critical f requency of the ear th , the sun, and as many of the planets as you can. Are any near this limit?

2.12. One theory for the formation of the asteroid belt (which is not accepted today) is that the asteroids are the result of the disruption of a planet . Le t us call this planet Kryp ton for definiteness, and argue as follows: Since the planet was near the ear th and Mars , it was presumably formed in the same way , and hence should ro ta te with about the same speed. If this were so , wha t would its density have to be to have it disrupt because of the mechanisms discussed in this chapter? Are there any materials of this densi ty known? Are they prevalent in the asteroid bel t?

REFERENCES

A discussion of stability of gravitating fluids is given in H. Lamb, Hydrodynamics (cited in Chapter 1). For more detailed presentations of the principles of stellar structure, see

S. Chandrasekar, An Introduction to the Study of Stellar Structure, Dover Publications, New York, 1957.

John P. Cox and R. T. Giugli, Principles of Stellar Structure, Gordon and Breach, New York, 1968.

3

The Idea of Stability

Bright star, were I as steadfast as thou art!

JOHN KEATS

Sonnet written on a blank page in Shakespeare's poems

A. INTRODUCTION

U p t o t h i s p o i n t w e h a v e o n l y b e e n c o n c e r n e d w i t h q u e s t i o n s r e l a t e d t o

t h e p o s s i b i l i t y of b a l a n c i n g f o r c e s in fluid m a s s e s . W e h a v e , in o t h e r w o r d s ,

l o o k e d o n l y fo r s i t u a t i o n s in w h i c h it w a s p o s s i b l e t o e s t a b l i s h e q u i l i b r i u m .

W e h a v e n o t a s k e d w h e t h e r t h e e q u i l i b r i u m c o n f i g u r a t i o n s w h i c h w e h a v e

f o u n d w e r e s t a b l e . F o r a s y s t e m t o b e in s t a b l e e q u i l i b r i u m , w e m u s t n o t

o n l y h a v e a s i t u a t i o n in w h i c h f o r c e s a r e in b a l a n c e , b u t w h e r e s m a l l

d e v i a t i o n s of t h e s y s t e m f r o m t h e e q u i l i b r i u m m u s t g e n e r a t e f o r c e s w h i c h

t e n d t o d r i v e t h e s y s t e m b a c k t o w a r d i t s e q u i l i b r i u m c o n f i g u r a t i o n , r a t h e r

t h a n f a r t h e r a w a y f r o m it . T h e c l a s s i c e x a m p l e of s u c h a s y s t e m is a m a s s

o n t h e e n d of a n u n s t r e t c h e d s p r i n g . A n y m o v e m e n t of t h e m a s s a w a y f r o m

t h i s e q u i l i b r i u m p o s i t i o n r e s u l t s in t h e s p r i n g e x e r t i n g a f o r c e p u l l i n g (o r

p u s h i n g ) t h e m a s s b a c k t o w a r d i t s o r i g i n a l p o s i t i o n .

A ba l l s i t t i ng o n t o p of a hill w o u l d b e a n e x a m p l e of a u n s t a b l e

e q u i l i b r i u m , s i n c e s m a l l c h a n g e s of p o s i t i o n w o u l d r e s u l t in t h e ba l l b e i n g

d r i v e n f a r t h e r a n d f a r t h e r f r o m e q u i l i b r i u m . A t h i r d t y p e of e q u i l i b r i u m —

n e u t r a l e q u i l i b r i u m — c a n b e d e f i n e d b e t w e e n t h e s e t w o . T h i s is a s i t u a t i o n

in w h i c h m o v e m e n t a w a y f r o m t h e e q u i l i b r i u m p o s i t i o n r e s u l t s in n o f o r c e s

b e i n g e x e r t e d a t al l . A ba l l o n a flat t a b l e t o p w o u l d b e a n e x a m p l e of s u c h a

s y s t e m . L e t u s n o w t r y t o f o r m u l a t e t h e s e i d e a s m o r e q u a n t i t a t i v e l y .

39

40 The Idea of Stability

L e t u s b e g i n w i t h a s y s t e m in w h i c h t h e k i n e t i c e n e r g y c a n b e n e g l e c t e d ,

a n d w h e r e t h e p o t e n t i a l e n e r g y c a n b e w r i t t e n a s V(qx... qt...) w h e r e t h e

qi a r e s o m e c o o r d i n a t e s . T h e n t h e c o n d i t i o n fo r e q u i l i b r i u m is t h a t t h e

f o r c e s o n t h e s y s t e m c a n c e l — i . e . t h a t a t e q u i l i b r i u m

T o i n v e s t i g a t e s t ab i l i t y , w e m u s t a s k h o w t h e s y s t e m b e h a v e s if w e m o v e

s l igh t ly a w a y f r o m e q u i l i b r i u m — i . e . if w e le t

qi (3 .A.2)

T o m a k e s u c h a n i n v e s t i g a t i o n , w e e x p a n d t h e p o t e n t i a l n e a r t h e

e q u i l i b r i u m p o i n t in a T a y l o r s e r i e s

V(qx . . . q l m . .) = V(ql0... qi0...)

(3 .A.3)

N o w f r o m t h e e q u i l i b r i u m c o n d i t i o n , t h e t e r m l i n e a r in t h e d i s p l a c e m e n t TJ,

is z e r o , s o w e s e e t h a t t h e c h a n g e in V a s w e m o v e a w a y f r o m e q u i l i b r i u m is

g o v e r n e d b y t h e s ign of t h e s e c o n d d e r i v a t i v e s of t h e p o t e n t i a l . If t h e t e r m

b i l i n e a r in TJ is p o s i t i v e , t h e n m o v i n g a w a y f r o m e q u i l i b r i u m t e n d s t o

i n c r e a s e t h e e n e r g y of t h e s y s t e m , s o t h a t t h e e q u i l i b r i u m is s t a b l e . If t h i s

t e r m is n e g a t i v e , h o w e v e r , t h e n s m a l l d e v i a t i o n s t e n d t o d e c r e a s e t h e e n e r g y

of t h e s y s t e m , a n d t h e e q u i l i b r i u m wil l b e u n s t a b l e .

T o fix t h e s e i d e a s m o r e firmly, l e t u s c o n s i d e r t h e c a s e of a p o t e n t i a l

w h i c h d e p e n d s o n o n l y o n e c o o r d i n a t e q a n d o n o n e o t h e r p a r a m e t e r A. I n

t h e e x a m p l e of t h e p a r t i c l e o n t h e s p r i n g , t h e s e w o u l d b e t h e p o s i t i o n of t h e

p a r t i c l e a n d t h e s p r i n g c o n s t a n t . I n t h i s s i m p l e e x a m p l e , t h e p o t e n t i a l a s a

f u n c t i o n of t h e t w o p a r a m e t e r s q a n d A w o u l d b e a s u r f a c e in t h r e e

d i m e n s i o n s . A n e x a m p l e of s u c h a s u r f a c e is p i c t u r e d in F i g . 3 . 1 .

T o b e g i n o u r d i s c u s s i o n of s t ab i l i t y , l e t u s c o n s i d e r o n l y t h o s e

p e r t u r b a t i o n s in w h i c h q is v a r i e d w h i l e A is h e l d fixed. W e wil l c o n s i d e r

o t h e r t y p e s of p e r t u r b a t i o n s l a t e r .

A t t h e p o i n t P , a p l a n e a t fixed A g i v e s a c u r v e of V v e r s u s q w h i c h l o o k s

l ike F i g . 3.2. T h u s , e i t h e r b y i n s p e c t i o n o r f r o m E q . 3 . A . 3 , w e s e e t h a t t h e

s y s t e m is s t a b l e a t t h e p o i n t P a g a i n s t p e r t u r b a t i o n s in w h i c h A is h e l d fixed.

Introduction 41

Fig. 3.1. Potential surface as a function of two parameters.

I n t e r m s of o u r s p r i n g e x a m p l e , a t t h i s p o i n t , t h e s p r i n g wi l l t e n d t o p u l l t h e

s y s t e m b a c k i n t o e q u i l i b r i u m .

A t t h e p o i n t Q, t h e s i t u a t i o n is s o m e w h a t d i f f e r en t . H e r e t h e c u t t h r o u g h

t h e p o t e n t i a l s u r f a c e a t c o n s t a n t A y i e l d s a g r a p h l i ke F i g . 3 .3 , s o t h a t t h e

s y s t e m is u n s t a b l e a g a i n s t p e r t u r b a t i o n s w i t h c o n s t a n t A a t t h i s p o i n t .

T h e t r a n s i t i o n b e t w e e n t h e s e t w o c a s e s o c c u r s a t A , w h e r e t h e p o t e n t i a l

s u r f a c e l o o k s l i ke F i g . 3.4. T h i s r e p r e s e n t s n e u t r a l e q u i l i b r i u m , w h e r e t h e

s e c o n d d e r i v a t i v e s of t h e p o t e n t i a l v a n i s h , s o t h a t d i s p l a c e m e n t s of t h e

s y s t e m d o n o t c h a n g e i t s e n e r g y a t al l .

T h u s , t h e p o t e n t i a l s u r f a c e w e h a v e d r a w n a s a n e x a m p l e i l l u s t r a t e s all of

t h e t y p e s of s t a b i l i t y d i s c u s s e d e a r l i e r . I t a l s o i l l u s t r a t e s a n o t h e r v e r y

i m p o r t a n t p o i n t a b o u t s t a b i l i t y . T o s e e t h i s p o i n t , l e t u s g o b a c k t o o u r

c o n s i d e r a t i o n of t h e p o i n t P . P r e v i o u s l y , w e h a d c o n s i d e r e d o n l y t h o s e

p e r t u r b a t i o n s in w h i c h w e c h a n g e d q s l i gh t ly , b u t h e l d A fixed. L e t u s n o w

c o n s i d e r t h e o t h e r a l t e r n a t i v e — l e t u s c o n s i d e r a p e r t u r b a t i o n in w h i c h q is

h e l d fixed a n d A is v a r i e d ( t h i n k , f o r e x a m p l e , of h o l d i n g t h e p o s i t i o n of t h e

Fig. 3.2. Potential surface at P for fixed A. Fig. 3.3. Potential surface at Q for fixed A.

q Q


A

Fig. 3.4. Potential surface at A for fixed A. Fig. 3.5. Potential surface at P for fixed q.

p a r t i c l e a t t h e e n d of a s p r i n g fixed, b u t h e a t i n g t h e s p r i n g s o t h a t t h e s p r i n g

c o n s t a n t c h a n g e s ) . A t P , t h i s c o r r e s p o n d s t o l o o k i n g a t a p l a n e p e r p e n d i c u -

l a r t o t h e q - a x i s , in w h i c h c a s e w e h a v e F i g . 3 .5 .

I n o t h e r w o r d s , t h e s y s t e m a t P w a s s t a b l e a g a i n s t t h e first t y p e of

p e r t u r b a t i o n b u t u n s t a b l e a g a i n s t t h e s e c o n d ! T h i s i s a v e r y i m p o r t a n t p o i n t

w h e n d i s c u s s i n g s t a b i l i t y — o n e m u s t a l w a y s s p e c i f y a g a i n s t w h i c h t y p e s of

p e r t u r b a t i o n t h e s y s t e m i s s t a b l e . T h e r e a r e m a n y s y s t e m s ( w e sha l l c o n s i d e r

o n e in t h e n e x t s e c t i o n ) w h i c h a r e s t a b l e a g a i n s t o n e t y p e of p e r t u r b a t i o n ,

b u t u n s t a b l e a g a i n s t a n o t h e r . I n o u r t w o - d i m e n s i o n a l e x a m p l e , t h e n , w e

w o u l d s a y t h a t a p o i n t w a s a p o i n t of s t ab i l i t y if a n d o n l y if t h e s e c o n d

d e r i v a t i v e s of V w e r e p o s i t i v e in e v e r y d i r e c t i o n a r o u n d t h e p o i n t , o r ,

e q u i v a l e n t l y , t h e p o t e n t i a l e x h i b i t e d a m i n i m u m in e v e r y p o s s i b l e p l a n e

d r a w n t h r o u g h t h e p o i n t .

If t h i s w e r e n o t t h e c a s e , s m a l l t h e r m a l f l u c t u a t i o n s w o u l d e v e n t u a l l y

m o v e t h e s y s t e m s l igh t ly in t h e d i r e c t i o n in w h i c h t h e p o t e n t i a l w o u l d b e

l o w e r , a n d , o n c e s t a r t e d , n o t h i n g c o u l d b r i n g it b a c k ( th i s i s s i m i l a r t o a ba l l

ro l l ing d o w n a h i l l—i t t a k e s o n l y a s m a l l p u s h t o s t a r t it g o i n g ) .

O u r e x a m p l e h a s c o n c e r n e d i t se l f o n l y w i t h a p o t e n t i a l w h i c h d e p e n d s o n

t w o v a r i a b l e s . I n g e n e r a l , p o t e n t i a l s wi l l d e p e n d o n m a n y m o r e v a r i a b l e s

t h a n t h i s . F o r e x a m p l e , a p a r t i c l e m o v i n g in t h r e e d i m e n s i o n s a t t a c h e d t o

t h r e e s p r i n g s w o u l d d e p e n d o n s ix v a r i a b l e s — t h e x, y, z c o o r d i n a t e s of t h e

p a r t i c l e a n d t h e t h r e e s p r i n g c o n s t a n t s . T h e p o t e n t i a l w o u l d t h e n b e a s u r f a c e

in a s e v e n - d i m e n s i o n a l s p a c e . T h e i d e a of finding m i n i m a a n d m a x i m a , a n d

t h e o t h e r p r o p e r t i e s of s t a b i l i t y d i s c u s s e d f o r t h e s i m p l e e x a m p l e a b o v e ,

h o w e v e r , is still a p p l i c a b l e , a n d p r o v i d e s a u s e f u l w a y t o v i s u a l i z e t h e

p r o b l e m .

L e t u s n o w t u r n t o a d i s c u s s i o n of t h e e v o l u t i o n of s y s t e m s in t i m e . W e g o

b a c k t o o u r s i m p l e t w o - d i m e n s i o n a l e x a m p l e , a n d s u p p o s e t h a t n o w w e d e a l

w i t h a p i e c e of t h e p o t e n t i a l s u r f a c e w h i c h l o o k s l i ke F i g . 3.6.

T h e l i ne XYZ n o w r e p r e s e n t s a l i ne of e x t r e m a of t h e s u r f a c e . T h e p o i n t Z

r e p r e s e n t s a s t a t e of t h e s y s t e m w h i c h is u n s t a b l e a g a i n s t a n y p e r t u r b a t i o n ,

w h i l e t h e p o i n t X r e p r e s e n t s a t r u l y s t a b l e s t a t e . N o w if w e s t a r t t h e s y s t e m

off a t s o m e p o i n t L , w h i c h is n o t n e c e s s a r i l y a p o i n t of e q u i l i b r i u m o r

s t ab i l i t y , t h e s y s t e m wil l e v o l v e in t i m e , j u s t a s a ba l l p l a c e d o n t h e s i d e of a

Stability of the Maclaurin Ellipsoid 43

9

Fig. 3.6. The evolution of a system along a potential surface.

hill wi l l s t a r t r o l l i ng . L e t t h e l i ne LM r e p r e s e n t t h e s t a t e s t h r o u g h w h i c h t h e

s y s t e m p a s s e s (it m i g h t b e h e l p f u l t o v i s u a l i z e t h i s in t e r m s of a p a r t i c l e o n a

s p r i n g — w h e n t h e s p r i n g c o n s t a n t is c h a n g e d b y h e a t i n g , f o r e x a m p l e , t h e

p o s i t i o n of t h e p a r t i c l e wi l l c h a n g e . T h i s l e a d s t o n e w v a l u e s of A a n d q, a n d

h e n c e t o a n e w s t a t e of t h e s y s t e m , r e p r e s e n t e d b y a n e w p o i n t o n t h e

s u r f a c e ) .

If t h e p o i n t M h a p p e n s t o fall o n t h e c u r v e XYZ b e t w e e n X a n d Y, t h e n

t h e s y s t e m h a s a c h a n c e of a c h i e v i n g s t ab i l i t y , w h i l e if it fa l l s b e t w e e n Y a n d

Z , it d o e s n o t . ( A g a i n , t h i n k i n g of t h e m o t i o n of t h e s y s t e m a s a b a l l r o l l i n g

a r o u n d o n t h e p o t e n t i a l s u r f a c e wi l l h e l p t o v i s u a l i z e t h i s p o i n t . )

I t is i n t e r e s t i n g t o a s k w h a t h a p p e n s if t h e p o i n t M fa l l s e x a c t l y o n t h e p o i n t

of n e u t r a l e q u i l i b r i u m Y. I n t h i s c a s e , t h e s y s t e m c a n " c h o o s e " s t a b i l i t y o r

i n s t a b i l i t y . T h e s i t u a t i o n is s i m i l a r t o b a l a n c i n g a ba l l o n a p o i n t a n d a s k i n g

w h i c h w a y it wi l l fa l l . T h e a n s w e r d e p e n d s o n a l a r g e n u m b e r of f a c t o r s — t h e

p r e c i s e w a y in w h i c h t h e ba l l w a s p l a c e d , s l igh t m o v e m e n t s of t h e a i r o r

v i b r a t i o n s of t h e f loor , e t c . S u c h e f f e c t s , w h i l e c a l c u l a b l e in p r i n c i p l e , a r e

u s u a l l y r e g a r d e d a s r a n d o m f a c t o r s b e y o n d t h e r a n g e of a n a l y s i s . B u t it is

c l e a r t h a t a t t h e p o i n t Y, a s l igh t d i s p l a c e m e n t of t h e s y s t e m t o w a r d Z wi l l

r e s u l t in i n s t a b i l i t y of t h e s y s t e m , w h i l e a s l igh t d e v i a t i o n t o w a r d X wi l l

r e s u l t in s t ab i l i t y .

B. STABILITY OF THE MACLAURIN ELLIPSOID

A s a n e x a m p l e of t h e d i s c u s s i o n of s t a b i l i t y in t h e p r e v i o u s s e c t i o n , w e

e x a m i n e t h e M a c l a u r i n e l l i p s o i d ' s s t a b i l i t y a g a i n s t a c e r t a i n t y p e of

p e r t u r b a t i o n . B e f o r e d o i n g s o , h o w e v e r , w e h a v e t o r e m e m b e r t h a t w h e n w e

a r e d e a l i n g w i t h a r o t a t i n g g r a v i t a t i n g fluid, t h e e n e r g y is m a d e u p b o t h of


k i n e t i c a n d p o t e n t i a l c o n t r i b u t i o n s , r a t h e r t h a n j u s t p o t e n t i a l e n e r g y , a s it

w a s in t h e s impl i f ied m o d e l w e c o n s i d e r e d in t h e p r e v i o u s s e c t i o n . T h u s , t h e

c o n d i t i o n f o r a n e q u i l i b r i u m b e c o m e s

(3 .B.1)

w h i l e t h e c o n d i t i o n fo r s t ab i l i t y is

(3 .B.2)

T h e s e n e w c o n d i t i o n s c o r r e s p o n d t o t h e f a c t t h a t e v e r y s y s t e m wil l t e n d t o

m o v e t o w a r d a s t a t e of l o w e s t t o t a l e n e r g y .

W e sha l l c o n s i d e r a v e r y r e s t r i c t e d c l a s s of p e r t u r b a t i o n s : t h o s e

p e r t u r b a t i o n s w h i c h

(i) c o n s e r v e a n g u l a r m o m e n t u m ,

(ii) p r e s e r v e t h a t g e o m e t r y of t h e M a c l a u r i n e l l i p s o i d ( i .e . t h o s e

p e r t u r b a t i o n s w h i c h k e e p t w o a x e s e q u a l ) ,

(iii) k e e p t h e d e n s i t y c o n s t a n t .

T h e first r e s t r i c t i o n is v e r y r e a s o n a b l e if w e t h i n k of t h i n g s l i ke s t e l l a r

b o d i e s , s i n c e a n y p e r t u r b a t i o n in s u c h a s y s t e m h a s t o c o m e f r o m w i t h i n t h e

s y s t e m i tself , a n d h e n c e p r e s e r v e a n g u l a r m o m e n t u m . T h e s e c o n d

r e s t r i c t i o n wil l b e i m p o s e d d u r i n g t h e c o u r s e of t h e d i s c u s s i o n fo r

m a t h e m a t i c a l s i m p l i c i t y .

T h e k i n e t i c e n e r g y of t h e s y s t e m in t e r m s of t h e a n g u l a r m o m e n t u m L is

j u s t 2

(3 .B.3)

w h e r e I is t h e m o m e n t of i n e r t i a a b o u t t h e a x i s of r o t a t i o n a n d is g i v e n b y

T h e p o t e n t i a l c a n b e c a l c u l a t e d in a s t r a i g h t f o r w a r d m a n n e r ( s e e P r o b l e m i n t n h e

(3 .B.4)

w h e r e t h e s y m b o l s a r e de f ined in C h a p t e r 2 .

L e t u s b e g i n b y n o t i n g t h a t t h e q u e s t i o n of s t ab i l i t y of a n e l l i p s o i d n o w

c o m e s d o w n t o finding m i n i m a in t h e f u n c t i o n E = T + V. I n g e n e r a l , t h i s is a

f u n c t i o n of a, b, a n d c. H o w e v e r , r e q u i r e m e n t (iii) m e a n s t h a t if a a n d b a r e

Stability of the Maclaurin Ellipsoid 45

c h a n g e d , t h e r e q u i r e m e n t of c o n s t a n t v o l u m e t h e n d e t e r m i n e s t h e v a l u e of c.

T h u s , E wi l l b e c o n s i d e r e d t o b e a f u n c t i o n of a a n d b o n l y . L a t e r , w e sha l l

i m p o s e r e s t r i c t i o n (ii) a n d c o n s i d e r t h e c a s e a = b o n l y . F o r t h e m o m e n t ,

h o w e v e r , le t u s k e e p t h e m o r e g e n e r a l c a s e u n d e r c o n s i d e r a t i o n .

W e c o u l d , of c o u r s e , c a l c u l a t e t h e v a l u e of E f o r e v e r y v a l u e of a a n d b

a n d l o o k f o r m i n i m a . W e c a n g e t a n a n s w e r in t h e c a s e of t h e M a c l a u r i n

e l l i p s o i d w i t h o u t s u c h a c o m p l i c a t e d p r o c e d u r e , h o w e v e r . W r i t e

N o w a s a - > o° , w e w o u l d h a v e a s i t u a t i o n in w h i c h t h e m a t e r i a l in t h e e l l i p se

w a s s p r e a d o u t o v e r all s p a c e , s o w e w o u l d e x p e c t V - » 0 . C l e a r l y , in t h i s l imi t

T -> 0 [ s e e E q . (3 .B .3)] a s w e l l , s o t h a t E -> 0. A s imi l a r a r g u m e n t h o l d s f o r t h e

l imi t b -»<».

F r o m t h e e x p r e s s i o n f o r V in E q . (3 .B .4 ) , w e s e e t h a t if e i t h e r a - » 0

o r b - » 0 , V - * 0 . If a ^ O , t h e n E(a,b)*Hb2 ( s i m i l a r l y , if b - * 0 ,

E(a, b) <* 1 /a2). T h i s m e a n s t h a t t h e f u n c t i o n E(a, b) m u s t l o o k l ike F i g . 3.7

in t h e r e g i o n s d e a l t w i t h a b o v e .

If w e r e s t r i c t o u r a t t e n t i o n t o t h e M a c l a u r i n e l l i p s o i d , w e w a n t o n l y t h e

p l a n e c o n t a i n i n g t h e l i ne a = b. N o t e t h a t b y r e s t r i c t i n g o u r a t t e n t i o n t o t h i s

p l a n e , w e a r e o n l y c o n s i d e r i n g s t ab i l i t y a g a i n s t p e r t u r b a t i o n s w h i c h l e a v e

t h e l e n g t h s of t h e t w o m a j o r a x e s e q u a l , a n d w e wil l b e u n a b l e t o s a y a n y t h i n g

a b o u t p e r t u r b a t i o n s w h i c h c h a n g e t h e s e l e n g t h s d i f f e r en t ly . H o w e v e r , in

t h i s p l a n e , t h e f u n c t i o n E(a,b) c a n b e s k e t c h e d o u t . W e k n o w t h a t it m u s t (1)

b e c o m e inf ini te a s a = b - > 0 , (2) g o t o z e r o a s a = b - » o ° , a n d (3) f r o m

S e c t i o n 2 . C , w e k n o w t h a t t h e r e is o n e a n d o n l y o n e p o i n t of e q u i l i b r i u m —

i .e . o n l y o n e p o i n t a t w h i c h dE/da = 0 . T h i s m e a n s t h a t E(a, b) in t h i s c a s e

m u s t l o o k l ike F i g . 3 .8 , w h i c h m e a n s t h a t t h e M a c l a u r i n e l l i p s o i d is s t a b l e

E(a, b)= T + V.

Fig. 3.7. The energy for a Maclaurin ellipsoid as a function of a and b.


•a =b

Fig. 3.8. The energy surface along the line a = b.

a g a i n s t p e r t u r b a t i o n s in w h i c h a = b. F r o m t h e a r g u m e n t so f a r , w e c a n

d r a w n o c o n c l u s i o n s a b o u t t h e s t ab i l i t y a g a i n s t o t h e r t y p e s of p e r t u r b a -

t i o n s .

I n f a c t , t h e M a c l a u r i n e l l i p s o i d is s t a b l e a g a i n s t all p e r t u r b a t i o n s i n v o l v i n g

b u l k c h a n g e s of t h e r e l a t i v e s i z e of t h e a x e s , a s a r e t h e J a c o b i e l l i p s o i d s . T h i s

m e a n s t h a t t h e m i n i m u m in t h e a = b p l a n e s h o w n a b o v e is a c t u a l l y a

m i n i m u m in t h e s u r f a c e E ( a , b), a n d n o t a s a d d l e p o i n t . O t h e r m i n i m a in t h e

s u r f a c e w o u l d c o r r e s p o n d , of c o u r s e , t o t h e J a c o b i e l l i p s o i d s .

F o r c o m p l e t e n e s s , it s h o u l d b e n o t e d t h a t t h e s e e l l i p s o i d s , w h i l e s t a b l e

a g a i n s t p e r t u r b a t i o n s w h i c h l e a v e t h e d e n s i t y of t h e fluid u n c h a n g e d , a r e

u n s t a b l e a g a i n s t fluctuations in t h i s d e n s i t y . T h i s i l l u s t r a t e s t h e p o i n t w h i c h

w a s m a d e e a r l i e r — t h a t it is p o s s i b l e f o r a s y s t e m t o b e s t a b l e a g a i n s t o n e t y p e

of p e r t u r b a t i o n b u t n o t a g a i n s t a n o t h e r .

SUMMARY

T h e q u e s t i o n of t h e s t a b i l i t y of a fluid s y s t e m w a s d i s c u s s e d . T h e g e n e r a l

r e q u i r e m e n t t h a t a s y s t e m b e in s t a b l e e q u i l i b r i u m is t h a t e v e r y p o s s i b l e

p e r t u r b a t i o n of t h e s y s t e m l e a d t o a s t a t e of h i g h e r t o t a l e n e r g y . I t is a l w a y s

p o s s i b l e , of c o u r s e , t h a t a s y s t e m c o u l d b e s t a b l e a g a i n s t o n e t y p e of

p e r t u r b a t i o n , b u t u n s t a b l e a g a i n s t a n o t h e r . T h e s t ab i l i t y of t h e M a c l a u r i n

e l l i p s o i d w a s i n v e s t i g a t e d , a n d it w a s s h o w n t h a t t h e e q u i l i b r i u m

c o n f i g u r a t i o n s d e r i v e d in t h e p r e v i o u s c h a p t e r w e r e i n d e e d s t a b l e a g a i n s t

p e r t u r b a t i o n s w h i c h k e e p t h e d e n s i t y of t h e fluid c o n s t a n t .

PROBLEMS

3.1. Given the expression for the potential inside of an ellipsoid from Appendix B , find the total gravitational potential energy of such a body, and hence verify Eq . (3.B.4).

3.2. A full discussion of the stability of the rings of Saturn would be a long undertaking. However , there is a relatively simple calculation that can be done to

E

Problems 47

show the instability of the rings if we a s sume that the rings are solid (clearly, if the rings cannot be stable if solid, they are unlikely to be stable if they are fluid). Consider a solid ring of circular cross section a, mass m, and radius D centered on an at tract ing body of mass M. Show that if the center of the ring is displaced slightly from the center of the at tract ing body, the energy of the system is lowered, so that the system is unstable .

3.3. Consider a Maclaurin ellipsoid of mass densi ty p and charge densi ty <x. (a) Calculate the total potential energy in such a sys tem, including bo th electrical

and gravitational contr ibut ions . (b) Unde r what condit ions will such an ellipsoid be stable? (Hint: You may wish to

refer to P rob lem 2.8.)

3.4. Consider a situation as shown in Fig. 3.9, in which a particle at the point (L, L ) is a t tached to t w o springs of equal spring cons tan ts k and uns t re tched length L.

(a) Calculate the potential energy of the sys tem if the particle is moved to an arbitrary point (X, Y) .

(b) Are there any other points of equilibrium in the plane? (c) Are these points stable or unstable equilibria?

3.5. Repea t the analysis of P rob lem 3.4 for the case when the particle carries a charge q, and a charge Q (of the same sign) is located at the origin.

3.6. An interesting kind of instability is occasionally encountered in dealing with binary star sys tems. Consider two s tars , of mass m and m l o c a t e d a dis tance JR apar t and rotat ing about the c o m m o n center of mass with f requency co.

(a) Give an argument leading to the conclusion that

(0,0) (0.L)

Fig. 3.9.

co =

(b) Show that the potent ial at any point in space is given by


where the arbitrary point is (X, Y, Z ) , and r and r' are the dis tances from the masses to the point.

(c) Show that if we define

X= r\, Y= r/x, Z = rv,

and

(d) Make a sketch of the potential in part (c) for various values of q. P roduce an argument that for some value of q, it should be possible for a particle to go from the gravitational field of one star to that of the other without expending energy. When this happens , we speak of having reached Roche' s limit, in which mass will be exchanged, be tween the s tars .

REFERENCES

For a general discussion of the stability of physical systems, see

Robert A. Becker, Introduction of Theoretical Mechanics, McGraw-Hill, New York, 1954 (Chapter 5).

S. Chandrasekar, Hydrodynamics and Hydromagnetic Stability, Clarendon Press, Oxford, 1961.

the potential becomes

4

Fluids in Motion

No man steps into the same river twice.

HERACLITUS

A. THE VELOCITY FIELD

U p t o t h i s p o i n t , w e h a v e b e e n c o n s i d e r i n g o n l y t h e c a s e of

h y d r o s t a t i c s , w h i c h d e a l s w i t h s t a t i o n a r y f lu ids . E v e n t h e c a s e of r o t a t i n g

s t a r s w a s t r e a t e d b y g o i n g t o a r o t a t i n g f r a m e of r e f e r e n c e , in w h i c h t h e

fluid w h i c h c o m p r i s e d t h e s t a r w o u l d n o t b e in m o t i o n . W e n o w t u r n o u r

a t t e n t i o n t o t h e m o r e g e n e r a l c a s e of m o v i n g f lu ids , t h e s t u d y of

h y d r o d y n a m i c s .

T h e first t h i n g w h i c h w e sha l l h a v e t o d e c i d e is h o w t o c h a r a c t e r i z e t h e

m o t i o n of t h e fluid. If w e t h i n k of t h e fluid a s b e i n g c o m p o s e d of

in f in i t e s imal v o l u m e e l e m e n t s , t h e n a v o l u m e e l e m e n t l o c a t e d a t c o o r d i -

n a t e s (x, y, z ) wil l h a v e s o m e v e l o c i t y v(x , y, z, t) ( s e e F i g . 4 .1) . T h i s

m e a n s t h a t t o e a c h p o i n t in s p a c e w e c a n a s s i g n a v e c t o r w h i c h c a n b e , in

g e n e r a l , a f u n c t i o n of b o t h p o s i t i o n a n d t i m e . T h i s c o l l e c t i o n of v e l o c i t i e s

is r e f e r r e d t o a s a velocity field.

I t is p o s s i b l e t o w r i t e d o w n t h e v e l o c i t y a t t h e p o i n t (x, y, z ) in t e r m s of

t h e v e l o c i t y v e c t o r a n d i t s d e r i v a t i v e s a t t h e o r ig in b y u s i n g a T a y l o r

e x p a n s i o n

49

(4 .A.1)

50 Fluids in Motion

(0,0,0)

Fig. 4.1. The velocity field.

If w e con f ine o u r a t t e n t i o n t o a sma l l n e i g h b o r h o o d n e a r t h e o r ig in , s o

t h a t x, y, a n d z a r e s m a l l , w e c a n i g n o r e h i g h e r - o r d e r t e r m s in t h i s

e x p a n s i o n a n d e x p r e s s t h e v e l o c i t y field n e a r t h e o r ig in in t e r m s of t h e

d e r i v a t i v e s of t h e v e l o c i t y . E q . (4 .A.1) c a n b e w r i t t e n in t h e f o r m

(in t h i s e q u a t i o n , t h e s u m m a t i o n c o n v e n t i o n is n o t u s e d ) . B y a d d i n g a n d

s u b t r a c t i n g t h e s a m e t h i n g t o t h e t e r m i n s i d e of t h e s u m m a t i o n , t h i s c a n b e

c a s t in t h e f o r m

(4 .A.2)

T h u s , t h e c h a n g e in v e l o c i t y a s w e m o v e f r o m o n e p o i n t in t h e fluid t o

a n o t h e r c a n b e w r i t t e n a s t h e s u m of t h r e e p a r t s ,

Vi(x, y,z)- 1 ^ ( 0 , 0 , 0 ) = Au, (4 .A.3)

= Dt + St + C, w h e r e

(4 .A.4)

is r e l a t e d t o t h e d i v e r g e n c e of t h e v e l o c i t y field ( i .e . dvjdxi is o n e p i e c e of

t h e d i v e r g e n c e V • vL

(4 .A.5)

The Velocity Field 51

is r e l a t e d t o t h e c u r l of t h e field, a n d t h e r e m a i n i n g t e r m ,

(4 .A.6)

wi l l j u s t b e c a l l e d t h e " s y m m e t r i c p a r t . "

T h e p u r p o s e of w r i t i n g in t h i s r a t h e r c u m b e r s o m e w a y is t o t r y t o

u n d e r s t a n d w h a t d i f f e ren t s o r t s of v e l o c i t y field c o r r e s p o n d t o in t e r m s of

p h y s i c a l m o v e m e n t of t h e fluid. F o r e x a m p l e , w e sha l l s e e t h a t t h e r e is a n

i n t i m a t e r e l a t i o n s h i p b e t w e e n t h e e x p r e s s i o n V x v a n d r o t a t i o n a l m o t i o n

in t h e fluid, a n d b e t w e e n t h e e x p r e s s i o n V • v a n d c h a n g e s of d e n s i t y .

T h u s , it wi l l b e p o s s i b l e t o g o f r o m t h e r a t h e r f o r m a l de f in i t ion of a

v e l o c i t y field w h i c h w e h a v e g i v e n a b o v e , in w h i c h e a c h p o i n t in s p a c e is

a s s o c i a t e d w i t h a v e c t o r , t o a p h y s i c a l p i c t u r e of w h a t s o r t of fluid m o t i o n

is a s s o c i a t e d w i t h v e l o c i t y fields w i t h d i f f e ren t k i n d s of p r o p e r t i e s .

T h e t e c h n i q u e w h i c h w e sha l l u s e t o a c c o m p l i s h t h i s wi l l b e t o c o n s i d e r

f o u r p o i n t s in t h e fluid a t t i m e t = 0 ( s e e F i g . 4 .2 ) . W e sha l l t h e n c o m p u t e

t h e v e l o c i t y a t e a c h c o r n e r of t h e s q u a r e in t e r m s of D , S, a n d C, w h i c h

sha l l b e c a l c u l a t e d f r o m t h e g i v e n v e l o c i t y field i tself . W e sha l l t h e n a s k

w h a t t h e f o u r p o i n t s l o o k l ike a n in f in i t e s ima l t i m e r l a t e r . E a c h p o i n t wil l

h a v e m o v e d a c e r t a i n in f in i t e s ima l d i s t a n c e . F o r e x a m p l e , t h e p o i n t (0 , L )

wil l h a v e m o v e d a d i s t a n c e

in t h e y - d i r e c t i o n . S i m i l a r r e s u l t s wi l l b e o b t a i n e d f o r e a c h of t h e o t h e r

p o i n t s , s o t h a t ( r e s t r i c t i n g o u r a t t e n t i o n t o t w o - d i m e n s i o n a l flow), a t t i m e

r , w e sha l l h a v e t h e s i t u a t i o n in F i g . 4 . 3 . T h u s , h a v i n g c a l c u l a t e d D , S, a n d

Ax = vx(0, L ) T

in t h e x-direction, a n d a d i s t a n c e

Ay = i ? y ( 0 , L ) r

(L,L) (0,L)

(0,0) (L,0)

Fig. 4.2. The initial square in a moving fluid.

52 Fluids in Motion

(vx{0, L)T, L + vy(0, L)r) (L + vx(L,L)r,L + vy(L,L)T)

| / ( M 0 , 0 ) T , ^ (0 ,0 )T) | /

f i — (L + VX(L, 0)T, vy(0, L)r)

Fig. 4.3. The final configuration of the square.

C f r o m t h e v e l o c i t y field, w e c a n i m m e d i a t e l y v i s u a l i z e t h e t y p e of m o t i o n

w h i c h i s b e i n g e x e c u t e d b y t h e fluid.

Of c o u r s e , w e c o u l d d o t h i s d i r e c t l y , w i t h o u t c a l c u l a t i n g D , S, a n d C, b y

t a k i n g t h e v e l o c i t i e s a t t h e p o i n t s of t h e s q u a r e d i r e c t l y f r o m t h e v e l o c i t y

field. W e sha l l s e e in l a t e r s e c t i o n s , h o w e v e r , t h a t t h e d i v e r g e n c e a n d t h e c u r l

of t h e v e l o c i t y field p l a y a s p e c i a l r o l e in d e s c r i b i n g fluid flow, a n d h e n c e it is

i m p o r t a n t t o d e s c r i b e fluid m o t i o n in t h e w a y w e h a v e a b o v e .

W e sha l l p r o c e e d b y l o o k i n g a t t h r e e e x a m p l e s , in w h i c h v e l o c i t y fields a r e

c h o s e n s o t h a t o n l y o n e of t h e t h r e e t e r m s in Avt i s n o n z e r o .

Example I

C o n s i d e r a v e l o c i t y field in t w o d i m e n s i o n s g i v e n b y

T h i s wi l l r e s u l t in a v e l o c i t y c o n f i g u r a t i o n l i ke t h a t s h o w n in F i g . 4 .4 . F o r t h i s

field, w e h a v e

vx = Cx,

Vy = 0 . (4 .A.7)

Fig. 4.4. The velocity field for Example I.


(L,0)

Fig. 4.5. The final configuration for Example I.

a n d

C, = Si = Dy = Dz = 0 .

T h u s , t h e x - c o m p o n e n t s of t h e v e l o c i t y a t e a c h of t h e f o u r p o i n t s a r e g i v e n

b y 1^(0,0) = 0,

t > x ( 0 , L ) = 0,

vx(L9L) = CL9

vx(L, 0) = C L .

(4 .A.8)

T h e s q u a r e a t t i m e r wi l l t h e n a p p e a r a s in F i g . 4 . 5 .

T h u s , a v e l o c i t y field w h i c h p o s s e s s e s a n o n z e r o d i v e r g e n c e wi l l g i v e r i s e

t o m o t i o n w h i c h c a n b e c h a r a c t e r i z e d a s a s t r e t c h i n g a l o n g o n e of t h e m a j o r

a x e s . T h i s i s q u i t e a r e a s o n a b l e r e s u l t , s i n c e w e k n o w t h a t f o r a n

i n c o m p r e s s i b l e fluid, t h e e q u a t i o n of c o n t i n u i t y b e c o m e s

V • v = 0 ,

s o t h a t t h e e x i s t e n c e of a d i v e r g e n c e i m p l i e s t h a t t h e r e m u s t b e a c h a n g i n g

d e n s i t y in o r d e r f o r c o n t i n u i t y t o b e sa t i s f ied . P i c t o r i a l l y , w e s e e t h a t s u c h a

c h a n g e of d e n s i t y m u s t o c c u r , t o o , s i n c e t h e a r e a b o u n d e d b y t h e l i n e s in t h e

a b o v e figure c h a n g e s , b u t n o fluid c r o s s e s t h e b o u n d a r i e s , s o t h a t t h e d e n s i t y

m u s t d e c r e a s e .

Example II

C o n s i d e r a v e l o c i t y field g i v e n b y

vx = Cy , (4 .A .9 )

DV = CX.

X Fig. 4.6. The velocity field for Example II.

T h i s wi l l r e s u l t in a v e l o c i t y c o n f i g u r a t i o n l ike t h a t s h o w n in F i g . 4 .6 , w h i c h

h a s

Sx=Cy,

Sy CX,

Q = Dt = SZ=0.

F o r s u c h a field, t h e x- a n d y - c o m p o n e n t s of v e l o c i t y a t t h e p o i n t s of t h e

s q u a r e a r e

i>,(0,0) = 0 = t > , ( 0 , 0 ) ,

vx(U 0) = 0 = t > y ( 0 , L ) , (4 . A . 10)

vx(0,L) = CL = vy(L,0),

vx(L,L)=CL = vy(UL).

T h e s q u a r e a t t i m e r wi l l t h e n a p p e a r a s in F i g . 4 .7 .

W e s e e , t h e n , t h a t a v e l o c i t y field c h a r a c t e r i z e d b y a n o n z e r o s y m m e t r i c

p a r t a l s o c o r r e s p o n d s t o a u n i f o r m s t r e t c h i n g of t h e fluid, b u t t h i s t i m e a l o n g

s o m e a x i s o t h e r t h a n a c o o r d i n a t e a x i s .

54 Fluids in Motion

y

I CiTJ

i

CLT

CLT

Fig. 4.7. The final configuration for Example II.

y


(4 . A . 11)

Example III

A s a final e x a m p l e , c o n s i d e r a v e l o c i t y field g i v e n b y

Vx = Cy,

Vy = - CX.

T h i s wi l l r e s u l t in a v e l o c i t y c o n f i g u r a t i o n l i ke t h a t s h o w n in F i g . 4 . 8 , a n d h a s

A = Si = 0,

C,= C y,

Cy=~C X.

\

Fig. 4.8. The velocity field for Example III.

T h e v e l o c i t i e s of t h e c o r n e r s of t h e s q u a r e a r e n o w

t>x(0,0) = 0 = ! > y ( 0 , 0 ) ,

vx(L, 0) = 0 = i ; y ( 0 , L ) ,

t>,(0, L) = CL = vx(L,L),

vy(L,0) = -CL = vy(L,L).

(4 . A . 12)

T h e s q u a r e a t t i m e t wi l l t h e n a p p e a r a s in F i g . 4 .9 .

S i m p l e g e o m e t r y s h o w s t h a t t h i s v e l o c i t y field c o r r e s p o n d s t o a r o t a t i o n of

t h e s q u a r e a r o u n d t h e o r ig in , w i t h n o c h a n g e in a r e a . T h u s , t h e e x i s t e n c e of

CLT [ CLT

r V * ' CLT

Fig. 4.9. The final configuration for Example III.

56 Fluids in Motion

t h e c u r l of a v e l o c i t y field c o r r e s p o n d s t o r o t a t i o n a l m o t i o n , j u s t a s t h e

e x i s t e n c e of a d i v e r g e n c e of a s y m m e t r i c p a r t c o r r e s p o n d s t o s t r e t c h i n g

m o t i o n .

W i t h t h i s u n d e r s t a n d i n g , w e c a n n o w l o o k a t s o m e g e n e r a l f e a t u r e s of fluid

flow.

B. THE VELOCITY POTENTIAL

W e h a v e s e e n t h a t if w e k n o w t h e v e l o c i t y of t h e fluid e l e m e n t s in a v o l u m e

a s a f u n c t i o n of t h e p o s i t i o n , w e c a n m a k e s o m e v e r y g e n e r a l s t a t e m e n t s

a b o u t t h e t y p e of fluid m o t i o n w h i c h o c c u r s . I n p a r t i c u l a r , if t h e field is s u c h

t h a t V - v = 0 , (4 .B .1)

t h e fluid d e n s i t y c a n n o t c h a n g e , a n d o n l y m o t i o n s w h i c h c o n s e r v e d e n s i t y

a r e a l l o w e d . O n t h e o t h e r h a n d , if t h e field is s u c h t h a t

V x v = 0 , (4 .B.2)

t h e n n o r o t a t i o n a l m o t i o n is a l l o w e d in t h e fluid. F o r o b v i o u s r e a s o n s , s u c h a

flow is c a l l e d irrotational flow.

I t is c l e a r t h a t if w e h a v e i r r o t a t i o n a l flow, t h e v e l o c i t y c a n b e w r i t t e n

v = V<£, (4 .B.3)

w h e r e </> is a s c a l a r f u n c t i o n c a l l e d t h e velocity potential. ( T h i s i s v e r y s imi l a r

t o t h e de f in i t ion of a m a g n e t i c s c a l a r p o t e n t i a l in e l e c t r o m a g n e t i c t h e o r y in

t h e s t a t i c c a s e w h e r e V x B = 0 ; s o B = V<£m.) T h u s , i r r o t a t i o n a l flow is

s o m e t i m e s r e f e r r e d t o a s potential flow. A l m o s t all of t h e e x a m p l e s w h i c h w e

sha l l c o n s i d e r wi l l i n v o l v e p o t e n t i a l flow, w h i c h is f o r t u n a t e , s i n c e t h e

i n t r o d u c t i o n of a v e l o c i t y p o t e n t i a l a l l o w s u s t o w o r k w i t h s c a l a r r a t h e r t h a n

v e c t o r q u a n t i t i e s .

A n i n t e r e s t i n g r e s u l t c a n b e w r i t t e n d o w n in t h e s p e c i a l c a s e of p o t e n t i a l

flow of a n i n c o m p r e s s i b l e fluid. F r o m E q s . ( 4 . B . 1) a n d (4 .B .3 ) , w e h a v e

V2<f> = 0 , (4 .B.4)

w h i c h is j u s t L a p l a c e ' s e q u a t i o n . W e sha l l h a v e r e p e a t e d r e c o u r s e t o t h i s

r e s u l t in f u t u r e e x a m p l e s .

L e t u s w r i t e d o w n t h e E u l e r e q u a t i o n in t e r m s of t h e v e l o c i t y p o t e n t i a l . If

w e s t a r t w i t h t h e E u l e r e q u a t i o n in t h e f o r m [ s e e E q . (1 .B .5) ]

The Velocity Potential 57

t h e n t h e s u b s t i t u t i o n of Vcp f o r v y i e l d s ( r e c a l l i n g t h a t t h e c u r l of t h e g r a d i e n t

v a n i s h e s )

(4 .B.5)

s o t h a t , in g e n e r a l ,

(4 .B.6)

w h e r e / ( t ) i s a n a r b i t r a r y f u n c t i o n of t i m e , a n d p l a y s t h e r o l e of a n i n t e g r a t i o n

" c o n s t a n t . " T o d e a l w i t h t h e f u n c t i o n f(t), w e n e e d t o n o t i c e a n i m p o r t a n t

p r o p e r t y of t h e v e l o c i t y p o t e n t i a l . If w e h a v e a p o t e n t i a l <f> w h i c h g i v e s r i s e

t o a v e l o c i t y field v, t h e n a n y p o t e n t i a l of t h e f o r m

4>' = <l> + J " / ( * ' ) * ' (4 .B .7)

wil l g i v e r i s e t o e x a c t l y t h e s a m e v e l o c i t y field. S i n c e it is o n l y v w h i c h c a n b e

m e a s u r e d , w e c a n a l w a y s a d d o r s u b t r a c t a n y f u n c t i o n of t i m e t o a n y v e l o c i t y

p o t e n t i a l w i t h o u t c h a n g i n g a n y of t h e p h y s i c s of t h e p r o b l e m . T h i s is

c o m p l e t e l y a n a l o g o u s t o t h e f a c t t h a t w e c a n a l w a y s a d d a c o n s t a n t t e r m t o a

g r a v i t a t i o n a l p o t e n t i a l w i t h o u t c h a n g i n g a n y f o r c e s , a n d c o r r e s p o n d s t o t h e

f r e e d o m t o p i c k t h e z e r o of a p o t e n t i a l w h e r e v e r w e l i ke , s i n c e o n l y

p o t e n t i a l d i f f e r e n c e s c a n b e m e a s u r e d . T h e r e f o r e , w i t h o u t l o s s of g e n e r a l -

i t y , w e c a n w r i t e

(4 .B.8) = c o n s t .

If, in a d d i t i o n t o b e i n g i r r o t a t i o n a l , t h e f low h a s a c h i e v e d s t e a d y s t a t e ( i .e . a

s i t u a t i o n w h e r e t h e v e l o c i t y a t a n y g i v e n p o i n t d o e s n o t d e p e n d e x p l i c i t l y o n

t h e t i m e , a l t h o u g h it m a y v a r y f r o m p o i n t t o p o i n t ) , t h e n dcp/dt = 0 , a n d t h i s

r e d u c e s t o

(4 .B.9)

w h i c h is a s p e c i a l c a s e of t h e Bernoulli equation. I n P r o b l e m 4 .4 , t h e p r o b l e m

of s h o w i n g t h a t t h e q u a n t i t y

is t h e s a m e e v e r y w h e r e a l o n g a s t r e a m l i n e in t h e fluid is g i v e n . T h i s is t h e

m o s t g e n e r a l f o r m of t h e B e r n o u l l i e q u a t i o n , a n d s t a t e s t h a t w h i l e t h i s

q u a n t i t y m u s t b e c o n s e r v e d a l o n g a s t r e a m l i n e , it c a n , in g e n e r a l , h a v e

58 Fluids in Motion

Fig. 4.10. The idea of stability.

d i f fe ren t v a l u e s f o r d i f f e r en t s t r e a m l i n e s . F o r t h e s p e c i a l c a s e of i r r o t a t i o n a l

m o t i o n , h o w e v e r , w e h a v e s h o w n t h a t t h i s q u a n t i t y m u s t n o t o n l y b e

c o n s e r v e d a l o n g a g i v e n s t r e a m l i n e , b u t m u s t b e t h e s a m e f o r e v e r y

s t r e a m l i n e in t h e fluid.

C. STABILITY OF FLOW

I n C h a p t e r 3 , w e s a w t h a t a v e r y i m p o r t a n t p r o p e r t y of fluid s y s t e m s in

e q u i l i b r i u m w a s s t ab i l i t y . W e s a w t h a t f o r s t a t i c o r s e m i - s t a t i c s y s t e m s , t h i s

c o u l d b e u n d e r s t o o d in t e r m s of t h e p r o p e r t i e s of t h e e n e r g y s u r f a c e . If t h e

s y s t e m w a s o n e in w h i c h t h e e n e r g y i n c r e a s e d a s w e m o v e d a w a y f r o m

e q u i l i b r i u m , t h e n it w a s s t a b l e , w h i l e if t h e e n e r g y d e c r e a s e d , it w a s u n s t a b l e .

T h i s s a m e s o r t of r e a s o n i n g c a n b e a p p l i e d t o fluid flow p a t t e r n s a s w e l l ,

a l t h o u g h it is u s u a l l y m o r e c o n v e n i e n t t o m a k e t h e c a l c u l a t i o n s w h i c h a l l o w

u s t o d e c i d e w h e t h e r a s y s t e m is s t a b l e o r u n s t a b l e in a d i f f e ren t w a y . T o

u n d e r s t a n d t h i s n e w l i ne of a t t a c k a n d c o n n e c t it t o t h e d i s c u s s i o n of C h a p t e r

3 , l e t u s c o n s i d e r t h e c a s e of a ba l l r o l l i ng o n a s u r f a c e ( s e e F i g . 4 .10) .

I n C h a p t e r 3 , w e w o u l d d e s c r i b e t h e s i t u a t i o n o n t h e lef t a s u n s t a b l e

b e c a u s e a s w e m o v e a w a y f r o m e q u i l i b r i u m , t h e e n e r g y of t h e s y s t e m is

l o w e r e d . T h e s i t u a t i o n o n t h e r i gh t , h o w e v e r , w o u l d b e s t a b l e , s i n c e

m o v e m e n t a w a y f r o m t h e e q u i l i b r i u m c o n f i g u r a t i o n r a i s e s t h e t o t a l e n e r g y .

T h e n e w r e a s o n i n g w h i c h w e sha l l a p p l y t o t h e fluid flow p r o b l e m is a s

f o l l o w s : I n t h e l e f t - h a n d d i a g r a m , a s m a l l d i s p l a c e m e n t of t h e s y s t e m f r o m

e q u i l i b r i u m wil l r e s u l t in t h e ba l l m o v i n g f a r a w a y f r o m t h e t o p of t h e hill

( s i n c e a s m a l l d i s p l a c e m e n t wi l l c a u s e it t o ro l l d o w n ) . T h u s , t h e e q u a t i o n s of

m o t i o n of t h e s y s t e m m u s t b e s u c h t h a t if I a l l o w s m a l l , t i m e - d e p e n d e n t

d e p a r t u r e s f r o m e q u i l i b r i u m ( t h e o r ig in of t h e s e s m a l l p e r t u r b a t i o n s is

d i s c u s s e d in C h a p t e r 3) , t h e n x(t), t h e p o s i t i o n of t h e b a l l , wi l l e v e n t u a l l y

b e c o m e q u i t e l a r g e . F o r t h e s t a b l e c o n f i g u r a t i o n s , h o w e v e r , t h e e q u a t i o n s of

m o t i o n a r e s u c h t h a t x(t) s t a y s s m a l l ( t y p i c a l l y , t h e s y s t e m wil l p e r f o r m

s m a l l - s c a l e o s c i l l a t i o n s a r o u n d t h e e q u i l i b r i u m p o i n t ) .

T h e a d v a n t a g e of t h i s t e c h n i q u e is t h a t it a l l o w s u s t o d e t e r m i n e t h e

q u e s t i o n of s t ab i l i t y d i r e c t l y f r o m t h e e q u a t i o n s of m o t i o n , w i t h o u t

Stability of Flow 59

c a l c u l a t i n g e n e r g y a t al l . F o r e x a m p l e , if w e a s s u m e d t h a t x(t) w a s of t h e

f o r m

x(t)~ei<ot, (4 .C.1)

t h e n f o r t h e s t a b l e c a s e , w h e n w e s o l v e d t h e e q u a t i o n s of m o t i o n f o r co, t h e y

w o u l d r e q u i r e t h a t co b e r e a l . F o r t h e u n s t a b l e c a s e , h o w e v e r , t h e y w o u l d

r e q u i r e t h a t co b e c o m p l e x , a n d of t h e f o r m

co = coR — i\coi\, (4 .C .2)

s o t h a t t h e t i m e d e p e n d e n c e of x(t) w o u l d b e

x i n - e ^ e 1 ^ ' , (4 .C .3)

a n d t h e s y s t e m w o u l d i n d e e d " r u n a w a y " w h e n a s m a l l p e r t u r b a t i o n w a s

a p p l i e d .

T o s e e h o w t h i s i d e a w o r k s in t h e c a s e of a fluid, l e t u s c o n s i d e r t h e

" t a n g e n t i a l i n s t a b i l i t y " in fluid flow. L e t t h e r e b e t w o fluids, of d e n s i t y px a n d

p 2 , w i t h t h e u p p e r fluid m o v i n g w i t h v e l o c i t y v 0 . A t t h e p l a n e z = 0, t h e t w o

fluids m e e t , a n d t h e e q u i l i b r i u m c o n f i g u r a t i o n is o b v i o u s l y t h e c a s e w h e r e

t h e i n t e r f a c e b e t w e e n t h e t w o fluids i s s i m p l y t h e z = 0 p l a n e . T h e q u e s t i o n

w h i c h w e c a n a s k c o n c e r n s t h e s t ab i l i t y of t h e e q u i l i b r i u m . If w e d i s t o r t t h e

i n t e r f a c e s l igh t ly , wi l l t h e d i s t o r t i o n s t e n d t o s m o o t h o u t o r s t a y s m a l l , o r wil l

t h e y g r o w a n d d i s r u p t t h e flow?

T o a n s w e r t h i s q u e s t i o n , le t u s a s s u m e t h a t t h e s u r f a c e is s l igh t ly d i s t o r t e d ,

a n d le t £ ( s e e F i g . 4 .11) m e a s u r e t h e d e v i a t i o n of t h e s u r f a c e f r o m i t s

e q u i l i b r i u m p o s i t i o n . W h e n t h e s u r f a c e is d i s t o r t e d , all of t h e o t h e r v a r i a b l e s

in t h e s y s t e m wil l c h a n g e b y a s m a l l a m o u n t a s w e l l , s o t h a t w e wi l l h a v e

P , = P,o + P'i,

P ' = P » + P ; ' (4 .C.4)

w h e r e P i a n d P 2 a r e t h e p r e s s u r e s in t h e r e g i o n s of fluids 1 a n d 2 . T h e

s u b s c r i p t " 0 " r e f e r s t o t h e e q u i l i b r i u m p r e s s u r e a n d t h e P ' t e r m s a r e t h e

Fig. 4.11. The deformation at the interface between two fluids.

60 Fluids in Motion

sma l l c h a n g e s in t h e e q u i l i b r i u m v a l u e c a u s e d b y t h e s m a l l d i s t o r t i o n of t h e

s u r f a c e . S i m i l a r l y , vx a n d v2 r e f e r t o t h e fluid v e l o c i t i e s , a l t h o u g h in t h i s c a s e

t h e e q u i l i b r i u m v e l o c i t y in r e g i o n 2 is z e r o .

I n g e n e r a l , all o f t h e s m a l l q u a n t i t i e s in E q . (4 .C .4) a r e c o m p l i c a t e d

f u n c t i o n s of t h e p o s i t i o n a n d t i m e . H o w e v e r , w e k n o w t h a t e a c h c a n b e

e x p a n d e d in a F o u r i e r s e r i e s , e a c h c o m p o n e n t of w h i c h h a s a b e h a v i o r

T h e r e f o r e , w i t h o u t l o s s of g e n e r a l i t y , w e c a n c o n s i d e r o n l y t h e c a s e

€(x, y, z, t) = f ( z ) e l ( t o — \

P'(x9y,z,t) = P'(z)eiikx-"\ (4 .C.5)

v ' (x , y, z, t) = V(z)eiikx'wt\

s i n c e a n y m o r e c o m p l i c a t e d f u n c t i o n s of x a n d t c a n b e e x p r e s s e d a s a s e r i e s

of t e r m s of t h i s t y p e . A s w e sha l l s e e , o u r final w o r k i n g e q u a t i o n s wi l l b e

l i nea r , s o if w e find a s o l u t i o n fo r t h e g e n e r a l t e r m in s u c h a s u m , t h e final

r e s u l t wil l s i m p l y b e a s u m of s u c h s o l u t i o n s ( s e e A p p e n d i c e s E a n d F ) .

T h e e q u a t i o n s of m o t i o n , a p p l i e d s e p a r a t e l y t o e a c h r e g i o n , a r e s i m p l y

V • v = 0 (4 .C.6)

a n d

V • v 0 = 0 ,

w e h a v e

V • v' = 0 . (4 .C.8)

S i m i l a r l y , t h e E u l e r e q u a t i o n is

(4 .C.9)

(4 .C.7)

W e sha l l s o l v e t h e s e e q u a t i o n s e x p l i c i t l y in r e g i o n 1, n o t i n g t h a t t h e

s o l u t i o n in r e g i o n 2 c a n b e o b t a i n e d f r o m t h i s b y l e t t i n g pi -> p 2 , Pi -> P2, a n d

s e t t i n g v0 = 0 . F o r n o t a t i o n a l s i m p l i c i t y , w e wi l l d r o p t h e s u b s c r i p t " 2 " w h i l e

s o l v i n g t h e e q u a t i o n , a n d wi l l r e i n t r o d u c e it a t t h e e n d of t h e s o l u t i o n .

T h e e q u a t i o n of c o n t i n u i t y b e c o m e s

V • v = V • v 0 4- V • v' = 0,

b u t s i n c e a t e q u i l i b r i u m ,


T h i s c a n b e c o n s i d e r a b l y s impl i f ied b y n o t i n g t h a t a t e q u i l i b r i u m

(4 .C .10)

s o t h a t

(4 .C .11)

T h e r e a r e t w o f u r t h e r s i m p l i f i c a t i o n s w h i c h c a n b e m a d e . F i r s t , w e n o t e t h a t vo is a c o n s t a n t , s o t h a t

S e c o n d l y , w e n o t e t h a t w e a r e d e a l i n g w i t h a s i t u a t i o n in w h i c h small

p e r t u r b a t i o n s t o e q u i l i b r i u m a r e b e i n g m a d e . T h e t e r m (v' • V)v ' in t h e a b o v e

e q u a t i o n is t h e r e f o r e of s e c o n d o r d e r in s m a l l n e s s , w h i l e al l of t h e o t h e r

t e r m s in t h e e q u a t i o n a r e of first o r d e r . T h u s , fo r s m a l l d e v i a t i o n s f r o m

e q u i l i b r i u m , w e c a n w r i t e

(v ' - V ) v , ~ 0

(4 .C .12)

t o g i v e

W e s e e t h a t t h e s m a l l p e r t u r b a t i o n a p p r o x i m a t i o n l e a v e s u s w i t h a l i n e a r

e q u a t i o n r e l a t i n g t h e v e l o c i t y a n d t h e p r e s s u r e , r a t h e r t h a n t h e o r i g i n a l

n o n l i n e a r o n e . Of c o u r s e , t h i s e q u a t i o n is m u c h e a s i e r t o s o l v e t h a n t h e

o r i g i n a l o n e . T h i s t e c h n i q u e , w h i c h w e h a v e u s e d h e r e in t h e c o n t e x t of a

h y d r o d y n a m i c s p r o b l e m , is c a l l e d l i n e a r i z a t i o n , a n d i s u s e d e x t e n s i v e l y

t h r o u g h o u t p h y s i c s .

If w e t a k e t h e d i v e r g e n c e of E q . ( 4 . C . 12) a n d u s e t h e c o n t i n u i t y c o n d i t i o n t h a t V • v ' = 0 , w e find a n e q u a t i o n f o r t h e s m a l l a d d i t i o n t o t h e e q u i l i b r i u m p r e s s u r e

V 2 P ' = 0 . (4 .C .13)

If w e s u b s t i t u t e t h e a s s u m e d f o r m of P ' f r o m E q . (4 .C .5) i n t o t h i s r e s u l t , w e find t h a t

(4 .C .14)

w h i c h m e a n s t h a t t h e m o s t g e n e r a l s o l u t i o n f o r P ' ( z ) is j u s t

P ' ( z ) = Ae~kz +Bek\ (4 .C .15)

62 Fluids in Motion

w h e r e A a n d B a r e u n d e t e r m i n e d c o n s t a n t s . A s in a n y d i f fe ren t i a l e q u a t i o n ,

t h e s e c o n s t a n t s m u s t b e d e t e r m i n e d b y t h e b o u n d a r y c o n d i t i o n s . O n e

b o u n d a r y c o n d i t i o n i s t h a t t h e p r e s s u r e m u s t s t a y finite, s o t h a t i n r e g i o n 1,

w e m u s t h a v e B = 0, a n d t h e p e r t u r b a t i o n o n t h e p r e s s u r e m u s t b e

P[ = A e ~ k V c k x _ f t > 0 . (4 .C .16)

S i m i l a r r e a s o n i n g in r e g i o n 2, w h e r e z is n e g a t i v e , g i v e s

P'2=Cekzeiikx-wt\ (4 .C.17)

w h e r e A a n d C a r e c o n s t a n t s sti l l t o b e d e t e r m i n e d .

I n o r d e r t o p r o c e e d f u r t h e r , it is n e c e s s a r y t o r e l a t e t h e p r e s s u r e t o t h e

d i s p l a c e m e n t of t h e s u r f a c e , £. W e b e g i n b y w r i t i n g d o w n t h e z - c o m p o n e n t

of t h e E u l e r e q u a t i o n [ E q . ( 4 . C . 12)] i n r e g i o n 1 ( a g a i n , d r o p p i n g t h e s u b s c r i p t

d u r i n g t h e d e r i v a t i o n ) , w h i c h , w i t h t h e a s s u m e d f o r m s fo r v' a n d Pf [ E q .

(4 .C.5) ] b e c o m e s

(4 .C.18)

T o r e l a t e t h i s t o t h e d i s p l a c e m e n t £, w e n o t e t h a t D £ IDt, t h e v e l o c i t y of t h e

s u r f a c e ( w h i c h is in t h e z - d i r e c t i o n , s i n c e £ is a v e c t o r in t h e z - d i r e c t i o n o n l y )

m u s t b e t h e s a m e a s v'z, t h e v e l o c i t y of a p a r t i c l e a t t h e s u r f a c e . I n o t h e r

w o r d s ,

(4 .C.19)

T a k i n g t h i s w i t h E q . (4 .C .18 ) , w e find t h a t t h e p r e s s u r e in r e g i o n 1 at the

surface m u s t b e j u s t

(4 .C .20)

A s imi l a r a r g u m e n t f o r r e g i o n 2 y i e l d s

(4 .C.21)

N o w a t t h e s u r f a c e , w e m u s t h a v e

P\ = P'2,

s o t h a t

— pi(a) — kvo)2 = co2p2,

(4 .C .22)

(4 .C .23)


w h i c h c a n b e s o l v e d f o r co t o g i v e

(4 .C .24) = a •+• ifi.

T h u s , t h e m o s t g e n e r a l f o r m of t h e t i m e d e p e n d e n c e of t h e q u a n t i t i e s P',

v', a n d § wi l l b e

s o t h a t f o r a n y v a l u e s of p i a n d p 2 e x c e p t t h e t r i v i a l c a s e w h e r e p i = 0 o r

p2 = 0 , a n y s m a l l p e r t u r b a t i o n of t h e s u r f a c e wi l l b e e x p e c t e d t o g r o w w i t h

t i m e a n d t h e s y s t e m wi l l b e u n s t a b l e .

T h u s , w e s e e t h a t it is i n d e e d p o s s i b l e t o d e t e r m i n e t h e s t a b i l i t y of a

s y s t e m d i r e c t l y f r o m t h e e q u a t i o n s of m o t i o n , s i m p l y b y a s s u m i n g s m a l l

t i m e - d e p e n d e n t d e v i a t i o n s f r o m e q u i l i b r i u m , a n d s e e i n g w h a t s o r t of t i m e

d e p e n d e n c e is i m p o s e d o n t h e s y s t e m b y t h e e q u a t i o n s a n d t h e b o u n d a r y

c o n d i t i o n s .

B e f o r e l e a v i n g t h i s t o p i c , t h e r e a r e a n u m b e r of p o i n t s w h i c h s h o u l d b e

e m p h a s i z e d . F i r s t , a s w a s d i s c u s s e d in C h a p t e r 3 , t h e q u e s t i o n of s t a b i l i t y

of a s y s t e m d e p e n d s o n t h e t y p e of a p p l i e d p e r t u r b a t i o n . I t is a l w a y s

p o s s i b l e f o r a s y s t e m t o b e s t a b l e a g a i n s t o n e t y p e of p e r t u r b a t i o n w h i l e

b e i n g u n s t a b l e a g a i n s t a n o t h e r .

S e c o n d , t h e f a c t t h a t w e h a v e s h o w n t h a t t h e t i m e d e p e n d e n c e of t h e

p e r t u r b a t i o n is e x p o n e n t i a l m a y a t first s igh t a p p e a r u n s e t t l i n g , s i n c e s u c h

a d e p e n d e n c e s e e m s t o i m p l y t h a t n o m a t t e r h o w s m a l l t h e in i t ia l

d e f l e c t i o n s of t h e s u r f a c e a r e , t h e d e v i a t i o n s f r o m e q u i l i b r i u m wil l

a p p r o a c h inf ini ty a f t e r a l o n g e n o u g h t i m e .

T h i s a c t u a l l y is n o t t h e c a s e , a s c a n b e s e e n b y e x a m i n i n g t h e v e l o c i t y

v'. If vo is t h e in i t ia l p e r t u r b a t i o n , t h e n a t a l a t e r t i m e , E q . (4 .C .24) w o u l d

g i v e

H o w e v e r , in o r d e r t o d e r i v e E q . (4 .C .24 ) , w e h a d t o m a k e t h e l i n e a r i z a t i o n

h y p o t h e s i s t o g e t E q . (4 .C .12 ) . C l e a r l y , f o r l a r g e t, t h i s a p p r o x i m a t i o n is

n o l o n g e r v a l i d , s o t h a t t h e e x p o n e n t i a l l y g r o w i n g s o l u t i o n wil l n o l o n g e r

b e v a l i d , e i t h e r .

T h e p o i n t is t h a t o u r l i n e a r i z e d e q u a t i o n s te l l u s h o w t h e s y s t e m

b e h a v e s in t i m e n e a r e q u i l i b r i u m , b u t o n c e t h e s y s t e m is f a r f r o m

e q u i l i b r i u m , w e h a v e t o g o b a c k t o t h e o r i g i n a l n o n l i n e a r e q u a t i o n s f o r a

s o l u t i o n . I n t e r m s of F i g . 4 .12 , o u r r e s u l t s te l l u s h o w t h e ba l l wi l l ro l l off

Deiatept +Eeiute'i (4 .C .25)

64 Fluids in Motion

t t = 0

Fig. 4.12. An illustration of a system which behaves differently near equilibrium than it does far from equilibrium.

of t h e hi l l , b u t o n c e w e g e t a w a y f r o m t h e hi l l , t h e s i t u a t i o n c h a n g e s , a n d

w e c a n n o t s a y t h a t t h e ba l l wi l l k e e p ro l l i ng f o r e v e r .

SUMMARY

T h e v e l o c i t y field is d e f i n e d . I t i s s h o w n t h a t v e l o c i t y fields w h i c h h a v e

n o c u r l c o r r e s p o n d t o fluid m o t i o n s in w h i c h n o r o t a t i o n is p r e s e n t , a n d

v e l o c i t y fields w i t h z e r o d i v e r g e n c e o r s y m m e t r i c p a r t c o r r e s p o n d t o

m o t i o n s in w h i c h t h e r e is n o c h a n g e in d e n s i t i e s .

T h e c o n c e p t of s t ab i l i t y of f low is i n t r o d u c e d , a n d t h e t e c h n i q u e of

e x a m i n i n g a fluid flow in e q u i l i b r i u m , i n t r o d u c i n g s m a l l , t i m e - d e p e n d e n t

p e r t u r b a t i o n s of e q u i l i b r i u m , a n d a p p l y i n g t h e e q u a t i o n s of m o t i o n t o t h e

p e r t u r b e d s y s t e m is d e v e l o p e d . I t is a r g u e d t h a t if t h e e q u a t i o n s i m p l y t h a t

a p e r t u r b a t i o n , o n c e i n t r o d u c e d , g r o w s w i t h t i m e , t h e n it i s u n s t a b l e . T h i s

t e c h n i q u e is a p p l i e d t o t h e t a n g e n t i a l flow i n s t a b i l i t y p r o b l e m .

PROBLEMS

4.1. Consider a container on the ear th which is filled to a height h with a fluid of density p, and has a small opening a dis tance z down from the top of the fluid, through which a fluid s t ream can emerge . Assuming irrotational flow, calculate the velocity of the s t ream just outside the en t rance (neglect the effect of the outflow on the height h).

4.2. Consider an imaginary surface 2 inside of a fluid, (a) Show that the total flow out through the surface is

where d/dn is the derivat ive normal to the surface.

Problems 65

(b) U s e (a) to show that cp cannot have a max imum or minimum anywhere inside of the fluid.

(c) H e n c e show that if t he fluid were of infinite extent , and <p w e r e not infinite anywhere , we would have to have

<p = const .

everywhere . This is a special case of Liouville Theorem of mathemat ica l analysis .

4 .3 . Carry out the pictorial analysis given in Section 4.A for the velocity field

vx = Cy 2 ,

Vy = CX2.

4.4. A streamline is defined to be a line which is eve rywhere tangent to the velocity of the fluid. It can b e pic tured easily by imagining a small needle inser ted into the moving fluid, and a thin s t ream of dye being emit ted from the needle . The dye will mark the fluid in a line which will have the proper ty of a s treamline.

Show from the Euler equat ion that , for general s teady-s ta te flow, the quant i ty

| p u 2 + P + pn

must be the same everywhere along a given streamline. F r o m your proof, does it follow that the cons tant in the above express ion must be the same for neighboring streamlines? (Hint: Wri te the Euler equat ion in the form of Eq . (1.B.5), and take the gradient of the equat ion in the direction of a streamline.)

4 .5. Consider a flow of fluid which is in the z -direction, and is axially symmetr ic , so that

vz = c(r),

Ve= Vr = 0,

where c(r) is an arbi trary function. Excep t for the case c = 0, show that it is not possible to define a velocity potential for such a flow.

4.6. W h y does a flag wave in the b reeze?

4.7. Le t us reconsider the rings of Saturn problem from the point of view of fluid stability. Consider the rings to be a flat sheet of th ickness 2c, centered on the x-y plane. Le t the densi ty of the fluid be p, and let the fluid exper ience a small per turbat ion such that each plane of the fluid which was level before the per turbat ion is n o w displaced by a dis tance TJ, where

7] = A cos mx.

(a) Show that the gravitational potential of the per turbed fluid is

V, = lirpcA sin mxe mc (emz + e~mz)

66 Fluids in Motion

inside the fluid and

V 2 = 2TTPCA sin mxe+mz(emc + e~mc) outside.

(b) Calculate the pressure in the fluid to be

P = 2irp(c2 - z 2 ) 4- lirpcA sin mx

x [2cm - 1 - e~2mc + e~mc(emz + e~mz)l

(c) H e n c e show that the sys tem is unstable if

Ac = — > 5 . 4 c .

4.8. Consider the two-dimensional flow of an incompressible fluid. Define a stream function if/ by the equat ions

(a) Show that such a definition automatically satisfies the equat ion of continuity. (b) Show that for irrotational flow, the equat ion for the s t ream function is

V V = 0.

(c) Show that the s t ream function is constant along any streamline.

4.9. W e can define a quanti ty called the circulation as

where the integral is unders tood to go over any closed path in the fluid. Show that if all of the forces acting on the fluid can be wri t ten as the gradient of a potential , that

i.e. that the circulation is conserved .

4.10. If we define a complex potential in te rms of the s t ream function and velocity potential as

(a) show that w is an analytic function. (b) H e n c e (or otherwise) show that the flow of fluid out of an aper ture extending

into the fluid in a large container (this is called Borda ' s mouthpiece) will contrac t half the width of the aper ture . (Hint: You will wan t to use complex variable techniques on this problem.)

4 .11 . Consider two planes meeting at an (acute) angle at the origin. Suppose an incompressible fluid is undergoing potential flow in the corner formed by these planes .

(a) Wri te down the boundary condit ions at the two planes .

w = <f> + iif/,

References 67

(b) Find the solution to the equat ions of motion and the boundary condit ions to lowest power in r, the radial coordinate .

(c) Calculate the velocities of the flow and sketch them out . (d) Find the streamlines by calculating the s t ream functions.

4.12. Show that the equat ion for the velocity potential for the two-dimensional potential flow of an incompressible fluid is

(a) Interpret the constant C in te rms of the p resence or absence of sources of fluid

in the sys tem.

(b) Show that if in this case we consider a two-dimensional electrical sys tem, and

make the ass ignments t;->crj, cp-*V,

where j is the current densi ty, a the conduct ivi ty and V is the voltage, we get equat ions which are identical to the hydrodynamic equat ion.

(c) H e n c e suggest an exper imental method for measur ing the flow of a fluid past

irregular obstacles .

4.13. Show that the s t ream function and velocity potential which are due to the

motion of a circular cylinder of radius a moving with velocity ( /para l le l t o the x-axis

are

(Hint: Consider a complex potential of the form w = A/Z . )

4.14. Consider now a sphere of radius a moving through a fluid with velocity v.

(a) Show that the velocity potential (assuming the fluid to be at rest far from the

sphere is ,

(b) Sketch the lines of flow around the sphere . (c) F r o m the Euler equat ion, calculate the pressure at the surface of the sphere . (d) Show that the equat ion of motion for the sphere in the fluid is just

where F is the external force, and a and p are the densit ies of the sphere and the fluid, respect ively. This says that in the absence of an external agent, there is no net force on the sphere . Does this seem reasonable (see Chapter 8)?

REFERENCES

All of the general texts cited in Chapter 1 contain discussions of the velocity potential. The author found the books by Ramsey and Lamb especially readable, and the discussion of tangential instabilities in the Landau and Lifschitz text particularly good.

5

Waves in Fluids

What dreadful noise of waters in mine ears!

WILLIAM SHAKESPEARE

King Richard III, Act I, Scene IV

A. LONG WAVES

O n e of t h e m o s t i m p o r t a n t a s p e c t s of t h e m o t i o n of f luids is t h e w i d e

v a r i e t y of w a v e s w h i c h c a n b e g e n e r a t e d a n d s u s t a i n e d in t h e m . I n t h i s

c h a p t e r , w e sha l l c o n s i d e r t h r e e s u c h w a v e m o t i o n s , b e g i n n i n g w i t h t h e

l o n g , o r t i d a l , w a v e s in t h i s s e c t i o n . T h e o r ig in of t h e n a m e " l o n g w a v e s "

wil l b e c o m e o b v i o u s l a t e r in t h e d i s c u s s i o n .

I n g e n e r a l , w e c a n t h i n k of w a v e m o t i o n a s t h e r e s u l t of t w o o p p o s i n g

f o r c e s a c t i n g o n a b o d y . C o n s i d e r a w e i g h t o n a s p r i n g , fo r e x a m p l e . If a

f o r c e is a p p l i e d w h i c h m o v e s t h e w e i g h t a w a y f r o m i t s e q u i l i b r i u m

p o s i t i o n , t h e w e i g h t wil l e x e r t a f o r c e w h i c h pu l l s t h e w e i g h t b a c k . If w e l e t

g o , t h e s p r i n g wil l r e t u r n t o i t s e q u i l i b r i u m p o s i t i o n , b u t w h e n it g e t s t h e r e , it

wi l l b e m o v i n g w i t h s o m e v e l o c i t y . T h u s it wil l o v e r s h o o t t h e e q u i l i b r i u m

p o s i t i o n , a n d m o v e o n un t i l t h e s p r i n g is c o m p r e s s e d e n o u g h t o c a u s e it t o

r e v e r s e i t s d i r e c t i o n . T h u s , t h e e x i s t e n c e of t h e r e s t o r i n g f o r c e in t h e s p r i n g

l e a d s t o t h e f a m i l i a r s i m p l e h a r m o n i c m o t i o n .

T h e s i t u a t i o n w i t h f luids i s q u i t e s imi l a r . L e t u s c o n s i d e r a b o d y of

u n i f o r m fluid w h o s e u n p e r t u r b e d h e i g h t is h ( s e e F i g . 5.1), b u t w h o s e

s u r f a c e is f o r s o m e r e a s o n p e r t u r b e d , s o t h a t t h e a c t u a l s u r f a c e is a t a h e i g h t

y s = h + r/.

68

Long Waves 69

L e t u s f u r t h e r m o r e s u p p o s e t h a t t h i s fluid is in a g r a v i t a t i o n a l field o n t h e

s u r f a c e of t h e e a r t h , s o t h a t t h e r e is a f o r c e p g p e r u n i t v o l u m e in t h e

y-direction. T h e n if 17 > 0 , t h e fluid e l e m e n t s in t h e s u r f a c e wi l l b e p u l l e d

d o w n w a r d b y g r a v i t y , w h i l e if 17 < 0 , t h e fluid p r e s s u r e wi l l t e n d t o e x e r t a n

u p w a r d f o r c e . T h u s , w e m i g h t e x p e c t t h a t w e w o u l d s e e h a r m o n i c m o t i o n

in t h i s s y s t e m .

T o m a k e t h e q u a n t i t a t i v e i d e a s i n t r o d u c e d in C h a p t e r 4 m o r e de f in i t e , w e

wil l a c t u a l l y w o r k o u t t h e p r o b l e m m e n t i o n e d a b o v e , w i t h o n e a d d i t i o n . L e t

u s c o n s i d e r w h a t h a p p e n s w h e n t h e r e is n o t o n l y a g r a v i t a t i o n a l f o r c e

a c t i n g o n t h e fluid, b u t a n a d d i t i o n a l f o r c e per unit volume F , w h o s e

c o m p o n e n t s ( s e e F i g . 5.1) a r e Fx a n d Fy. W e wil l n e e d t h e s e r e s u l t s in

C h a p t e r 6 w h e n w e d i s c u s s t h e t h e o r y of t h e t i d e s , in w h i c h c a s e t h e e x t r a

f o r c e w o u l d b e t h e g r a v i t a t i o n a l a t t r a c t i o n of t h e m o o n .

L e t u s c o n s i d e r a n in f in i t e s ima l v o l u m e e l e m e n t of fluid a t a h e i g h t y in

t h e fluid ( s e e F i g . 5.1). T h e y - c o m p o n e n t of t h e E u l e r e q u a t i o n is t h e n

T h e s e e q u a t i o n s a s t h e y s t a n d a r e p r e t t y c o m p l i c a t e d . T h e m o s t

i m p o r t a n t diff iculty is t h a t t h e y a r e n o n l i n e a r . T h a t i s , t h e y c o n t a i n t e r m s in

t h e c o n v e c t i v e d e r i v a t i v e w h i c h a r e p r o p o r t i o n a l t o b o t h v a n d v2. S u c h

e q u a t i o n s a r e v e r y difficult t o s o l v e , a n d t h e f a c t t h a t t h e E u l e r e q u a t i o n is

n o n l i n e a r is t h e m a i n r e a s o n t h a t a d v a n c e s in h y d r o d y n a m i c s a r e s o

difficult t o m a k e ( s e e P r o b l e m 5.2).

T o g e t a r o u n d t h i s p r o b l e m , w e a r e g o i n g t o h a v e t o a p p e a l t o s o m e of t h e

p h y s i c s in t h e p r o b l e m s w e a r e t r y i n g t o s o l v e . T h e q u a n t i t y v w h i c h

a p p e a r s in t h e E u l e r e q u a t i o n r e f e r s t o t h e m o t i o n of a v o l u m e e l e m e n t in a

(5 .A.1)

(5 .A.2)

w h i l e t h e x-component is

Fig. 5.1. The perturbed surface of a fluid.

70 Waves in Fluids

L i k e t h e r e a s o n i n g l e a d i n g t o E q . (5 .A .3 ) , t h i s a p p r o x i m a t i o n c a n b e m o s t

e a s i l y a n a l y z e d a f t e r w e h a v e s o l v e d t h e a p p r o x i m a t e e q u a t i o n s . P h y s i -

c a l l y , t h i s r e d u c e s E q . (5 .A.4) t o a h y d r o s t a t i c e q u a t i o n , a n d a m o u n t s t o

s a y i n g t h a t t h e m o t i o n in t h e y - d i r e c t i o n is s o s l o w t h a t w e c a n t a k e it t o

b e s u c h t h a t h y d r o s t a t i c e q u i l i b r i u m is m a i n t a i n e d a t all t i m e s a s f a r a s t h e

y - m o t i o n is c o n c e r n e d . T h i s is s o m e t i m e s c a l l e d a quasi-static a p p r o x i -

m a t i o n . W e sha l l s e e t h a t t h i s is a v a l i d a p p r o x i m a t i o n p r o v i d e d t h a t t h e

d e p t h of t h e fluid is m u c h l e s s t h a n t h e w a v e l e n g t h of t h e w a v e .

W i t h t h i s final a p p r o x i m a t i o n , t h e l e f t - h a n d s i d e of E q . (5 .A.4) v a n i s h e s ,

s o t h a t t h e e q u a t i o n c a n b e i n t e g r a t e d d i r e c t l y t o g i v e

P-Po = g P ( h + v - y ) , (5 .A.6)

fluid. N o w t h i s v e l o c i t y c a n b e q u i t e s m a l l , e v e n t h o u g h t h e v e l o c i t y of t h e

w a v e in t h e fluid m a y b e l a r g e . T h i s c a n b e s e e n b y t h i n k i n g a b o u t a w a v e

t r a v e l i n g a l o n g a r o p e . A n y g i v e n s e g m e n t of t h e r o p e m o v e s o n l y a s m a l l

a m o u n t u p a n d d o w n a s t h e w a v e g o e s b y , b u t t h e w a v e i tse l f m a y m o v e

v e r y q u i c k l y . W e a r e g o i n g t o a s s u m e t h a t a s i m i l a r s i t u a t i o n h o l d s in

d e a l i n g w i t h w a v e s in fluids, a n d w e wil l w r i t e

(5 .A.3)

in t h e E u l e r e q u a t i o n . T h i s c o r r e s p o n d s t o s a y i n g t h a t s i n c e v is s m a l l , w e

c a n d r o p t e r m s of o r d e r v2. I t is a n a p p r o x i m a t i o n w h i c h wil l b e m a d e m a n y

t i m e s in t h i s t e x t . W e wil l s e e e x a c t l y w h a t p h y s i c a l c o n d i t i o n is i m p l i e d b y

E q . (5 .A.3) l a t e r in t h i s s e c t i o n .

If w e a r e d e a l i n g w i t h a s y s t e m l ike t h e t i d e s , t h e n t h e t e r m s Fy in E q .

(5 .A.1) w h i c h r e p r e s e n t t h e a t t r a c t i o n of t h e m o o n wil l b e q u i t e s m a l l

c o m p a r e d t o t h e g r a v i t a t i o n a l f o r c e of t h e e a r t h , s o t h a t E q . (5 .A.1) wi l l b e

g i v e n b y

(5 .A.4)

N o w if w e c o n f i n e o u r a t t e n t i o n t o s y s t e m s l i ke t h e t i d e s , t h e r e is still

a n o t h e r a p p r o x i m a t i o n w h i c h w e c a n m a k e o n t h i s e q u a t i o n . If w e t h i n k

of t h e t i d e s , w e r e a l i z e t h a t t h e fluid wi l l m o v e , t y p i c a l l y , a d i s t a n c e of

s e v e r a l y a r d s in t h e y - d i r e c t i o n o v e r a c o u r s e of m a n y h o u r s . T h u s , t h e

v e l o c i t y in t h e y - d i r e c t i o n is q u i t e s m a l l , a n d w e c a n e x p e c t t h e r a t e of

c h a n g e of t h a t v e l o c i t y t o b e e v e n s m a l l e r . T h e r e f o r e , it m a k e s s e n s e t o

se t

(5 .A.5)

Long Waves 71

w h e r e P 0 is t h e p r e s s u r e of t h e m e d i u m a b o v e t h e fluid. I n m o s t c a s e s , t h i s

wil l j u s t b e t h e a t m o s p h e r i c p r e s s u r e .

I n E q . ( 5 .A .6 ) , w e h a v e a r e a d y i n c o r p o r a t e d o n e b o u n d a r y c o n d i t i o n ,

w h i c h is t h a t in t h i s c a s e t h e p r e s s u r e m u s t b e a c o n s t a n t a t y = h + T J .

T h i s s h o u l d b e f a m i l i a r f r o m t h e d i s c u s s i o n of s t e l l a r s t r u c t u r e in C h a p t e r

2 .

W e c a n d i f f e r e n t i a t e E q . (5 .A.6) w i t h r e s p e c t t o JC t o g e t

(5 .A.7)

T h e l e f t - h a n d s i d e of t h i s e x p r e s s i o n is p r e c i s e l y w h a t a p p e a r s o n t h e

r i g h t - h a n d s i d e of E q . (5 .A .2 ) , s o t h a t w e c a n e l i m i n a t e t h e p r e s s u r e

b e t w e e n t h e s e t w o e q u a t i o n s t o g e t

(5 .A.8)

T h i s e q u a t i o n still c o n t a i n s t w o u n k n o w n s , vx a n d 17. W e c a n e l i m i n a t e

o n e of t h e m b y r e c o u r s e t o t h e r e m a i n i n g c o n d i t i o n w h i c h w e c a n a p p l y t o

f luids in g e n e r a l , t h e c o n d i t i o n of c o n t i n u i t y . W e c o u l d , of c o u r s e , s i m p l y

w r i t e it d o w n a s in E q . (1 .C .4 ) . H o w e v e r , b e c a u s e w e w a n t i n f o r m a t i o n

a b o u t t h e v a r i a b l e 17, w e wil l find it e a s i e r t o g o t h r o u g h t h e d e r i v a t i o n of

t h e e q u a t i o n f o r t h e p a r t i c u l a r g e o m e t r y in F i g . 5 . 1 .

C o n s i d e r a w a v e m o v i n g b y a p o i n t x ( s e e F i g . 5.2), a n d c o n s i d e r t w o

p l a n e s a d i s t a n c e dx a p a r t . T h e m a s s of fluid c o n t a i n e d b e t w e e n t h e

p l a n e s p e r u n i t l e n g t h in t h e z - d i r e c t i o n is j u s t (h +17 ) p dx s o t h a t t h e t i m e

r a t e of c h a n g e of m a s s in t h e v o l u m e is g i v e n b y

(5 .A.9)

x ' x x + dx

Fig. 5.2. The idea of continuity and the perturbed surface.

72 Waves in Fluids

H o w c a n t h e m a s s c h a n g e ? If w e a r e d e a l i n g w i t h a n i n c o m p r e s s i b l e

fluid, t h e o n l y w a y t h e a m o u n t of m a s s c a n c h a n g e is f o r s o m e fluid t o flow

o u t a c r o s s t h e p l a n e s . T h i s , in t u r n , wil l c a u s e t h e l e v e l of fluid,

r e p r e s e n t e d b y 17, t o d r o p .

T h e a m o u n t of fluid flowing a c r o s s t h e l e f t - h a n d p l a n e is

[h + r](x)]pvx(x) « hpvx(x),

w h e r e w e h a v e d r o p p e d t h e t e r m rjvx a s b e i n g s e c o n d o r d e r in s m a l l

p a r a m e t e r s . T h e a m o u n t flowing a c r o s s t h e r i g h t - h a n d p l a n e is s im i l a r l y

w h e r e w e h a v e d r o p p e d h i g h e r - o r d e r t e r m s in t h e T a y l o r s e r i e s e x p a n s i o n

of vx. T h u s , t h e n e t in f low o r o u t f l o w is t h e d i f f e r e n c e b e t w e e n t h e s e t w o

q u a n t i t i e s , a n d m u s t b e t h e r a t e of c h a n g e of m a s s in E q . (5 .A .4 ) . E q u a t i n g

t h e s e q u a n t i t i e s g i v e s

(5 .A.10)

f o r t h e e q u a t i o n of c o n t i n u i t y f o r t h e i n c o m p r e s s i b l e fluid in t e r m s of vx

a n d TJ.

If w e d i f f e r e n t i a t e E q . (5 .A . 10) w i t h r e s p e c t t o t a n d E q . (5 .A.8) w i t h

r e s p e c t t o x, w e c a n e l i m i n a t e vx f r o m o u r e q u a t i o n s , a n d g e t

(5 .A.11)

I n t h e c a s e w h e r e t h e r e is n o f o r c e e x c e p t t h e e a r t h ' s g r a v i t a t i o n a l field,

t h i s b e c o m e s

(5 . A . 12)

w h i c h is s i m p l y t h e w a v e e q u a t i o n fo r a w a v e w h o s e v e l o c i t y is

(5 .A.13) c = vgh.

T h e e q u a t i o n h a s f o r i t s s o l u t i o n a n y f u n c t i o n of t h e t r a v e l i n g w a v e f o r m ,

s o t h a t

V(x,t) = f(x-ct\ (5 .A.14)

w h e r e / is a n y w a v e s h a p e .

T h u s , w e s e e t h a t t h e E u l e r e q u a t i o n a n d t h e e q u a t i o n of c o n t i n u i t y l e a d

d i r e c t l y t o a w a v e e q u a t i o n f o r t h e d e v i a t i o n of t h e s u r f a c e of a fluid f r o m

Long Waves 73

w h e r e A is t h e w a v e l e n g t h of t h e w a v e . T h u s , w e wi l l h a v e

p r o v i d e d t h a t

T,max<Â. ( 5 . A . 18)

I n o t h e r w o r d s , w h e n e v e r t h e w a v e l e n g t h of t h e w a v e is l o n g c o m p a r e d

t o t y p i c a l d i s t a n c e s w h i c h p a r t i c l e s in t h e fluid m o v e w h i l e t h e w a v e g o e s

b y , w e c a n d r o p t h e t e r m in (v • V)v. S i n c e t h i s c o n d i t i o n is e a s i l y m e t b y

m o s t w a v e s , w e sha l l n o t r e f e r t o t h i s a p p r o x i m a t i o n a g a i n , b u t w e wi l l

u s e it t h r o u g h o u t t h e r e m a i n d e r of t h e d i s c u s s i o n .

T h e s e c o n d i m p o r t a n t a p p r o x i m a t i o n w a s s t a t e d in E q . ( 5 .A .5 ) , w h e r e

w e a s s u m e d t h a t t h e y - e q u a t i o n c o u l d b e t r e a t e d in t h e q u a s i - s t a t i c l imi t .

T o e x a m i n e t h i s a p p r o x i m a t i o n , w e n o t e t h a t h a d w e n o t u s e d E q . ( 5 . A . 5 ) ,

w e w o u l d h a v e t o r e p l a c e E q . (5 .A.6) b y

P - P 0 = gp(h+r1-y) + p I (5 . A . 19)

(5 . A . 17)

s i n c e t h e v e l o c i t y g o e s f r o m z e r o t o v in t i m e t.

B y a s i m i l a r a r g u m e n t , w e w o u l d h a v e t y p i c a l l y

(5 . A . 16)

I t is r e a s o n a b l e t o s u p p o s e t h a t vx wi l l b e of t h i s o r d e r of m a g n i t u d e a s

w e l l . T h e n w e e x p e c t t h a t t h e first t e r m in t h e c o n v e c t i v e d e r i v a t i v e wi l l

b e r o u g h l y

(5 .A .15)

i t s flat e q u i l i b r i u m c o n f i g u r a t i o n w h e n t h a t fluid is u n d e r t h e i n f l u e n c e of

i t s o w n p r e s s u r e a n d g r a v i t y .

T h e n e x t q u e s t i o n w h i c h w e m u s t e x a m i n e is t h e v a l i d i t y of t h e

a p p r o x i m a t i o n s w h i c h l ed u s t o t h i s r e s u l t . L e t u s b e g i n w i t h E q . ( 5 .A .3 ) ,

w h i c h a l l o w e d u s t o d r o p t h e n o n l i n e a r t e r m s in t h e E u l e r e q u a t i o n . I s t h i s

a p p r o x i m a t i o n r e a l l y v a l i d ?

T o e x a m i n e t h i s q u e s t i o n , c o n s i d e r a w a v e g o i n g b y a g i v e n p o i n t in t h e

fluid. L e t r b e t h e t i m e it t a k e s a w a v e t o g o p a s t t h e p o i n t , a n d l e t T / m a x b e

t h e m a x i m u m h e i g h t a b o v e h w h i c h t h e s u r f a c e a t t a i n s . T h e n a t y p i c a l

v e l o c i t y f o r a p a r t i c l e a t t h e s u r f a c e w o u l d b e

74 Waves in Fluids

w h e r e t h e s e c o n d i n e q u a l i t y f o l l o w s f r o m t h e f a c t t h a t t h e e x p r e s s i o n

(h +17 - y ) h a s i t s l a r g e s t p o s s i b l e v a l u e a t y = 0.

O n t h e o t h e r h a n d , t h e first t e r m o n t h e l e f t - h a n d s i d e of E q . (5 .A . 19)

wi l l h a v e i t s m i n i m u m n e a r t h e s u r f a c e , a n d i t s m i n i m u m v a l u e wi l l b e of

o r d e r gprj . T h u s , w e c a n a l w a y s d r o p t h e c o r r e c t i o n t e r m p r o v i d e d t h a t

s i n c e in t h a t c a s e , t h e m a x i m u m c o r r e c t i o n is l e s s t h a n t h e m i n i m u m of

t h e t e r m t o w h i c h it is b e i n g c o m p a r e d .

W e c a n n o w p r o c e e d u s i n g t h e s a m e t y p e of a r g u m e n t s t h a t w e r e u s e d

b e f o r e . If t h e t y p i c a l a c c e l e r a t i o n is

w h e r e , b y de f in i t ion , T = A / c , t h e n E q . 5 .A .20 b e c o m e s

w h i c h , u s i n g E q . (5 .A . 13), w e c a n finally w r i t e a s

T h u s , t h e q u a s i - s t a t i c a p p r o x i m a t i o n is v a l i d p r o v i d e d t h a t t h e

w a v e l e n g t h of t h e w a v e s in q u e s t i o n a r e m u c h g r e a t e r t h a n t h e d e p t h of

t h e fluid. T h e r e a r e m a n y e x a m p l e s of s u c h c a s e s ( s o m e of w h i c h a r e

g i v e n in t h e p r o b l e m s a t t h e e n d of t h e c h a p t e r ) . F o r e x a m p l e , if w e w e r e

d e a l i n g w i t h t i d e s , t h i s w o u l d c l e a r l y b e a va l i d a p p r o x i m a t i o n , s i n c e t h e

l e n g t h of t h e t i da l b u l g e is o n t h e o r d e r of t h e c i r c u m f e r e n c e of t h e e a r t h ,

w h i l e t h e d e p t h of t h e o c e a n is o n l y a f e w k i l o m e t e r s . A n o t h e r e x a m p l e

w o u l d b e w a v e s a p p r o a c h i n g a b e a c h , s i n c e a t s o m e p o i n t t h e d e p t h of t h e

fluid wil l b e c o m e s m a l l e n o u g h t o s a t i s fy E q . (5 .A .21 ) .

B. SURFACE WAVES IN FLUIDS

I n t h e p r e v i o u s s e c t i o n , w e s a w t h a t if w e m a d e a s e r i e s of a p p r o x i m a -

t i o n s o n t h e E u l e r e q u a t i o n s a n d t h e e q u a t i o n s of c o n t i n u i t y , w e c o u l d

d e r i v e a w a v e e q u a t i o n f o r n , t h e d i s p l a c e m e n t of t h e s u r f a c e f r o m

(5 .A.20)

A >h. (5 .A.21)

w h e r e t h e l a s t t e r m r e p r e s e n t s t h e ef fec t of vy. N o w if w e d e n o t e t h e

m a x i m u m a c c e l e r a t i o n of a p a r t i c l e in t h e y - d i r e c t i o n b y /3, t h e n w e h a v e

Surface Waves in Fluids 75

e q u i l i b r i u m . T h e m o s t i m p o r t a n t a s s u m p t i o n w a s t h e l o n g - w a v e a p p r o x i -

m a t i o n , w h i c h , in t h e f o r m of E q . (5 .A .3 ) a l l o w s u s t o n e g l e c t t h e v e r t i c a l

m o t i o n of t h e fluid e l e m e n t s . T h i s a s s u m p t i o n is v a l i d in m a n y c a s e s of

i n t e r e s t , b u t it i s c l e a r t h a t t h e r e a r e m a n y c a s e s w h e r e it i s n o t .

If w e w e r e t o c o n s i d e r w a v e s o n t h e o c e a n o r a l a k e , f o r e x a m p l e , t h e

l o n g - w a v e l e n g t h a p p r o x i m a t i o n w o u l d n o t a p p l y . S u c h w a v e s t y p i c a l l y

h a v e w a v e l e n g t h s of t h e o r d e r of t e n s o r h u n d r e d s of f e e t , w h i c h is m u c h

l e s s t h a n t h e d e p t h of t h e w a t e r . T h i s m e a n s t h a t w e wil l h a v e t o g o

t h r o u g h t h e d e r i v a t i o n w i t h o u t t h e bene f i t of E q . (5 .A .3 ) .

F o r p r o b l e m s of t h i s t y p e , it is v e r y c o n v e n i e n t t o u s e t h e v e l o c i t y

p o t e n t i a l d e f i n e d in S e c t i o n 4 . B . L e t u s a s s u m e t h a t w e a r e d e a l i n g w i t h

i r r o t a t i o n a l flow of a n i n c o m p r e s s i b l e fluid, s o t h a t t h e e q u a t i o n f o r t h e

p o t e n t i a l is

V 2 < / > = 0 . (5 .B.1)

T h e u s e of t h i s e q u a t i o n a l r e a d y i n c o r p o r a t e s t h e e q u a t i o n of c o n -

t i n u i t y , s o t h a t t h e o n l y o t h e r e q u a t i o n w h i c h w e n e e d t o w r i t e d o w n is t h e

E u l e r e q u a t i o n . I n t e r m s of t h e v e l o c i t y p o t e n t i a l , t h i s is g i v e n in E q .

(4 .B .5 ) . If w e m a k e t h e u s u a l a s s u m p t i o n t h a t w e c a n d r o p s e c o n d - o r d e r

t e r m s in t h e v e l o c i t y , t h i s is j u s t

- f c 7 ( y ) = 0, (5 .B.4)

(5 .B.2)

H o w e v e r , t h e r o l e w h i c h t h i s e q u a t i o n wil l n o w p l a y is s o m e w h a t

d i f f e r en t t h a n in t h e p r e v i o u s s e c t i o n . T h e r e , w e c o m b i n e d t h e E u l e r

e q u a t i o n a n d t h e e q u a t i o n of c o n t i n u i t y t o d i s p l a y t h e w a v e e q u a t i o n

e x p l i c i t l y . I n t h i s s e c t i o n , w e sha l l p r o c e e d b y assuming t h a t <f>

h a s a w a v e - l i k e s o l u t i o n , a n d v e r i f y t h a t t h i s is i n d e e d t h e c a s e b y d i r e c t

s u b s t i t u t i o n i n t o t h e a b o v e t w o e q u a t i o n s . W e sha l l s e e t h a t in t h i s c a s e ,

t h e E u l e r e q u a t i o n e n t e r s o n l y in t h a t it d e t e r m i n e s t h e b o u n d a r y

c o n d i t i o n s a t t h e fluid s u r f a c e .

L e t u s , t h e n , g u e s s t h a t it is p o s s i b l e t o find s o l u t i o n s of E q s . (5 .B .1) a n d

(5 .B.2) of t h e f o r m

4> = f(y) c o s (kx - cot), (5 .B.3)

w h e r e / ( y ) is s o m e f u n c t i o n t o b e d e t e r m i n e d . P u t t i n g t h i s i n t o t h e

L a p l a c e e q u a t i o n g i v e s

76 Waves in Fluids

(5 .B.8)

(5 .B.9)

w h i c h m e a n s t h a t t h e m o s t g e n e r a l f o r m of / ( y ) is

f(y) = Aeky + Be~k\ (5 .B.5)

T o p r o c e e d f u r t h e r , w e n e e d t o i m p o s e t h e b o u n d a r y c o n d i t i o n s . A t t h e

b o t t o m of t h e fluid, a t y = 0, w e k n o w t h a t vy — d<frl dy — 0 , s i n c e , b y

de f in i t i on , n o fluid c a n c r o s s t h e b o t t o m b o u n d a r y . T h i s m e a n s t h a t

s o A = - B , a n d

/ ( y ) = 2 A c o s h ( f c y ) . (5 .B.6)

T h e s e c o n d b o u n d a r y c o n d i t i o n is j u s t a s s i m p l e p h y s i c a l l y , b u t

s o m e w h a t m o r e difficult m a t h e m a t i c a l l y . I t s t a t e s t h a t a t t h e s u r f a c e of

t h e fluid, t h e p r e s s u r e m u s t b e e q u a l t o P 0 , t h e a t m o s p h e r i c p r e s s u r e ,

w h i c h w e t a k e t o b e a c o n s t a n t . T o s t a t e t h i s c o n d i t i o n a t t h e p e r t u r b e d

s u r f a c e , w e sha l l h a v e t o m a k e u s e of t h e E u l e r e q u a t i o n in t h e f o r m

(5 .B .2 ) .

A t t h e s u r f a c e of t h e e a r t h , w e c a n t a k e

ft = gy 4 - c o n s t . , (5 .B.7)

s o t h a t a t y = h +17, E q . (5 .B.2) b e c o m e s

+ gh + gr] = c o n s t

N o w t h e s e c o n d t e r m o n t h e lef t c a n b e e x p a n d e d

a n d if w e d r o p all b u t t h e first t e r m in t h e e x p a n s i o n a s b e i n g of s e c o n d

o r d e r in s m a l l q u a n t i t i e s , a n d a t t h e s a m e t i m e de f ine t h e c o n s t a n t in E q .

(5 .B.7) a p p r o p r i a t e l y , w e find

o r , d i f f e r e n t i a t i n g

N o w if w e c o n s i d e r a v o l u m e e l e m e n t j u s t a t t h e s u r f a c e of t h e fluid, it

h a s a v e l o c i t y g i v e n b y vy = d(pldy)y=h+^ B u t t h e v o l u m e e l e m e n t a t t h e

s u r f a c e m u s t b e m o v i n g w i t h j u s t t h e v e l o c i t y i tself , w h i c h is 4-drj/dt.

Surface Waves in Fluids 77

T h u s , w e h a v e a t t h e s u r f a c e

o r

(5 .B .10)

(5 .B .11)

(5 .B .16)

T h u s , in t h e l o n g w a v e l e n g t h l imi t , w e r e c o v e r t h e l o n g - w a v e r e s u l t

w h i c h w e d e r i v e d in t h e l a s t s e c t i o n . H o w e v e r , w h e n t h e d e p t h of t h e fluid

w h e r e , a s in t h e d e r i v a t i o n of E q . (5 .B .8 ) , w e h a v e r e p l a c e d all q u a n t i t i e s

w h i c h a r e t o b e e v a l u a t e d a t t h e s u r f a c e y = h + TJ b y q u a n t i t i e s e v a l u a t e d

a t t h e s u r f a c e of e q u i l i b r i u m y = h.

E q u a t i o n (5 .B .11 ) , t h e n , is t h e b o u n d a r y c o n d i t i o n a t t h e u p p e r s u r f a c e

( t h e a n a l o g u e of vy(0) = 0 , t h e b o u n d a r y c o n d i t i o n a t t h e l o w e r s u r f a c e )

w h i c h o u r a s s u m e d s o l u t i o n h a s t o s a t i s f y . I n s e r t i n g t h e s o l u t i o n in E q s .

(5 .B.8) a n d (5 .B .6 ) , w e find t h a t w e c a n s a t i s fy E q . (5 .B .11) p r o v i d e d t h a t

ay2 = gk t a n h ( k h ) . (5 .B .12)

T h u s , t h e g e n e r a l s o l u t i o n f o r t h e v e l o c i t y p o t e n t i a l i s j u s t

<t> =2A c o s h (ky) c o s (kx - cot) (5 .B.13)

a n d , s i n c e t h e s o l u t i o n s t o L a p l a c e ' s e q u a t i o n a r e u n i q u e , t h i s is t h e o n l y

s o l u t i o n . T o find t h e s u r f a c e d i s p l a c e m e n t , w e u s e t h e b o u n d a r y c o n d i t i o n

E q . (5 .B .8) t o g e t

(5 .B .14) c o s h (ky) s in (kx - cot)

w h i c h d e s c r i b e s a w a v e t r a v e l i n g in t h e J C - d i r e c t i o n , a s w e e x p e c t e d . T h e

v e l o c i t y is g i v e n b y

(5 .B .15)

w h e r e w e h a v e w r i t t e n k = 2rr/\.

R e c a l l i n g t h a t

x > 1

x < 1

A >h

t a n h x = 1 .x

c

w e find

78 Waves in Fluids

is c o m p a r a b l e t o o r s h o r t e r t h a n t h e w a v e l e n g t h , w e find t h a t t h e v e l o c i t y

d e p e n d s o n t h e w a v e l e n g t h i tself , w h i c h is a r e s u l t w h i c h w e h a v e n o t

e n c o u n t e r e d b e f o r e .

A q u e s t i o n w h i c h w e m i g h t we l l a s k a t t h i s s t a g e is w h y t h e r e l a t i o n of

t h e d e p t h t o t h e w a v e l e n g t h of t h e w a v e s h o u l d b e i m p o r t a n t . T o a n s w e r

t h i s q u e s t i o n , le t u s c a l c u l a t e t h e v e l o c i t i e s of v o l u m e e l e m e n t s in t h e fluid

a t s o m e d e p t h y [ th i s v e l o c i t y is n o t t o b e c o n f u s e d w i t h t h e v e l o c i t y of

t h e w a v e , w h i c h is g i v e n b y E q . (5 .B .15 ) ] . F r o m t h e de f in i t ion of t h e

v e l o c i t y p o t e n t i a l ,

vy = 2kA s i n h (ky) c o s (kx — cot), (5 .B .17)

vx = — 2kA c o s h (ky) s in (kx - cot),

a t a n a r b i t r a r y p o i n t in t h e fluid. T h u s , e a c h p a r t i c l e is s e e n t o d e s c r i b e a n

e l l i p se in t h e x-y p l a n e , w i t h t h e a x i s in t h e y - d i r e c t i o n b e i n g p r o p o r t i o n a l

t o s i n h ky, a n d t h e a x i s in t h e x - d i r e c t i o n t o c o s h ky ( s e e P r o b l e m 5.3).

T h e r e a r e s e v e r a l c o n c l u s i o n s w h i c h c a n b e d r a w n f r o m t h i s . F i r s t , a t

y = 0, t h e v e r t i c a l m o v e m e n t v a n i s h e s ( th i s w a s t o b e e x p e c t e d , s i n c e it

w a s o u r first b o u n d a r y c o n d i t i o n ) . M o r e i m p o r t a n t , w e s e e t h a t t h e

d i s t u r b a n c e a s s o c i a t e d w i t h t h e w a v e fa l ls off l ike a h y p e r b o l i c f u n c t i o n

a s w e g o b e l o w t h e s u r f a c e , a n d t h e l e n g t h a s s o c i a t e d w i t h t h i s fall off is

l/k, o r A / 2 7 T . T h u s , t h e d i s t u r b a n c e is c o n f i n e d t o s o m e t h i n g l ike a

d i s t a n c e of o n e w a v e l e n g t h f r o m t h e s u r f a c e . T h i s is t h e o r ig in of t h e

n a m e " s u r f a c e w a v e " a n d of t h e d e p e n d e n c e of t h e s o l u t i o n of t h e

e q u a t i o n s of fluid m e c h a n i c s o n t h e r e l a t i o n b e t w e e n d e p t h a n d

w a v e l e n g t h . O n e c o u l d s a y t h a t t h e e x i s t e n c e of a w a v e r e q u i r e s t h e

c o o p e r a t i o n of t h e fluid a t t h e s u r f a c e t o a d e p t h a b o u t e q u a l t o t h e

w a v e l e n g t h of t h e w a v e . I n t h e l o n g w a v e l e n g t h l imi t , t h i s m e a n s t h a t w e

m u s t h a v e t h e e n t i r e fluid i n v o l v e d in t h e w a v e .

I n S e c t i o n 12 .E , w e sha l l s e e t h a t t h i s s u r f a c e w a v e p h e n o m e n o n is n o t

u n i q u e t o fluids, b u t e x i s t s in so l i d s a s w e l l .

C. SURFACE TENSION AND CAPILLARY WAVES

U p t o t h i s p o i n t , w e h a v e c o n s i d e r e d o n l y p r e s s u r e a n d e x t e r n a l f o r c e s

a c t i n g o n p a r t i c l e s of t h e fluid. W h i l e t h i s m a y b e a p e r f e c t l y a d e q u a t e

d e s c r i p t i o n in t h e i n t e r i o r of t h e fluid, it is w e l l k n o w n t h a t t h e r e a r e

f o r c e s o n t h e s u r f a c e of a fluid w h i c h t e n d t o o p p o s e a n y i n c r e a s e in

s u r f a c e a r e a — a n y " s t r e t c h i n g " of t h e s u r f a c e . T h i s f o r c e is u s u a l l y c a l l e d

t h e " s u r f a c e t e n s i o n , " T, a n d is de f i ned b y t h e w o r k n e c e s s a r y t o i n c r e a s e

Surface Tension and Capillary Waves 79

Fig. 5.3. Molecular forces and surface tension.

t h e a r e a of a s u r f a c e b y a n a m o u n t dS b y t h e r e l a t i o n

dW=TdS. (5 .C.1)

T h e s i m p l e s t w a y t o p i c t u r e t h e r e a s o n f o r t h i s f o r c e is t o n o t e ( s e e F i g .

5.3) t h a t t h e r e a r e u s u a l l y a t t r a c t i v e ( c o h e s i v e ) f o r c e s o n t h e m o l e c u l a r

l e v e l in a fluid w h i c h t e n d s t o m a k e it s t a y t o g e t h e r . F o r a m o l e c u l e in t h e

i n t e r i o r , t h e s e f o r c e s a r e e x e r t e d in all d i r e c t i o n s , a n d t h e r e f o r e c a n c e l o u t

o n t h e a v e r a g e . F o r a m o l e c u l e o n t h e s u r f a c e , h o w e v e r , t h e s e f o r c e s a r e

all d i r e c t e d i n w a r d t o w a r d t h e b o d y of t h e f luid, a n d t h e r e is a n e t i n w a r d

f o r c e . I n c r e a s i n g t h e s u r f a c e a r e a c o r r e s p o n d s t o p u t t i n g m o r e p a r t i c l e s

i n t o t h e s u r f a c e , a n d h e n c e w o r k m u s t b e d o n e a g a i n s t t h e a t t r a c t i v e

f o r c e s , g i v i n g r i s e t o r e l a t i o n (5 .C .1) a b o v e . W e s h o u l d n o t e t h a t in t e r m s

of t h i s p i c t u r e , t h e e x i s t e n c e of s u r f a c e t e n s i o n is s t r i c t l y a g e o m e t r i c a l

e f f e c t — i t a r i s e s b e c a u s e a s u r f a c e , b y de f in i t i on , d i v i d e s a r e g i o n filled

w i t h fluid f r o m a r e g i o n e m p t y of t h e fluid. T h u s , w h e t h e r t h e f o r c e is

m o l e c u l a r in o r i g i n ( a s in t h e s e c t i o n ) o r is a c o n s e q u e n c e of n u c l e a r

i n t e r a c t i o n s ( a s i s t h e c a s e of t h e l i qu id d r o p m o d e l of t h e n u c l e u s w h i c h

w e sha l l d i s c u s s l a t e r ) wi l l n o t a f fec t t h e e x i s t e n c e of a s u r f a c e f o r c e .

I n o r d e r t o q u a n t i f y t h e a b o v e r e m a r k s o n s u r f a c e t e n s i o n , l e t u s

e x a m i n e t h e f o l l o w i n g p r o b l e m : A s u r f a c e finds i t se l f w i t h p r e s s u r e Pi o n

o n e s i d e a n d P2 o n t h e o t h e r . T h e i m b a l a n c e of p r e s s u r e s c a u s e s t h e

s u r f a c e t o e x p a n d . I n t h e p r o c e s s , w o r k m u s t b e d o n e a g a i n s t T, t h e

s u r f a c e t e n s i o n . C o n s i d e r t h e e l e m e n t t o h a v e u n p e r t u r b e d l e n g t h s 811 a n d

8l2 a n d r a d i i of c u r v a t u r e Ri a n d R2 ( s e e F i g . 5.4) a n d le t t h e l e n g t h s of t h e

s i d e s a f t e r s t r e t c h i n g b e g i v e n b y 8U(1 + a) a n d 6 / 2 ( l + j3). T h e n , s i n c e

w e h a v e

W=TdA = TdUdl2(a+p).

80 Waves in Fluids

dx1 d/i(1 +a)

Fig. 5.4. The displacement of a surface by pressure differentials.

O n t h e o t h e r h a n d , t h e w o r k d o n e b y t h e p r e s s u r e in d i s p l a c i n g t h e

s u r f a c e a d i s t a n c e dx is j u s t

T h u s , w e find t h a t t h e s u r f a c e f o r c e is q u i t e l a r g e w h e n t h e s u r f a c e is

s h a r p l y c u r v e d . T h i s n e w f o r c e i n t r o d u c e s a r a t h e r d i f f e ren t p r o b l e m a t

t h e s u r f a c e . U p t o t h i s p o i n t , w e h a v e a l w a y s u s e d t h e c o n d i t i o n t h a t a

s u r f a c e w a s c h a r a c t e r i z e d b y a c o n s t a n t v a l u e of t h e p r e s s u r e . B u t t h e

e x i s t e n c e of a f o r c e in t h e s u r f a c e w h i c h c o u l d b a l a n c e a f o r c e d u e t o a n

i m b a l a n c e in p r e s s u r e m e a n s t h a t w e m u s t b e m o r e c a r e f u l . E q u a t i o n

(5 .C.2) n o w te l l s u s t h a t t h e c o n d i t i o n a t t h e s u r f a c e is n o l o n g e r t h a t P is

c o n s t a n t , b u t t h a t p r e s s u r e d i f f e r e n c e s a r e r e l a t e d t o t h e c u r v a t u r e of t h e

s u r f a c e , a n d t h a t c h a n g e s in c u r v a t u r e a l o n g t h e s u r f a c e wi l l n e c e s s i t a t e

c h a n g e s in t h e p r e s s u r e d i f f e r e n c e a c r o s s it .

T u r n i n g o u r a t t e n t i o n n o w t o t h e e f fec t of s u r f a c e t e n s i o n o n t h e t y p e of

w a v e s w e h a v e b e e n d i s c u s s i n g , le t u s c o n s i d e r t h e s i t u a t i o n s h o w n in F i g .

5 .5 , w h e r e t w o semi - in f in i t e fluids of d e n s i t i e s p a n d p ' h a v e a n i n t e r f a c e

a t t h e p l a n e y = 0. If w e le t <f> a n d <f>' r e p r e s e n t t h e v e l o c i t y p o t e n t i a l s in

t h e t w o fluids, t h e n , a s b e f o r e , t h e b a s i c e q u a t i o n s g o v e r n i n g t h e

( P 2 - P 0 dhdhdx, s o t h a t

(5 .C.2)

P', </>'

y = 0

Fig. 5.5. The perturbed interface between two fluids.

Surface Tension and Capillary Waves 81

p o t e n t i a l s a r e

V > = 0 ,

a n d , f o l l o w i n g t h e s t e p s in t h e p r e v i o u s s e c t i o n , w e find

4>' = (C'eky + Ce~ky) c o s (kx - cot),

<f>=(Deky + D ' * T k y ) c o s (kx - cot).

(5 .C .3)

(5 .C.4)

A s u s u a l , w e wi l l d e t e r m i n e t h e c o n s t a n t s C a n d C \ D a n d D' f r o m t h e

b o u n d a r y c o n d i t i o n s . F r o m t h e r e q u i r e m e n t t h a t t h e v e l o c i t i e s s t a y finite

a t y = ± °°, w e find

C ' = D ' = 0. (5 .C.5)

If, a s b e f o r e , w e d e n o t e b y 17 t h e d e v i a t i o n of t h e s u r f a c e f r o m

e q u i l i b r i u m , a n d w e a s s u m e , f o l l o w i n g t h e p r o c e d u r e of t h e p r e v i o u s

s e c t i o n , t h a t

17 = A s in (kx - cot), (5 .C.6)

t h e n t h e c o n d i t i o n t h a t a n e l e m e n t in t h e s u r f a c e m o v e a t t h e s a m e

v e l o c i t y a s t h e s u r f a c e i t se l f g i v e s

(5 .C .8)

W i t h t h e s e s o l u t i o n s f o r t h e v e l o c i t y p o t e n t i a l s , w e c a n n o w s o l v e t h e

E u l e r e q u a t i o n f o r t h e p r e s s u r e o n e a c h s i d e of t h e s u r f a c e . W e find

(5.C.1

w h i c h y i e l d s

(5 .C.9) )A s in (kx — cot)

J A s in (kx — cot). (5 .C .10)

a n d

B y E q . (5 .C .2 ) , t h i s is s u p p o s e d t o b e r e l a t e d t o t h e s u r f a c e t e n s i o n a n d

t h e c u r v a t u r e of t h e s u r f a c e . F r o m F i g . 5 .5 , w e s e e t h a t t h e r a d i u s of

c u r v a t u r e in t h e z-direction is j u s t

(5 .C .11)

82 Waves in Fluids

w h i c h is p r e c i s e l y t h e r e s u l t f o r s u r f a c e w a v e s in a fluid of inf in i te d e p t h

[ s ee E q . (5 .B .16 ) ] . T h i s g i v e s u s s o m e c o n f i d e n c e t h a t o u r r e s u l t s a r e

c o r r e c t , s i n c e o u r i n t u i t i o n t e l l s u s t h a t t h e p r o b l e m w e a r e w o r k i n g in t h i s

s e c t i o n s h o u l d r e d u c e t o t h e p r o b l e m of t h e p r e v i o u s s e c t i o n in t h i s l imi t .

A r e l a t e d c o n s e q u e n c e c o m e s if w e n o t e t h a t fo r v e r y l a r g e

w a v e l e n g t h s , t h e s e c o n d t e r m in E q . (5 .C .15) wil l b e c o m e u n i m p o r t a n t ,

a n d t h e w a v e wil l l o o k l ike a n o r d i n a r y s u r f a c e w a v e , r e g a r d l e s s of t h e

p r e s e n c e of s u r f a c e t e n s i o n . O n t h e o t h e r h a n d , a t v e r y s m a l l

w a v e l e n g t h s , t h e s e c o n d t e r m wil l d o m i n a t e c o m p l e t e l y , a n d w e wil l h a v e

A p i c t o r i a l w a y of r e p r e s e n t i n g t h i s is t o p l o t c2 v e r s u s w a v e l e n g t h ( s e e

(5 .C.17)

F i g . 5.6).

s i n c e b y h y p o t h e s i s , n o t h i n g d e p e n d s o n t h e z - c o o r d i n a t e . F r o m P r o b l e m

5.7, o r f r o m e l e m e n t a r y c a l c u l u s , w e k n o w t h a t t h e o t h e r r a d i u s i s g i v e n

b y

(5 .C.12)

w h e r e t h e s e c o n d a p p r o x i m a t e e q u a l i t y is t r u e f o r s m a l l d e f o r m a t i o n s of

t h e s u r f a c e . S u b s t i t u t i n g E q s . (5 .C.9) a n d (5 .C .10) i n t o E q . (5 .C.2) g i v e s ,

a f t e r s o m e c a n c e l l a t i o n , t h e c o n d i t i o n t h a t

(5 .C.13)

If w e r e c a l l t h a t t h e v e l o c i t y of t h e w a v e , a s o p p o s e d t o v e l o c i t y of t h e

fluid p a r t i c l e s , is g i v e n b v

(5 .C .14)

w e s e e t h a t

(5 .C.15)

w h e r e t h e s e c o n d e q u a l i t y f o l l o w s f r o m t h e de f in i t ion k = lirlk. T h e r e a r e a n u m b e r of i n t e r e s t i n g c o n s e q u e n c e s of t h i s r e s u l t . W e s e e

t h a t if w e t a k e t h e l imi t

p ' = T = 0. w e g e t

(5 .C .16)

Problems 83

c 2

A

Fig. 5.6. A plot of velocity versus wavelength. The small wavelength part corresponds to capillary waves, and the long wavelength part to surface waves.

I n r e g i o n 2, w e h a v e t h e o r d i n a r y s u r f a c e w a v e s d i s c u s s e d in t h e

p r e v i o u s s e c t i o n . F o r s u c h w a v e s , t h e e x i s t e n c e of s u r f a c e t e n s i o n is

l a r g e l y i r r e l e v a n t . I n r e g i o n 1, w e h a v e a n e w t y p e of w a v e , w h o s e

e x i s t e n c e is a d i r e c t c o n s e q u e n c e of t h e e x i s t e n c e of s u r f a c e f o r c e s . T h i s

t y p e of w a v e is g e n e r a l l y c a l l e d a capillary wave, o r ripple.

T h e r e a d e r h a s p r o b a b l y a l r e a d y o b s e r v e d c a p i l l a r y w a v e s in n a t u r e .

W h e n a w i n d is b l o w i n g o n a l a k e o r t h e o c e a n , o n e o f t e n s e e s t h e u s u a l

l a r g e w a v e s , b u t w i t h s m a l l ruffles s u p e r i m p o s e d o n t h e m . T h e ruffles a r e ,

in f a c t , c a p i l l a r y w a v e s w h i c h a r e c a u s e d b y t h e w i n d ( s e e P r o b l e m 5.16) .

W e wil l s e e o t h e r e x a m p l e s of s u r f a c e t e n s i o n e f f ec t s in C h a p t e r 8, w h e n

w e d i s c u s s n u c l e a r f i ss ion , a n d in C h a p t e r 14, w h e n w e d i s c u s s s o m e

a p p l i c a t i o n s t o m e d i c i n e .

SUMMARY

W e h a v e s e e n t h a t a s a c o n s e q u e n c e of t h e E u l e r e q u a t i o n a n d t h e

e q u a t i o n of c o n t i n u i t y t h a t t h e r e a r e a w i d e v a r i e t y of w a v e s p o s s i b l e in

f lu ids . T h e s e i n c l u d e l o n g w a v e s , in w h i c h t h e v e r t i c a l m o t i o n of t h e fluid

c a n b e i g n o r e d s o l o n g a s t h e d e p t h of t h e fluid is m u c h l e s s t h a n t h e

w a v e l e n g t h ; s u r f a c e w a v e s , in w h i c h t h e d i s t u r b a n c e of t h e w a v e

d i m i n i s h e s w i t h d e p t h in t h e fluid, a n d c a p i l l a r y w a v e s , w h i c h d e p e n d o n

t h e e x i s t e n c e of s u r f a c e t e n s i o n , a n d a r e t y p i c a l l y of s h o r t w a v e l e n g t h .

T h i s d o e s n o t e x h a u s t t h e n u m b e r of p o s s i b l e w a v e s in f lu ids , b u t

r e p r e s e n t s t h e t y p e s of w a v e s m o s t c o m m o n l y e n c o u n t e r e d in p h y s i c a l

s i t u a t i o n s .

PROBLEMS

5 . 1 . Show that there are wavelike disturbances (for long waves) possible on a canal of rectangular cross section and uniform depth, where the frequency of the wave is

84 Waves in Fluids

(a) Show that if / , and f2 are solutions to this equat ion,

/ = / l + / 2

is not necessari ly a solution. (b) Show that if f^x) is a solution,

/ = C/ 1 (x) ,

where C is a constant , is not necessari ly a solution. (c) Could we solve such an equat ion by solving for one Four ier component , and

then adding componen t s together?

5.3. Given the equat ions for the velocity of fluid e lements of a surface wave in Eq . (5.B.17), show that the motion descr ibed by a fluid e lement is indeed the ellipse descr ibed in the text .

5.4. In many par ts of this text , we shall use the incompressible fluid approxima-tion. That is, we shall wri te the equat ion of continuity as

V - v = 0.

The physical reason for this is based on the fact that the volume of most fluids is relatively insensit ive to changes in pressure . Convince yourself that this is t rue by looking at several different fluids, including water .

5.5. (a) Show that in the case of a canal in which the breadth b and depth h vary along the length of the canal , the equat ion for long waves becomes

TJ = A cos at,

W e can make a simple model of a tidal inlet, or river es tuary , as a sys tem in which the depth varies uniformly from h0 at the ocean to zero at a dis tance a from the ocean, and whose breadth varies from bo to zero over the same range. Show that if the elevation at the ocean is given by

where / is the length and h the depth of the canal , and n is an integer. An oscillation of this type on a surface, which can be excited by ear thquakes , for example , is called a seiche, and is similar to the phenomenon of water sloshing around in a ba th tub .

Using a reasonable approximat ion scheme, calculate the period of a seiche in (1) Lake Geneva , Switzerland, and (2) Lake Eyre , Australia.

5.2. In Section 5.A, we discussed nonlinear equat ions briefly. T o see why such equat ions are difficult to solve, consider the equat ion

Problems 85

5.10. An important type of wave which can propagate in a fluid is the sound wave. Unlike the waves considered in the text , these waves can exist only in compress i -ble fluids.

(a) Assume that the densi ty of a fluid is given by

p = p 0 ( l + s ) ,

where s < 1. Show that if the velocities are small, the equat ion of continuity is

the elevation in the es tuary is given by

where x is the coordinate measuring dis tance from the ocean, and

(b) Show that if the breadth is constant , but the depth varies as above , the

elevation varies as

17 = AJQ(2V/CX) cos at.

(c) Consider the sloping bo t tom of a beach as the canal of variable depth in this problem. Suppose that 10-foot b reakers are coming in off the ocean at the ra te of one every 10 seconds . What would the slope of the beach have to be so that (1) the long-wave solution is valid, and (2) the surf near the beach is at least three feet high? 5.6. In Sect ions 5.B and 5.C, we assumed a form for the velocity potential and the surface d is turbance of the form cos (kx - cot) or sin (kx - cot). Another commonly used form would be e "*"" 0 . Show that the final results in E q s . (5.B.15) and (5.C.15) are unchanged if we use this exponent ia l form.

5.7. Show that the radius of curva ture of a curve y(x) is given by

5.8. Consider the case of the type shown in Fig. 5.5, in which the upper region is a vacuum, so that p ' = 0, while the surface tension of the lower fluid is T. Suppose also that the depth of the lower fluid is h. Der ive an express ion for the velocity of the wave in this case , and show that it r educes to Eq . (5.C. 15) in the limit T -* 0.

5.9. Equat ion (5.B.10) can be derived in another way . Consider an e lement of surface dS which will move an amount AT} in the vertical direction in t ime At. F r o m the conservat ion of mass , show that

86 Waves in Fluids

(b) F r o m the Euler equat ion, show that

where c2 = dp /dp ) 0 . (c) Show that a wave of the type derived in part (b) is, in fact, a wave in which

the densi ty of the fluid is changing periodically with t ime. H e n c e show that c must be the velocity of sound in the fluid.

5.11. Show that the only sound waves that can exist in a closed tube of length L are those for which the displacement of the particles at a point x is given by

Discuss the construct ion of an organ pipe.

5.12. Suppose that there are two media, separated by the plane x = 0. Suppose further that the velocity potential , densi ty, and velocity of sound in the first medium are cp\, pi, and c, with similar definitions for the second medium.

(a) Wri te the equat ions governing cp and s in each medium, and the boundary condit ions which can be expec ted to hold at the interface (see Prob lem 5.10).

(b) Suppose that a plane wave of f requency co is incident at an angle 0 to the normal from the upper medium. If 0i is the angle of the refracted wave , show that

(c) If A, A ' , and Ai are the ampli tudes of the incident, reflected, and refracted waves at the interface, show that

(d) When will there be no reflected wave?

5.13. Wri te down the equat ion which governs the propagat ion of a sound wave in

a spherically symmetr ic uniform medium (see Prob lem 5.10). (a) Show that the equat ions for <f> and s yield, at large r, a wave for which

v = cs. Show that this same equat ion holds t rue for plane waves . (b) Show that if a source at r = 0 causes a velocity potential which varies as

l n lot

<p ~ e , that the velocity potential for an outgoing wave will have the form

(c) Show that mean work done by the source is

References 87

5.14. Consider a fluid of densi ty p and surface tension T in a box of depth h with a flexible bo t tom. Suppose that the bo t tom is manipulated so that its d isplacement from a plane is given by

7} = A cos (cot - kx).

Show that the surface of the fluid will be given by

y = A' cos ((ot — kx),

where

REFERENCES

All of the general texts cited in Chapter 1 contain discussions of fluid waves. In particular, the text by Lamb, in Chapters 8, 9, and 10, contains a large number of physically interesting examples of wave motion, including the ship's wake and tidal waves.

5.15. You have probably had the exper ience of walking somewhere with a cupful of coffee and have observed the standing waves which can b e set u p in such a system.

(a) If the cup is of circular cross section, radius a, and of depth h, show that the general standing wave on it is of the form

r] ~ AJn(kr) cos nO cos at.

(b) Determine the values of k which satisfy the boundary condi t ions. (c) Dete rmine the frequencyôf oscillation of the w a v e s , given the k n o w n

velocity of long waves c = Vgh. (d) H o w would you prevent the coffee from spilling over?

5.16. Re t race the deve lopment in Section 5.C for the case in which the upper medium is moving with velocity U with respect to the lower medium.

(a) Show the Eq . (5.C.15) is now replaced by

(b) This is clearly a model for waves generated by the wind. Can we ever get a situation in which waves t ravel against the wind? In terpre t this resul t .

(c) What is the value of U for which the per turbat ion at the surface will be unstable? Show that for water , U ^ 6.5 m/sec will cause the waves to be blown into spindrift.

5.17. Show that a fluid in space (with no gravitational field around) will form itself into a sphere . H e n c e comment on the prospec ts of manufactur ing ball bearings in satteli tes.

The Theory of the Tides

A ring from his finger he hastily drew Saying, "Take it, dearest Nellie, that your heart may be true. For the good ship stands waiting for the next flowing tide And if ever I return again, I will make you my bride."

Traditional English Ballad

A. THE TIDAL FORCES

T h e t i d e s h a v e a l w a y s p l a y e d a n i m p o r t a n t r o l e in h u m a n af fa i r s . I n t h e

l a s t c h a p t e r , w e s h o w e d t h a t t h e e q u a t i o n s g o v e r n i n g t h e m o t i o n of f luids

a d m i t w a v e l i k e s o l u t i o n s , b u t w e d i d n o t a d d r e s s o u r s e l v e s t o t h e q u e s t i o n

of h o w s u c h m o t i o n s m i g h t b e g e n e r a t e d . I n t h i s c h a p t e r , w e wil l l o o k a t

o n e t y p e of w a v e — t h e l o n g w a v e — a n d s h o w h o w t h e w a v e s a r e

g e n e r a t e d a n d h o w t h e y m i g h t b e e x p e c t e d t o b e h a v e in s o m e s i m p l e

m o d e l s of t h e o c e a n s .

I t i s g e n e r a l l y k n o w n t h a t t h e t i d e s a r e c a u s e d b y t h e e f f ec t s of t h e

m o o n ' s g r a v i t a t i o n a l a t t r a c t i o n o n t h e w a t e r in t h e o c e a n s . L e t u s b e g i n

o u r c o n s i d e r a t i o n of t h e t h e o r y of t h e t i d e s b y w o r k i n g o u t a n a p p r o x i -

m a t e e x p r e s s i o n f o r t h e p o t e n t i a l w h i c h d e s c r i b e s t h i s a t t r a c t i o n . C o n -

s i d e r t h e g e o m e t r y s h o w n in F i g . 6 . 1 . T h e g r a v i t a t i o n a l p o t e n t i a l a t t h e

p o i n t P d u e t o t h e m o o n is j u s t

88

6

w h e r e r is t h e d i s t a n c e f r o m t h e c e n t e r of t h e e a r t h t o P . H o w e v e r , t h i s is

n o t t h e p o t e n t i a l w h i c h w e w o u l d h a v e t o u s e if w e w i s h t o c a l c u l a t e t h e

t i d e s . T h e r e a s o n f o r t h i s is t h a t in a d d i t i o n t o e x e r t i n g a f o r c e o n t h e w a t e r

a t t h e e a r t h ' s s u r f a c e , t h e m o o n a l s o a c c e l e r a t e s t h e e a r t h a s a w h o l e . I t is

(6 .A.1)

Tides at the Equator 89

Mr

Fig. 6.1. The configuration of the earth and the moon.

w h e r e w e h a v e le t r = a in t h e final s t e p , a n d t h u s r e s t r i c t e d o u r a t t e n t i o n

t o t h e s u r f a c e of t h e e a r t h . W e h a v e a l s o s e t t h e z e r o of H D a t - G M / D . I t

is t h i s p o t e n t i a l w h o s e d e r i v a t i v e s a r e t h e " e x t r a " f o r c e s w h i c h w e r e

i n t r o d u c e d in E q s . (5 .A.1) a n d (5 .A .2 ) . I n f a c t , w e h a v e

B. TIDES AT THE EQUATOR

A s a first e x a m p l e of a t h e o r y of t h e t i d e s , l e t u s c o n s i d e r a c a s e in

w h i c h t h e g e o m e t r y is a s s i m p l e a s p o s s i b l e , s o t h a t w e c a n s e e t h e

p h y s i c s of t h e s i t u a t i o n c l e a r l y . L e t u s c o n s i d e r a n o b s e r v e r a t t h e

D

(6 .A.5)

o n l y t h e n e t a c c e l e r a t i o n , of c o u r s e , w h i c h w o u l d b e m e a s u r e d b y a n

o b s e r v e r a t t h e s u r f a c e of t h e e a r t h . T h e a c c e l e r a t i o n of t h e c e n t e r of t h e

e a r t h b e c a u s e of t h e p r e s e n c e of t h e m o o n is

(6 .A.2)

w h e r e w e h a v e w r i t t e n t h e f o r c e a s t h e d e r i v a t i v e of a f u n c t i o n , w h i c h w e

c a n n o w r e g a r d a s a p o t e n t i a l , t h a t t a k e s i n t o a c c o u n t t h e m o t i o n of t h e

e a r t h , a n d x i s a u n i t v e c t o r in t h e x - d i r e c t i o n . T h u s , t h e n e t g r a v i t a t i o n a l

p o t e n t i a l a t P—the n e t p o t e n t i a l w h i c h wil l a c t u a l l y b e fe l t b y t h e

w a t e r — i s j u s t

(6 .A.3)

W e h a v e w r i t t e n t h i s a s H D , t h e d i s t u r b i n g p o t e n t i a l , t o d i s t i n g u i s h it f r o m

f l M , t h e p o t e n t i a l a t P d u e t o t h e m o o n . N o w in p r a c t i c e , w e k n o w t h a t

r/D ^ 1 , s o w e c a n e x p a n d H D t o l o w e s t o r d e r in r/D t o g e t

(6 .A.4)

90 The Theory of the Tides

e q u a t o r , a n d le t u s a s s u m e t h a t t h e m o o n l ies d i r e c t l y a b o v e t h e e q u a t o r a t

all t i m e s . L e t u s f u r t h e r m o r e n e g l e c t t h e d y n a m i c a l e f f ec t s of t h e e a r t h ' s

r o t a t i o n ( i .e . n e g l e c t c e n t r i f u g a l a n d C o r i o l i s f o r c e s ) , a n d l e t t h e o n l y

ef fec t of t h i s r o t a t i o n b e a n a p p a r e n t m o v e m e n t of t h e m o o n ( a s s e e n b y

o u r o b s e r v e r ) a r o u n d t h e e a r t h o n c e e a c h d a y . W e wil l a l s o a s s u m e t h a t

t h e e a r t h is a u n i f o r m s p h e r e c o v e r e d w i t h a n o c e a n of u n i f o r m d e p t h , a n d

i g n o r e t h e p r e s e n c e of l a n d m a s s e s .

I n t h i s c a s e , t h e a n g l e S w h i c h a p p e a r s in E q . ( 6 . A . 4 ) — t h e a n g l e

b e t w e e n t h e v e c t o r t o t h e p o i n t P a t w h i c h t h e t i d e s a r e b e i n g m e a s u r e d

a n d t h e v e c t o r t o t h e m o o n — w i l l l ie in t h e p l a n e of t h e e q u a t o r . T h i s

g r e a t l y s impl i f ies t h e g e o m e t r y , s i n c e t h e a n g l e ® n o w c o r r e s p o n d s t o t h e

a n g l e of l o n g i t u d e a t t h e e q u a t o r ( s e e F i g . 6.2) . T h e c o m p l i c a t i o n s w h i c h

a r i s e w i t h t h e m o r e g e n e r a l c a s e wil l b e d i s c u s s e d in t h e n e x t s e c t i o n s . W e

sha l l s e e , in f a c t , t h a t t h e m a i n m a t h e m a t i c a l c o m p l i c a t i o n s w h i c h a p p e a r

in t h e L a p l a c e t h e o r y of t h e t i d e s h a v e t o d o w i t h t h e f a c t t h a t t h e a n g l e ©

b e t w e e n t h e r a d i u s t o t h e p o i n t of o b s e r v a t i o n a n d t h e r a d i u s t o t h e m o o n

is n o t , in g e n e r a l , s o e a s i l y e x p r e s s i b l e in t e r m s of o t h e r a n g l e s in t h e

p r o b l e m .

I n d e r i v i n g t h e l o n g - w a v e e q u a t i o n , E q . (5 .A .11 ) , w e u s e d C a r t e s i a n

c o o r d i n a t e s . F o r a n o b s e r v e r o n t h e s u r f a c e of t h e e a r t h , t h e a p p a r e n t

v e r t i c a l a n d h o r i z o n t a l w o u l d b e t h e x- a n d y - a x e s s h o w n in F i g . 6 .2 .

S i n c e it is o n l y t h e x-component of t h e e x t r a f o r c e w h i c h e n t e r s E q .

(5 .A . 11), w e h a v e

w h e r e w e h a v e u s e d t h e g e o m e t r i c a l i d e n t i t y dx = ad<f> a n d s e t 0 = <j>. If w e

i n s e r t E q . (6 .B.1) i n t o E q . (5 .A .11 ) , w e find

(6 .B .1)

(6 .B .2)

y

D

Fig. 6.2. The coordinates for the discussion of long waves.

Tides at the Equator 91

F r o m t h e t h e o r y of i n h o m o g e n e o u s d i f f e ren t i a l e q u a t i o n s ( s e e A p p e n -

d i x E ) , w e k n o w t h a t t h e m o s t g e n e r a l s o l u t i o n of E q . (6 .B .2) c a n b e

w r i t t e n

7] = 7]H + TJP, (6 .B.3)

w h e r e r)h r e p r e s e n t s t h e s o l u t i o n t o t h e e q u a t i o n w i t h Fx = 0 ( t h e

h o m o g e n e o u s s o l u t i o n ) a n d TJP r e p r e s e n t s t h e p a r t i c u l a r s o l u t i o n f o r t h e

e q u a t i o n w i t h t h e f o r c i n g t e r m .

I n w h a t f o l l o w s w e sha l l n o t i n c l u d e t h e t e r m TJH in o u r s o l u t i o n s ,

b u t l o o k o n l y f o r t h e p a r t i c u l a r s o l u t i o n s t o t h e e q u a t i o n s . T h e r e a s o n

f o r t h i s l i es w i t h o u r p h y s i c a l i n t u i t i o n , a n d n o t w i t h t h e m a t h e m a t i c s .

W e k n o w t h a t w e h a v e i g n o r e d p r o c e s s e s ( s u c h a s f r i c t i o n a n d v i s -

c o s i t y ) b y w h i c h a r e a l fluid wi l l l o s e e n e r g y . W e k n o w , t h e r e f o r e , t h a t

a d i s t u r b a n c e in t h e fluid wil l t e n d t o d i e o u t u n l e s s s o m e o u t s i d e a g e n c y is

p r e s e n t w h i c h a d d s e n e r g y c o n t i n u o u s l y t o t h e s y s t e m . I n t h e c a s e w e a r e

c o n s i d e r i n g , t h i s o u t s i d e a g e n c y i s , of c o u r s e , t h e m o o n . T h u s , w e k n o w

t h a t t h e o n l y l o n g - t e r m d i s t u r b a n c e s w h i c h wil l b e p r e s e n t in t h e o c e a n s

wil l b e t h o s e r e p r e s e n t e d b y T J p , w h i l e t h e d i s t u r b a n c e s r e p r e s e n t e d b y f)h

will t e n d t o d i e o u t w i t h t i m e . I t s h o u l d b e n o t e d t h a t t h i s s a m e s o r t of

t r e a t m e n t of l o n g - a n d s h o r t - t e r m e f f ec t s i s o f t e n e n c o u n t e r e d in e l e c t r i c a l

c i r c u i t s , w h e r e t h e h o m o g e n e o u s s o l u t i o n s a r e c u s t o m a r i l y r e f e r r e d t o a s

t r a n s i e n t s , a n d t h e p a r t i c u l a r s o l u t i o n s a r e r e f e r r e d t o a s s t e a d y - s t a t e

s o l u t i o n s .

If w e le t co b e t h e f r e q u e n c y of t h e m o o n a b o u t t h e e a r t h a s o b s e r v e d

f r o m t h e p o i n t P , t h e n

</> = cot, (6 .B.4)

w h e r e w e s e t t h e z e r o of t i m e w h e n t h e m o o n is d i r e c t l y o v e r t h e p o i n t P .

co, of c o u r s e , s h o u l d c o r r e s p o n d t o a p e r i o d of 24 h o u r s . E q u a t i o n (6 .B.4)

m e a n s t h a t w e c a n e l i m i n a t e t h e v a r i a b l e cp f r o m E q . (6 .B.2) a n d g e t

(6 .B.5)

w h i c h is e a s i l y s o l v e d t o g i v e

(6 .B.6)

T h e r e a r e t w o i m p o r t a n t f e a t u r e s of t h i s s o l u t i o n of t h e t i d a l e q u a t i o n s

a t t h e e q u a t o r w h i c h w e s h o u l d n o t e . F i r s t , w e o b s e r v e t h a t t h e w a t e r

l eve l a t a p a r t i c u l a r p o i n t wil l r e a c h i t s m a x i m u m v a l u e t w i c e a d a y — e v e n

t h o u g h t h e m o o n t r a v e r s e s i t s p a t h o n l y o n c e in t h e s a m e p e r i o d of t i m e .


T h i s f e a t u r e of t h e t i d e s — t h a t t h e y a r e s e m i - d i u r n a l — w i l l r e a p p e a r w h e n

w e d i s c u s s t h e L a p l a c e t h e o r y l a t e r .

P e r h a p s m o r e i n t e r e s t i n g is t h e f a c t t h a t if w e l o o k a t t = 0, t h e t i m e

w h e n t h e m o o n is d i r e c t l y o v e r h e a d , TJ will a t t a i n e i t h e r i t s m a x i m u m o r

m i n i m u m v a l u e , d e p e n d i n g o n t h e s ign of gh—eo2a2. W e r eca l l t h a t

c 2 = gh is t h e v e l o c i t y of a l o n g w a v e , a n d w e s e e t h a t 00a is t h e v e l o c i t y

of t h e m o o n ' s " s h a d o w " o n t h e e a r t h . If w e t a k e t h e e a r t h t o h a v e a r a d i u s

of 4 0 0 0 k m , a n d t h e a v e r a g e d e p t h of t h e o c e a n a s 4 k m , w e e a s i l y s e e t h a t

c2<oj2a2, (6 .B.7)

s o t h a t , in f a c t , t h e t i d e is i n v e r t e d — i . e . , w e h a v e a l o w t i d e w h e n t h e m o o n

is d i r e c t l y o v e r h e a d . T h e r e a s o n fo r t h i s is s i m p l y t h e f a c t t h a t a s t h e

m o o n g o e s a r o u n d , it a t t r a c t s t h e w a t e r t o w a r d it , f o r m i n g a t i da l b u l g e o n

t h e e a r t h . T h i s t ida l b u l g e , h o w e v e r , c a n n o t k e e p u p w i t h t h e m o o n , a n d

l a g s b e h i n d . O u r c a l c u l a t i o n s g i v e a l ag of 180°, s o t h a t l o w t i d e o c c u r s

w h e n t h e m o o n is d i r e c t l y o v e r h e a d .

T h u s , f o r a n o c e a n of u n i f o r m d e p t h a n d a m o o n c o n s t r a i n e d t o o r b i t

e x a c t l y o v e r t h e e q u a t o r , t h e e q u a t o r i a l t i d e s w o u l d b e s e m i - d i u r n a l a n d

i n v e r t e d . I n f a c t , w e k n o w t h a t t h e m a j o r t i d e s a r e s e m i - d i u r n a l , a l t h o u g h

t h e p r e s e n c e of v a r i a b l e d e p t h in t h e o c e a n a n d l a n d m a s s e s c o m p l i c a t e s

t h e c a l c u l a t i o n of r e a l t i d e s c o n s i d e r a b l y . B u t t h e m a i n f e a t u r e s of t h e

d i s c u s s i o n in t h i s s e c t i o n , w h i c h i n v o l v e t h e e f fec t of t h e l u n a r d i s t u r b i n g

p o t e n t i a l o n t h e l o n g w a v e s in t h e o c e a n , wil l c a r r y t h r o u g h in t h e m o r e

c o m p l i c a t e d c a l c u l a t i o n s d o n e in l a t e r s e c t i o n s .

I t s h o u l d b e p o i n t e d o u t t h a t a l t h o u g h w e h a v e a l w a y s r e f e r r e d t o

" l u n a r f o r c e s , " in p o i n t of f a c t e v e r y b o d y c a p a b l e of e x e r t i n g a g r a v i t a -

t i o n a l a t t r a c t i o n a t t h e e a r t h ' s s u r f a c e is c a p a b l e of c a u s i n g a t i d e , a n d , in

f a c t , s o l a r t i d e s a r e e a s i l y s e e n . T h i s is t r e a t e d in m o r e d e t a i l in P r o b l e m

6 . 1 .

C. THE EQUATIONS OF MOTION WITH ROTATION

I n t h e t r e a t m e n t of t h e e q u a t o r i a l t i d e s in t h e p r e v i o u s s e c t i o n , t w o

i m p o r t a n t a s p e c t s of t h e p r o b l e m of t i d e s h a v e b e e n i g n o r e d . O n e of

t h e s e , t h e c o m p l i c a t e d d e p e n d e n c e of t h e a n g l e © o n t h e c o o r d i n a t e s of

t h e p r o b l e m , wil l b e t r e a t e d in t h e n e x t s e c t i o n . T h e o t h e r i m p o r t a n t

e f fec t s w h i c h w e m u s t c o n s i d e r a r e t h e d y n a m i c a l c o n s e q u e n c e s of t h e

r o t a t i o n of t h e e a r t h . I n S e c t i o n 2 .A , w e s a w t h a t if w e w e n t t o a

c o o r d i n a t e s y s t e m w h i c h w a s r o t a t i n g w i t h a b o d y , a n e x t r a f o r c e

a p p e a r e d . I n t h e s t a t i c c a s e , t h i s w a s t h e f a m i l i a r c e n t r i f u g a l f o r c e . S i n c e

The Equations of Motion with Rotation 93

t h e m e a s u r e m e n t of t i d e s i n v o l v e s m o v i n g f luids o n t h e s u r f a c e of t h e

e a r t h , w e wil l h a v e t o e x p a n d t h i s c o n c e p t s o m e w h a t .

W e k n o w t h a t if a f o r c e a c t s in a n i n e r t i a l s y s t e m in s u c h a w a y a s t o

p r o d u c e a n a c c e l e r a t i o n a 0 , t h e n t h a t s a m e f o r c e a c t i n g in a r o t a t i n g

c o o r d i n a t e s y s t e m wil l p r o d u c e a n a p p a r e n t a c c e l e r a t i o n g i v e n b y

w h e r e co is t h e f r e q u e n c y of r o t a t i o n of t h e c o o r d i n a t e s y s t e m . If w e a r e

s i t t ing in a c o o r d i n a t e s y s t e m fixed o n t h e s u r f a c e of t h e e a r t h , t h e n w e

c a n t a k e dco/dt = 0. T h e t w o " e x t r a " t e r m s in t h e a b o v e e q u a t i o n a r e t h e n

t h e f a m i l i a r c e n t r i f u g a l a n d C o r i o l i s f o r c e s . I t is c u s t o m a r y t o t r e a t t h e s e

t e r m s , w h i c h a c t u a l l y a r i s e b e c a u s e of t h e a c c e l e r a t i o n of t h e c o o r d i n a t e

s y s t e m , a s f o r c e s ( u s u a l l y g i v e n s o m e n a m e l ike a p p a r e n t o r ficticious

f o r c e s ) w h e n w e w r i t e N e w t o n ' s l a w s . O n c e t h e s e e x t r a f o r c e s a r e

i n c l u d e d , w e c a n e a s i l y s e e t h a t t h e E u l e r e q u a t i o n , w h i c h is j u s t N e w t o n ' s

s e c o n d l a w , b e c o m e s

T o u n d e r s t a n d t h i s e q u a t i o n , c o n s i d e r t h e s y s t e m s h o w n in F i g . 6 .3 . T h e

p o i n t P r e p r e s e n t s t h e s p o t a t w h i c h t h e t i d e s a r e b e i n g m e a s u r e d , t h e

r a d i u s of t h e e a r t h is t a k e n t o b e a, a n d t h e a n g l e s 6 a n d cf> g i v e t h e

l o c a t i o n of P . T h e l e n g t h co is t h e p e r p e n d i c u l a r d i s t a n c e f r o m t h e a x i s of

r o t a t i o n ( t a k e n t o b e t h e z - a x i s ) t o P . T h i s s o m e w h a t c u m b e r s o m e

n o t a t i o n i s , u n f o r t u n a t e l y , s t a n d a r d f o r t h i s t y p e of s y s t e m .

a = a 0 - 2 co x v - w X ( w X r ) - (6 .C .1)

(6 .C.2)

0)

Fig. 6.3. Polar coordinates for the discussion of tidal waves.


I n S e c t i o n 2 . A , w e t r e a t e d t h e c e n t r i f u g a l f o r c e b y d i r e c t i n t e g r a t i o n of

t h e E u l e r e q u a t i o n f o r t h e s t a t i c c a s e . I n t h i s p r o b l e m , t h e fluid is in

m o t i o n r e l a t i v e t o t h e s u r f a c e of t h e e a r t h , s o w e c a n n o t i n t e g r a t e s o

e a s i l y . W e c a n , h o w e v e r , p e r f o r m a n e q u i v a l e n t o p e r a t i o n b y n o t i n g t h a t

w X ( w X r ) = OJ2 COP, (6 .C.3)

w h e r e p is a u n i t v e c t o r p e r p e n d i c u l a r t o t h e z - a x i s in t h e d i r e c t i o n of P .

A s i m p l e m a n i p u l a t i o n ( s e e P r o b l e m 6.3) t h e n g i v e s

(6.C.4)

T h u s , t h e c e n t r i f u g a l f o r c e t e r m c a n b e w r i t t e n a s a g r a d i e n t , a n d

c o m b i n e d w i t h o t h e r t e r m s o n t h e r i g h t - h a n d s i d e of E q . (6 .A .2 ) . If w e

p r o c e e d a s in S e c t i o n 5 .A a n d d r o p t h e (v • V)v t e r m in t h e c o n v e c t i v e

d e r i v a t i v e , t h e E u l e r e q u a t i o n c a n b e w r i t t e n

(6.C.5)

w h e r e t h e p o t e n t i a l is a c t u a l l y t h e s u m of t w o t e r m s

ft = ne + n D . (6.C.6)

W e h a v e w r i t t e n f l e f o r t h e p o t e n t i a l d u e t o t h e e a r t h ' s g r a v i t a t i o n , a n d f l D

is t h e d i s t u r b i n g p o t e n t i a l d u e t o t h e p r e s e n c e of t h e m o o n d e r i v e d in

S e c t i o n 6 .A.

I n t h e c a s e of l o n g w a v e s ( s e e S e c t i o n 5 .A) , w e f o u n d it v e r y

c o n v e n i e n t t o d i s c u s s t h e y - c o m p o n e n t o n t h e E u l e r e q u a t i o n first . T h e

g e n e r a l s c h e m e of t h i n g s is t o s o l v e t h e y - e q u a t i o n f o r t h e q u a n t i t y w h o s e

g r a d i e n t a p p e a r s o n t h e r i g h t - h a n d s i d e of E q . (6 .C .5), a n d t h e n i n s e r t t h i s

i n t o t h e r e m a i n i n g e q u a t i o n s . T h e y - e q u a t i o n is

vJ.C.7)

If w e n o w i n v o k e t h e l o n g - w a v e a p p r o x i m a t i o n s t h a t w e r e i n t r o d u c e d

in S e c t i o n 5 .A, w e wil l s e t t h e l e f t - h a n d s i d e of t h i s e q u a t i o n e q u a l t o z e r o .

T h i s c o r r e s p o n d s t o a s s u m i n g t h a t t h e m o t i o n in t h e y - d i r e c t i o n is s l o w

e n o u g h t o b e r e g a r d e d a s q u a s i - s t a t i c . If w e t h e n i n t e g r a t e t h e r i g h t - h a n d

s i d e f r o m s o m e a r b i t r a r y p o i n t y t o t h e p o i n t y = h + t ] ( w h i c h w e a g a i n

t a k e t o b e t h e s u r f a c e of t h e fluid), w e h a v e

(6 .C.8)

The Equations of Motion with Rotation 95

f o r t h e - c o m p o n e n t , w h e r e w e h a v e d r o p p e d t e r m s in vy.

T h e r e m a i n i n g e q u a t i o n of m o t i o n w h i c h m u s t b e w r i t t e n d o w n is

c o n t i n u i t y . I n S e c t i o n 5 .A, w e s a w t h a t it w a s s i m p l e r t o d e r i v e t h e

e q u a t i o n f r o m t h e s t a r t f o r t h e p a r t i c u l a r g e o m e t r y in q u e s t i o n . T h e

d e r i v a t i o n c o n s i s t e d of c a l c u l a t i n g t h e a m o u n t of fluid in a n in f in i t e s ima l

s l i ce of v o l u m e , a n d t h e n n o t i n g t h a t a n y fluid w h i c h e n t e r s o r l e a v e s t h e

v o l u m e m u s t r e s u l t in a c h a n g e of h e i g h t ( a n d t h e r e f o r e a c h a n g e of 17) of

t h e fluid in t h e v o l u m e .

T h e s a m e t e c h n i q u e c a n b e a p p l i e d f o r t h e g e o m e t r y a p p r o p r i a t e t o t h e

s u r f a c e of t h e e a r t h , a l t h o u g h it i s a l i t t le m o r e difficult t o v i s u a l i z e in t h i s

c a s e . W e c a n i m a g i n e t h e in f in i t e s ima l v o l u m e e l e m e n t , w h i c h w a s a

s i m p l e t w o - d i m e n s i o n a l s l i ce in S e c t i o n 5 .A t o b e a b o d y e x t e n d i n g

u p w a r d r a d i a l l y f r o m t h e s u r f a c e of t h e e a r t h , s o t h a t i t s h e i g h t is

m e a s u r e d in t e r m s of t h e c o o r d i n a t e y. I n t h e u n p e r t u r b e d s t a t e , t h i s b o d y

w o u l d b e filled t o a h e i g h t h w i t h fluid. L e t t h e p e r i m e t e r of t h e b o d y b e

d e l i n e a t e d b y a r c s ( s e e F i g . 6 .4) , o n e c o r r e s p o n d i n g t o a n i n f in i t e s ima l

i n c r e m e n t in 0, a n d t h e o t h e r t o a n in f in i t e s ima l i n c r e m e n t in <p. T h i s

A n u m b e r of p o i n t s c a n b e m a d e a b o u t t h i s r e s u l t . F i r s t , j u s t a s w e

d r o p p e d Fy in E q . (5 .A . 11), w e wil l i g n o r e £lD w i t h r e s p e c t t o O e in t h i s

e q u a t i o n . S e c o n d , t h e q u a n t i t y f l e - \co2co2 is t h e p o t e n t i a l w h i c h w o u l d b e

fel t b y a s t a t i o n a r y b o d y a t t h e s u r f a c e of t h e e a r t h , a n d is u s u a l l y r e f e r r e d

t o a s t h e " a p p a r e n t g r a v i t y . " If w e e x p a n d t h i s q u a n t i t y a t y = h + TJ in a

T a y l o r s e r i e s a b o u t y = h, w e h a v e

(6 .C.9) = c o n s t . + grj .

T h e final r e s u l t f o r t h e i n t e g r a t e d y - c o m p o n e n t of t h e E u l e r e q u a t i o n

( E q . (16 .C.8) ) is t h e n j u s t

(6 .C .10)

S u b s t i t u t i n g t h i s r e s u l t i n t o t h e r i g h t - h a n d s i d e of E q . (6 .C .5 ) , w e find

t h e 6 c o m p o n e n t of t h e E u l e r e q u a t i o n t o b e

(6 .C .11)

(6 .C .12)

a n d


Fig. 6.4. The idea of continuity in polar coordinates.

f o r t h e e q u a t i o n of c o n t i n u i t y ( s e e P r o b l e m 6.4) .

D. TIDES AT THE SURFACE OF THE EARTH

I n t h e e q u a t o r i a l t h e o r y of t h e t i d e s , w e a s s u m e d t h a t b o t h t h e m o o n

a n d t h e p o i n t a t w h i c h t h e t i d e s w e r e t o b e o b s e r v e d w e r e o n t h e e q u a t o r ,

s o t h a t t h e a n g l e & in F i g . 6.2 c o u l d b e iden t i f i ed w i t h t h e a n g l e cf> in o u r

n e w c o o r d i n a t e s y s t e m . F o r t h e g e n e r a l p r o b l e m of finding t h e t i d e s a t a n

a r b i t r a r y p o i n t o n t h e s u r f a c e of t h e e a r t h , t h i s is n o l o n g e r p o s s i b l e . I n

f a c t , if w e s a y t h a t t h e d i r e c t i o n of t h e r a d i u s v e c t o r t o t h e m o o n is g i v e n

b y t h e a n g l e s A a n d a w h i l e t h e r a d i u s v e c t o r t o t h e p o i n t P i s g i v e n b y 6

a n d cf> ( s e e F i g . 6.5), t h e n

c o s 0 = c o s A c o s 6 + s in A s in </> c o s ( a + c/>). (6 .D.1)

W e c a n n o w i n s e r t t h i s i n t o t h e e q u a t i o n f o r t h e d i s t u r b i n g p o t e n t i a l ,

E q . (6 .A .4 ) , a n d p u t t h e r e s u l t i n g e x p r e s s i o n f o r Q D i n t o E q s . (6 .C .11) a n d

(6 .C .12) t o g e t t h e e q u a t i o n s g o v e r n i n g t h e t i d e s . B e f o r e d o i n g s o ,

h o w e v e r , it wi l l b e p r o f i t a b l e t o d i s c u s s t h e f o r m f o r t h e d i s t u r b i n g

s h o u l d b e f a m i l i a r t o t h e r e a d e r , s i n c e it is t h e s t a n d a r d v o l u m e e l e m e n t in

s p h e r i c a l c o o r d i n a t e s . W i t h t h i s g e o m e t r y , it is r e l a t i v e l y s t r a i g h t f o r w a r d

t o r e p e a t t h e d e r i v a t i o n of S e c t i o n 5 .A t o g e t

(6 .C .13)

Tides at the Surface of the Earth 97

Fig. 6.5. The angles involved in the general theory of the tides.

= F , + F 2 + F 3 . (6 .D.4)

T h u s , w e s e e t h a t t h e d i s t u r b i n g p o t e n t i a l c a n b e t h o u g h t of a s

c o n s i s t i n g of t h r e e s e p a r a t e t e r m s . S i n c e e a c h of t h e s e t e r m s p l a y s t h e

s a m e r o l e a s t h e i n h o m o g e n e o u s t e r m in E q . (6 .B .2 ) , it i s r e a s o n a b l e t o

s u p p o s e t h a t e a c h is a s s o c i a t e d w i t h a s e p a r a t e m o t i o n of t h e fluid, a n d

t h e t o t a l m o t i o n of t h e fluid wi l l b e t h e s u m of t h e t h r e e s e p a r a t e m o t i o n s .

T h i s p r o p e r t y of d i f f e ren t i a l e q u a t i o n s is d i s c u s s e d in A p p e n d i x E , a n d w e

wil l s e e s o m e e x p l i c i t e x a m p l e s l a t e r in t h i s s e c t i o n . F o r t h e m o m e n t ,

h o w e v e r , le t u s a s s u m e t h a t t h i s is t h e c a s e a n d p r o c e e d w i t h t h e

d i s c u s s i o n .

W e s e e t h a t t h e t i m e d e p e n d e n c e s of Fu F 2 , a n d F 3 a r e al l q u i t e

d i f fe ren t . If P is f ixed , t h e n 6 a n d cp d o n o t v a r y w i t h t h e t i m e . S i n c e a is

p o t e n t i a l w h i c h r e s u l t s f r o m t h i s m a n i p u l a t i o n . R e c a l l i n g t h a t

(6 .D.2)

w h e r e Me i s t h e m a s s of t h e e a r t h , a n d de f in ing

(6 .D.3)

w e h a v e

H s in 2A s in 26 c o s ( a + cp)

H s i n 2 A s i n 2 6 c o s 2(a + cp)


t h e p r o j e c t i o n of t h e m o o n ' s s h a d o w o n t o t h e e q u a t o r i a l p l a n e , w e m u s t

h a v e

a ~ cot, (6 .D .5)

w h e r e co is t h e f r e q u e n c y of t h e e a r t h ' s r o t a t i o n . O v e r t h e p e r i o d of a d a y

o r s o , A, t h e a n g l e of d e c l i n a t i o n of t h e m o o n , i s a p p r o x i m a t e l y c o n s t a n t .

T h u s , t o a first a p p r o x i m a t i o n , Fx is a c o n s t a n t t e r m , F2 is a t e r m w h i c h

v a r i e s a s c o s cot a n d w o u l d h e n c e g i v e r i s e t o a o n c e - a - d a y ( d i u r n a l ) t i d e ,

w h i l e F 3 v a r i e s a s c o s 2cot, a n d is a s s o c i a t e d w i t h t h e t w i c e - a - d a y

( s e m i - d i u r n a l ) t i d e .

I n f a c t , w e k n o w t h a t t h e e a r t h ' s r o t a t i o n a x i s is t i l t ed a t a b o u t 23° w i t h

r e s p e c t t o t h e p l a n e of t h e m o o n ' s o r b i t , s o t h e a n g l e A wi l l h a v e a t i m e

d e p e n d e n c e w h o s e f r e q u e n c y wi l l b e a b o u t a m o n t h . I n a d d i t i o n , t h e

" c o n s t a n t " H c o n t a i n s a f a c t o r 1 / D 3 , w h e r e D i s t h e d i s t a n c e t o t h e

m o o n . D i t se l f c h a n g e s w i t h t i m e o v e r o n e l u n a r r e v o l u t i o n , c o r r e s p o n d -

i n g t o t h e f a c t t h a t t h e m o o n a n d t h e e a r t h d e s c r i b e e l l ip t i ca l o r b i t s a b o u t

t h e i r c o m m o n c e n t e r of m a s s . T h u s , t h e s i m p l e s t a t e m e n t s g i v e n a b o u t

t i m e d e p e n d e n c e s in t h e a b o v e p a r a g r a p h a r e n o t s t r i c t l y t r u e . I t i s c l e a r ,

h o w e v e r , t h a t w h a t e v e r t h e t i m e d e p e n d e n c e of t h e a n g l e A a n d t h e

p a r a m e t e r H, t h e y a r e v e r y s l o w c o m p a r e d t o t h e t i m e d e p e n d e n c e of t h e

a n g l e a. H e n c e , in c a l c u l a t i n g t h e t i d e s d u e t o F2 a n d F 3 , w e c a n r e g a r d

b o t h of t h e s e a s c o n s t a n t s , b u t a s c o n s t a n t s w h o s e v a l u e s m a y c h a n g e

o v e r m a n y p e r i o d s of t h e t i d e , a n d w h i c h m u s t t h e r e f o r e b e a d j u s t e d

b e f o r e u n d e r t a k i n g n u m e r i c a l c a l c u l a t i o n s . T h i s i d e a is u s u a l l y e x p r e s s e d

b y w r i t i n g

F 2 = H " c o s ( « + </>) (6 .D.6)

a n d

F 3 = H" c o s 2(a 4- <f>), (6 .D .7 )

w h e r e H" a n d Hf" a r e a p p r o x i m a t e l y c o n s t a n t s . I n w h a t f o l l o w s , w e sha l l

i g n o r e t h e m o n t h l y t i d e s a s s o c i a t e d w i t h F i , a l t h o u g h t h e y a r e k n o w n t o

e x i s t a n d h a v e b e e n m e a s u r e d .

W e k n o w t h a t in d e a l i n g w i t h c o m p l i c a t e d e q u a t i o n s , it is o f t e n b e s t t o

i s o l a t e v a r i o u s t e r m s f o r c o n s i d e r a t i o n . W e wi l l t h e r e f o r e c o n s i d e r t h e

E u l e r e q u a t i o n in w h i c h n D / g in E q . (6 .D.4) is r e p l a c e d b y e i t h e r F 2 o r F 3 ,

w h i c h , f o r t h e s a k e of c o n v e n i e n c e , w e wi l l w r i t e a s F*. L e t u s d e n o t e b y

Tj, t h e d i s p l a c e m e n t of t h e s u r f a c e a s s o c i a t e d w i t h t h e t e r m F*. T h e E u l e r

e q u a t i o n s a r e

(6 .D.8)


a n d

I n g e n e r a l , t h e t a n d <p d e p e n d e n c e of F f wi l l b e of t h e f o r m

Ft ~ e ^ ' + v ^

w h e r e w e u s e d t h e de f i n i t i ons of t h e s i n e a n d c o s i n e in t e r m s of

e x p o n e n t i a l s , a n d E q . (6 .D .5 ) w h i c h g i v e s t h e a n g l e a a s a f u n c t i o n of

t i m e . o~i a n d St a r e

C 7 2 = co, s2 = 1, (6 .D .10 )

cr 3 = 2 co, s 3 = 2 .

f o r F 2 a n d F 3 , r e s p e c t i v e l y .

I t b e c o m e s n a t u r a l , t h e r e f o r e , t o l o o k f o r s o l u t i o n s of t h e f o r m ( d r o p -

p i n g t h e s u b s c r i p t i f o r c o n v e n i e n c e )

ve(6, <t>,t) = ve(0)eH"t+s*\

v+(6, 4>, t) = v+(6)eiiat+M+\ (6 .D .11)

r , ( 0 , ct>,t) = 71(6)ei(Tt+s+\

w h i c h , u p o n s u b s t i t u t i o n i n t o E q s . (6 .D.8) a n d (6 .D.9) y i e l d s

iave — Itov* c o s 6

itJVj, + 2coi?E c o s 6

(6 .D.12)

w h e r e w e h a v e d e f i n e d

6 .D.13)

T h e s e e q u a t i o n s a r e n o w a l g e b r a i c , a n d t h e r e f o r e q u i t e e a s y t o s o l v e .

T h e n e t e f fec t of t h e a s s u m p t i o n w h i c h w e m a d e a b o u t t h e f o r m of t h e

s o l u t i o n , w e s e e , w a s t o r e d u c e t h e c o m p l e x i t y of t h e E u l e r e q u a t i o n s .

S i m p l e a l g e b r a ( s e e P r o b l e m 6.5) t h e n y i e l d s

a n d

(6 .D.9)

(6 .D.14)

(6 .D.15)



a n d

6 .D.16)

(6 .D.17)

w h e r e w e h a v e a s s u m e d t h a t h, t h e d e p t h of t h e o c e a n a t t h e p o i n t P , i s a

f u n c t i o n of 6 o n l y . T h i s a p p r o x i m a t i o n is n o t va l i d f o r t h e r e a l e a r t h , of

c o u r s e , a n y m o r e t h a n t h e a p p r o x i m a t i o n of u n i f o r m d e p t h in E q . (6 .B.2)

w a s .

T h e g e n e r a l p r o b l e m of t h e s o l u t i o n of t h e t i d e s c a n n o w b e s e e n t o

i n v o l v e s o l v i n g E q . (6 .D .18 ) , o r i t s m o r e g e n e r a l f o r m w h i c h i n c l u d e s a </>

d e p e n d e n c e in t h e d e p t h , t o g e t h e r w i t h t h e E u l e r e q u a t i o n s , (6 .D.14) a n d

(6 .D .15) . F o r a n a r b i t r a r y d e p t h l a w ( b y w h i c h w e m e a n t h e d e p e n d e n c e

of h o n 6 a n d </>), it is n o t p o s s i b l e t o d o t h i s e x p l i c i t l y a l t h o u g h it c a n b e

d o n e n u m e r i c a l l y .

T h e r e i s , h o w e v e r , o n e d e p t h l a w w h i c h d o e s a l l o w e x p l i c i t s o l u t i o n s

f o r b o t h r]2 a n d 173. S u p p o s e w e c o n s i d e r a n o c e a n w h o s e d e p t h is g i v e n b y

h(6) = h0sin2e. (6 .D.19)

T h i s is a c t u a l l y n o t a b a d a p p r o x i m a t i o n t o t h e o c e a n s o n t h e e a r t h — a t

l e a s t it k e e p s t h e i d e a of t h e o c e a n s a t t h e p o l e s b e i n g s h a l l o w e r t h a n

t h o s e a t t h e e q u a t o r .

L e t u s b e g i n b y c a l c u l a t i n g T J 2 , t h e d i u r n a l t i da l d i s p l a c e m e n t . F o r t h i s

c a s e , w e h a v e

( N o t e t h a t m is n o t a m a s s . )

B u t of c o u r s e , w e m u s t d o m o r e t h a n j u s t s o l v e t h e E u l e r e q u a t i o n s if

w e a r e t o h a v e a s o l u t i o n . W e m u s t s o l v e a n d sa t i s fy t h e e q u a t i o n of

c o n t i n u i t y a s w e l l . If w e p u t o u r a s s u m e d f o r m s of t h e s o l u t i o n i n t o E q .

(6 .C .13 ) , w e find

(h(0)ve(6) s in 0 ) + isft(0)i?*(0)J, (6 .D .18)

S i n c e

F 2 = H" s in 0 c o s 0eiia*++\ (6 .D.20)

it is n a t u r a l t o a s s u m e a f o r m of s o l u t i o n

T ? 2 = C c o s 0 s in 6eiia*++\ (6 .D.21)


P u t t i n g t h i s i n t o t h e E u l e r e q u a t i o n s in t h e f o r m of E q s . (6 .D.14) a n d

(6 .D.15) q u i c k l y y i e l d s

icrC , _ _ ve= (6 .D .22)

m a n d

f o r t h e d i s p l a c e m e n t d u e t o t h e s e m i - d i u r n a l t i d e .

T h e t o t a l d i s p l a c e m e n t a t t h e p o i n t P wi l l , of c o u r s e , b e g i v e n b y

T/P = r / 2 + r /3 , (6 .D .27)

s o t h a t a n o b s e r v e r wi l l s e e b o t h a d a i l y a n d a t w i c e d a i l y t i d e . C o m p a r i n g

E q s . (6 .D.23) a n d (6 .D.26) w e w o u l d e x p e c t t h e s e t i d e s t o b e r o u g h l y of

e q u a l i m p o r t a n c e , b u t t h i s q u e s t i o n is e x a m i n e d in m o r e d e t a i l in P r o b l e m

6.8 .

W e n o t e t h a t t h e s e m i - d i u r n a l t i d e is sti l l i n v e r t e d , s o t h a t r / 3 is n e g a t i v e

w h e n t h e m o o n is d i r e c t l y o v e r h e a d . T h i s i s n o t a g e n e r a l r e s u l t f o r all

h(f3, h o w e v e r . F o r e x a m p l e , c a l c u l a t i o n s of s e m i - d i u r n a l t i d e s in a n

o c e a n of u n i f o r m d e p t h g i v e n o n i n v e r t e d t i d e s f o r s o m e l a t i t u d e s . I n

P r o b l e m 6.7, t h i s p r o b l e m is d e a l t h w i t h f u r t h e r .

I n s e r t i n g t h i s i n t o E q . ( 6 .D .18 ) , r e c a l l i n g t h e de f in i t ion in E q . ( 6 .D .13 ) , w e

find

(6 .D.23) -H" s in 6 c o s 0eiiajt++\ a

T u r n i n g n o w t o t h e s e m i - d i u r n a l t i d e a s s o c i a t e d w i t h F 3 , w e c a n

p r o c e e d in a n a l o g y t o E q . (6 .D.21) t o a s s u m e t h a t

r)3 = B s i n 2 6 ei(2*"+2*>. (6 .D.24)

If w e a g a i n t u r n t o t h e E u l e r e q u a t i o n s , w e find s i n c e / = \ t h a t

(6 .D.25)

a n d

P r o c e e d i n g a s b e f o r e a n d i n s e r t i n g t h e s e i n t o t h e e q u a t i o n of

c o n t i n u i t y , w e find

( 6 . D . 2 6 ;


F i n a l l y , w e k n o w f o r r e a l t i d e s o n t h e r e a l e a r t h , t h e d i u r n a l a n d

s e m i - d i u r n a l t i d e s a r e n o t of e q u a l i m p o r t a n c e . T h e m a j o r t i d e s c o m e

t w i c e a d a y . ( T h e a u t h o r , b o r n a n d r a i s e d in t h e M i d w e s t , l e a r n e d t h i s f a c t

w h e n h e b e g a n s t u d y i n g t i d e s b y l i s t e n i n g t o l a t e - n i g h t r a d i o r e p o r t s f r o m

N o r f o l k . ) C a n w e u n d e r s t a n d t h i s f e a t u r e of t h e t i d e s o n t h e b a s i s of o u r

s i m p l e t h e o r i e s ?

I n P r o b l e m 6.6, w e s h o w t h a t t h e a n a l y s i s p r e s e n t e d a b o v e a p p l i e d t o a n

o c e a n w h o s e d e p t h is g i v e n b y

h ( 0 ) = hod ~ q c o s 2 0 ) (6 .D .28)

y i e l d s a d i u r n a l t i d e f o r w h i c h

SUMMARY

T h e n e t g r a v i t a t i o n a l a t t r a c t i o n a t t h e s u r f a c e of t h e m o o n is g i v e n b y

t h e d i s t u r b i n g p o t e n t i a l . T h i s a t t r a c t i o n is t h e c a u s e of t h e t i d e s . S o m e

s i m p l e g e o m e t r y s h o w s t h a t t i d e s a t a n a r b i t r a r y p o i n t wi l l b e of t h r e e

t y p e s — a m o n t h l y t i d e , a d a i l y t i d e , a n d a s e m i - d i u r n a l t i d e . F o r s o m e

s i m p l e f o r m s of t h e d e p t h l a w f o r t h e o c e a n s , it is p o s s i b l e t o s o l v e f o r

t h e s e t i d e s e x p l i c i t l y , t a k i n g i n t o a c c o u n t t h e r o t a t i o n of t h e e a r t h . W e find

t h a t t h e s e m i - d i u r n a l t i d e s a r e t h e m o s t i m p o r t a n t .

PROBLEMS

6.1. For the case of equitorial t ides, compare the maximum tide due to the moon with tides due to (a) the sun, (b) Jupiter , and (c) Alpha Centaur i .

6.2. Would equatorial t ides be inverted on Venus or on Mars (assuming that they had oceans of the same dep ths as our own)?

6.3. Verify Eq . (6.C.4) and show how it is related to Eq . (2.A.4).

(6 .D.29)

W e n o t e i m m e d i a t e l y t h a t f o r a n o c e a n of u n i f o r m d e p t h , w h e r e q = 0 ,

t h e r e is no d i u r n a l t i d e a t al l . T h u s , t h e d i u r n a l t i d e e x i s t s o n l y i n s o f a r a s

t h e o c e a n d e p a r t s f r o m c o m p l e t e u n i f o r m i t y . S i n c e t h e o c e a n s a r e

a p p r o x i m a t e l y u n i f o r m , w e w o u l d e x p e c t t h a t t h e i m p o r t a n c e of t h e

d i u r n a l t i d e s h o u l d b e g r e a t l y d i m i n i s h e d . T h i s e x p l a n a t i o n w a s o n e of t h e

g r e a t t r i u m p h s of t h e L a p l a c e t h e o r y . I t a l s o e x p l a i n s w h y n o d i u r n a l t i d e s

a p p e a r e d in S e c t i o n 6 . B , w h e n w e c o n s i d e r e d e q u a t o r i a l t i d e s in a n o c e a n

of c o n s t a n t d e p t h .

Problems 103

6.4. Consider the problem of continuity in spherical coordinates , as shown in Fig. 6.4.

(a) Show that the amount of fluid in the body at any t ime is

M = pa dOco dcp (h +r/).

(b) Show that the net flux through the walls of length a dO is

6.7. For the depth law in Eq . (6.D.19), find the smallest value of h0 such that the

tide is not inverted.

6.8. For the depth law of Eq . (6.D.19), calculate the rat io of the maximum values of the diurnal and semi-diurnal t ides as a function of longitude. Make a rough sketch of the resul ts .

6.9. Consider the ear th to be a sphere of radius a which is covered by an ocean of uniform depth h which is much less than a. Le t 17 be the deviat ion of the dep th of the ocean from uniformity.

(a) Using the methods of Prob lem 6.4, show that the equat ion of continuity is

(b) If we neglect Coriolis and centrifugal forces , show that the 0- and cp-components of the Euler equat ion are

and

respectively.

(vehco) dd dep.

(c) Show that the net flux through the walls of length a sin 0 dcp is

-(v^ha) dd dep.

(d) H e n c e verify Eq . (6.C.13).

6.5. Verify E q s . (6.D.14) and (6.D.15).

6.6. For a depth law of the form

h(6) = foo(l-42cos20),

show that the diurnal d isplacement is given by

H" s i n 0 c o s 0 e , ( a , t + d ) .


(Hint: U s e the expansion in spherical harmonics discussed in Appendix F.) Evalua te aim for the first three values of /.

(c) Using the theory of inhomogeneous equat ion outlined in Appendix E , find the part icular solution to this equat ion for the first three / values . Sketch the resul ts .

REFERENCES

H. Lamb, Hydrodynamics (cited in Chapter 1) has an excellent discussion of the theory of the tides. The following texts are also quite useful.

R. A. Becker, Introduction to Theoretical Mechanics, McGraw-Hill, New York, 1954.

(c) H e n c e show that

6.10. (a) For the wave equat ion derived in P rob lem 6.9, show that using the

technique of separat ion of variables outl ined in Appendix F , the solution for 17 will be of the form

r) * Ylm(e,<j>)e^,

where Yim is the spherical harmonic defined in Appendix F . (b) If we define = (a I toe)2, show that the only solutions which are possible

are those for which /3 = /(/ + l).

(Hint: Consider the case where cos 8 = + 1 , and use the recurs ion relation for Legendre polynomials given in Appendix F to show that T/ will be infinite unless the Legendre series terminates.)

(c) These allowed frequencies are associated with the normal modes of

oscillation. Calculate the frequencies for the first four modes for the ear th . (d) Consider a plane through the ear th at <f> = 0. Sketch the value of 17 as a

function of 6 for the first few normal modes .

6.11. (a) Continuing with the example of the flooded ear th in the previous

problems, show that if a disturbing potential is present , the Euler equat ions in

Prob lem 6.9 will have a te rm

added, respect ively, to the 6 and <f> equat ions . (b) Derive the new wave equat ion corresponding to the new Euler equat ions ,

and show that it can be wri t ten in the form

References 105

In Chapter 11, there is a readable and complete discussion of equations of motion in accelerated frames, and of the Coriolis and centrifugal forces.

Walter Kauzmann, Quantum Chemistry, Academic Press, New York, 1957. Chapter 3 of this text contains a very nice description of the use of spherical harmonics applied to the problem of tides on the earth.

William S. von Arx, An Introduction to Physical Oceanography, Addison-Wesley, New York, 1962.

An excellent descriptive text on the motion of the oceans, currents, and waves, along with a discussion of how measurements are made.

C. Eckart, Hydrodynamics of Oceans and Atmospheres, Pergamon Press, New York, 1962. Contains an excellent discussion of the tidal equations with rotation, and of the actual structure of the ocean.

7

Oscillations of Fluid Spheres: Vibrations of the Earth and Nuclear Fission

He felt the earth move out and away from under them.

ERNEST HEMINGWAY

For Whom the Bell Tolls

A. FREE VIBRATIONS OF THE EARTH

I n S e c t i o n 2 . D , w e s a y t h a t f o r s o m e p u r p o s e s it is r e a s o n a b l e t o t r e a t

t h e e a r t h a s if it w e r e a u n i f o r m l i qu id . If w e i g n o r e t h e r o t a t i o n of t h e

e a r t h f o r t h e d i s c u s s i o n in t h i s s e c t i o n , t h e n t h e e q u i l i b r i u m c o n f i g u r a t i o n

of t h e e a r t h w o u l d b e a s p h e r e . I t i s r e a s o n a b l e t o a s k w h a t w o u l d h a p p e n

t o s u c h a s p h e r e if, f o r s o m e r e a s o n , it w e r e s l igh t ly d e f o r m e d (e .g . , b y a n

e a r t h q u a k e ) a n d t h e n a l l o w e d t o r e s p o n d . W e wi l l s h o w in t h i s s e c t i o n

t h a t a l i q u i d s p h e r e w o u l d b e e x p e c t e d t o p e r f o r m o s c i l l a t i o n s a b o u t i t s

e q u i l i b r i u m c o n f i g u r a t i o n . T h i s p h e n o m e n o n , s i m i l a r t o t h e r i n g i n g of a

b e l l , h a s r e c e n t l y b e e n m e a s u r e d b y g e o p h y s i c i s t s .

If t h e e a r t h in i t s u n p e r t u r b e d s t a t e i s a s p h e r e of r a d i u s a a n d d e n s i t y p ,

t h e n in a p e r t u r b e d s t a t e , t h e d i s t a n c e f r o m t h e c e n t e r t o t h e p e r t u r b e d

s u r f a c e wi l l b e ( s e e F i g . 7.1)

r = a + ao,<f>)- (7 .A .1)

I t is a l w a y s p o s s i b l e t o e x p a n d t h e f u n c t i o n £ ( 0 , cp) in t e r m s of s p h e r i c a l

h a r m o n i c s ( s e e A p p e n d i x F )

106

(7 .A.2)

Free Vibrations of the Earth 107

(7 .A.5)

(7 .A.6)

w h e r e w e h a v e i n c o r p o r a t e d t h e s u m o v e r m i n t o t h e de f in i t i on of

A s in S e c t i o n 4 . B , t h e e q u a t i o n w h i c h d e t e r m i n e s t h e v e l o c i t y p o t e n t i a l

is j u s t

V2<f> = 0 , (7 .A.3)

w h i c h f o r a s p h e r i c a l g e o m e t r y h a s t h e s o l u t i o n ( s e e A p p e n d i x F )

S i n c e w e w a n t $ t o b e b o u n d e d a t r = 0 , w e m u s t h a v e Bim = 0 ( th i s

c o r r e s p o n d s t o u s i n g t h e " b o t t o m " b o u n d a r y c o n d i t i o n in E q . ( 5 . B . 6 ) — i n

t h i s c a s e t h e c o n d i t i o n is t h a t a t t h e o r i g i n t h e v e l o c i t y is finite). T h u s , w e

c a n w r i t e

(7 .A.4)

w h e r e st is d e f i n e d in a m a n n e r s i m i l a r t o £i.

T h e b o u n d a r y c o n d i t i o n a t t h e o u t e r s u r f a c e is g i v e n b y t h e E u l e r

e q u a t i o n a s i n E q . (5 .B .10) t o b e


w h e r e , a s in t h e d e v e l o p m e n t of s u r f a c e w a v e s in S e c t i o n 5 . B , w e h a v e

e v a l u a t e d t h e b o u n d a r y e q u a t i o n a t t h e u n p e r t u r b e d s u r f a c e .

W e n o w t u r n t o t h e p r o b l e m of finding w h e t h e r o r n o t w e c a n find

w a v e l i k e s o l u t i o n s f o r £, t h e d e v i a t i o n of t h e s u r f a c e of t h e s p h e r e f r o m

108 Oscillations of Fluid Spheres

e q u i l i b r i u m a t a g i v e n p o i n t . If s u c h s o l u t i o n s a r e f o u n d , t h e n a n o b s e r v e r

a t t h a t p o i n t w o u l d o b s e r v e t h e o s c i l l a t i o n s in t h e e a r t h t h a t w e a r e

d i s c u s s i n g . A s in t h e d e v e l o p m e n t of s u r f a c e w a v e s [ s e e E q . (5 .B .2 ) ] , w e

u s e t h e E u l e r e q u a t i o n in t h e f o r m

t o d e t e r m i n e t h e e x i s t e n c e of o s c i l l a t i o n s a n d t h e i r f r e q u e n c y . I n o r d e r t o

u s e t h i s e q u a t i o n , h o w e v e r , w e h a v e t o find t h e p o t e n t i a l d u e t o a d i s t o r t e d

s p h e r e . W e wi l l s o l v e t h e g e n e r a l p r o b l e m of finding t h e p o t e n t i a l a t a

p o i n t r j u s t a b o v e t h e u n p e r t u r b e d s u r f a c e of t h e s p h e r e , a n d l a t e r l e t

r - H > a ( a s is a p p r o p r i a t e f o r o u r p r o c e d u r e of e v a l u a t i n g all b o u n d a r y

c o n d i t i o n s a t t h e u n p e r t u r b e d s u r f a c e ) .

W e sha l l find t h e p r o b l e m t o b e c o n s i d e r a b l y s impl i f ied if w e b r e a k t h e

p o t e n t i a l i n t o t w o p a r t s ( s e e F i g . 7 . 2 ) — o n e t h e p o t e n t i a l a t r d u e t o a n

u n p e r t u r b e d s p h e r e of r a d i u s a, a n d t h e o t h e r t h e p o t e n t i a l a t r d u e t o a

t h i n s p h e r i c a l she l l of v a r i a b l e d e n s i t y p s ( 0 , cp). L a t e r , w e sha l l s e e h o w t o

r e l a t e t h i s v a r i a b l e d e n s i t y t o t h e d i s p l a c e m e n t of t h e s u r f a c e , £. F o r t h e

m o m e n t , h o w e v e r , w e s i m p l y n o t e t h a t t h e d e n s i t y of t h e she l l c a n b e

e i t h e r p o s i t i v e o r n e g a t i v e , d e p e n d i n g o n w h e t h e r t h e a c t u a l s u r f a c e is

a b o v e o r b e l o w t h e u n p e r t u r b e d s u r f a c e a t a g i v e n p o i n t . T h u s ,

w h e r e t h e first t e r m of t h e r i g h t - h a n d s i d e is t h e p o t e n t i a l a t r d u e t o t h e

s p h e r e , a n d t h e s e c o n d t e r m (sti l l t o b e c a l c u l a t e d ) r e p r e s e n t s t h e

(7 .A.7)

(7 .A.8)

\

/

\ \

shell

Fig. 7.2. Coordinates for breaking the sphere into a central core plus a shell.

Free Vibrations of the Earth 109

Fig. 7.3. Situation at the surface of the distorted sphere.

p o t e n t i a l a t r d u e t o t h e r e s t of t h e she l l . T h i s i s

(7 .A.9)

w h e r e , b e c a u s e w e a r e c o n s i d e r i n g o n l y s m a l l v i b r a t i o n s , a n d h e n c e v e r y

t h i n s h e l l s , w e c a n r e g a r d p s (0 ' ,< /> ' ) a s a s u r f a c e m a s s d e n s i t y . T h e

s u r f a c e d e n s i t y , b e i n g a f u n c t i o n of 6' a n d <£>', c a n b e e x p a n d e d in t e r m s

of s p h e r i c a l h a r m o n i c s , j u s t a s w e e x p a n d e d £ in E q . (7 .A .2).

Ps (0 \ = H YLM(0\ 4>') = 2 P^- (7 .A .10)

JL M L

I t i s a s t a n d a r d m a t h e m a t i c a l r e s u l t t h a t t h e t e r m \/R w h i c h a p p e a r s in

E q . (7 .A .9) c a n b e w r i t t e n a s

(7 . A . 11)

w h e r e y is t h e a n g l e b e t w e e n r a n d r' ( s e e F i g . 7 .2) .

If w e t a k e t h e s e r e s u l t s a n d p u t t h e m b a c k i n t o E q . ( 7 .A .9 ) , w e c a n u s e

t h e r e s u l t s of P r o b l e m 7.1 t o c a r r y o u t t h e i n t e g r a l s o v e r t h e a n g l e s 0 ' a n d

<f)'. L e t t i n g r = a, w e find t h a t n S HEII e v a l u a t e d a t t h e u n p e r t u r b e d s u r f a c e

of t h e s p h e r e is

(7 . A . 12)

Al l w e n e e d t o d o n o w is d e t e r m i n e t h e s u r f a c e d e n s i t y of t h e she l l p{l\

a n d w e wi l l h a v e t h e p o t e n t i a l d u e t o a d i s t o r t e d s p h e r e . C o n s i d e r F i g . 7 . 3 .

T h e s h a d e d a r e a r e p r e s e n t s t h e e x c e s s m a s s in t h e s u r f a c e e l e m e n t d u e t o

t h e d i s t o r t i o n of t h e s u r f a c e . T h e a m o u n t of e x c e s s m a s s is j u s t p £ d o \


A n o b s e r v e r a t t h e s u r f a c e , t h e n , wi l l s e e o s c i l l a t i o n s c o r r e s p o n d i n g t o t h e

a b o v e f r e q u e n c i e s if f o r s o m e r e a s o n t h e s u r f a c e of t h e e a r t h is e v e r

d i s t o r t e d .

Of c o u r s e , in a g e n e r a l e x c i t a t i o n , w e w o u l d e x p e c t all p o s s i b l e

f r e q u e n c i e s t o b e e x c i t e d , a n d t h e a c t u a l d i s p l a c e m e n t of t h e s u r f a c e

w o u l d b e s o m e s o r t of s e r i e s , w h e r e t h e f r e q u e n c i e s of e a c h t e r m in t h e

s e r i e s a r e g i v e n b y t h e a b o v e e q u a t i o n . L e t u s l o o k a t t h e first f e w t e r m s in

(7 . A . 18)

w h i c h w e m u s t e q u a t e t o p ( s ) D A , t h e m a s s in a s u r f a c e e l e m e n t of t h e

she l l . T h u s ,

(7 . A . 13)

w h e r e O f is de f ined in E q . (7 .A . 12). T h i s m e a n s t h a t

(7 . A . 14)

i s t h e t o t a l p o t e n t i a l a t t h e p o i n t r. L e t t i n g r i n t h e first t e r m b e a + f, a n d

t h e n k e e p i n g o n l y first-order t e r m s in £, w e find

(7 . A . 15)

w h e r e w e h a v e u s e d t h e i d e n t i t y \ irap = g. W e l e a v e a s a n e x e r c i s e f o r

t h e r e a d e r t h e p r o b l e m of w h y w e s e t r = a in t h e c a l c u l a t i o n of ftsheii, b u t

h a d t o s e t r = a + £ in E q . (7 .A . 14).

W e a r e n o w r e a d y t o u s e t h e E u l e r e q u a t i o n a t t h e s u r f a c e t o d e t e r m i n e

t h e e q u a t i o n f o r U s i n g o u r p r e s c r i p t i o n of e v a l u a t i n g all t e r m s a t t h e

u n p e r t u r b e d s u r f a c e , a n d u s i n g t h e c o n d i t i o n t h a t t h e p r e s s u r e a t t h e

s u r f a c e m u s t b e a c o n s t a n t , w e find

(7 . A . 16)

D i f f e r e n t i a t i n g t h i s e q u a t i o n w i t h r e s p e c t t o t i m e , a n d u s i n g t h e

b o u n d a r y c o n d i t i o n in E q . (7 .A .6 ) , w e find

(7 . A . 17)

w h i c h i s , i n d e e d , t h e e q u a t i o n of a h a r m o n i c o s c i l l a t o r , w i t h f r e q u e n c y of

o s c i l l a t i o n g i v e n b y

The Liquid Drop Model of the Nucleus 111

Fig. 7.4. The distortion corresponding to / = 1.

s u c h a s e r i e s ( c o r r e s p o n d i n g t o t h e l o w e s t v a l u e s of / ) . F o r / = 0 , to = 0

a n d n o t i m e - d e p e n d e n t d i s p l a c e m e n t s w o u l d b e o b s e r v e d .

F o r / = 1, to = 0 a l s o . T h i s t y p e of d i s p l a c e m e n t of t h e s u r f a c e w o u l d

c o r r e s p o n d t o r = a + ax c o s 0 w h i c h w o u l d c o r r e s p o n d t o a n o v e r a l l

d i s p l a c e m e n t of t h e s p h e r e , a n d c o u l d n o t b e d e t e c t e d b y a n o b s e r v e r a t

t h e s u r f a c e ( s e e F i g . 7 .4) .

T h u s , t h e l o w e s t o b s e r v a b l e o s c i l l a t i o n w o u l d c o r r e s p o n d t o / = 2 , o r

r = a + a 2 ( 3 c o s 2 0 - 1 ) ( w e wi l l i g n o r e t h e d e p e n d e n c e o n <j> f o r s i m p l i c -

i t y ) . T h i s c o r r e s p o n d s t o a d i s t o r t i o n s u c h a s t h a t s h o w n in F i g . 7 . 5 , w h i c h

h a s a f r e q u e n c v

Fig. 7.5. The lowest observable oscillation for a liquid sphere.

T h i s c o r r e s p o n d s t o a t i m e b e t w e e n p u l s e s a t t h e s u r f a c e of t h e e a r t h of a b o u t _

w h i c h is c l o s e t o t h e 3 - 6 0 m i n u t e p u l s e s o b s e r v e d a f t e r t h e C h i l e a n e a r t h q u a k e of 1960!

B. THE LIQUID DROP MODEL OF THE NUCLEUS

T h r o u g h o u t t h i s t e x t , w e h a v e e m p h a s i z e d t h e f a c t t h a t h y d r o d y n a m i c s

is a s u b j e c t w h i c h c a n b e a p p l i e d o v e r a w i d e r a n g e of p h y s i c a l

p h e n o m e n a . P e r h a p s n o w h e r e is t h a t f a c t s o s u r p r i s i n g a s in t h e

r e a l i z a t i o n t h a t s o m e of t h e e a r l i e s t i d e a s a b o u t t h e a t o m i c n u c l e u s w e r e


b a s e d o n c o n c e p t s of fluid m e c h a n i c s . I t is t o t h e s e m o d e l s t h a t w e n o w

t u r n o u r a t t e n t i o n .

I t m a y s e e m s t r a n g e t h a t t h e c l a s s i c a l t h e o r y of f luids s h o u l d h a v e

a n y t h i n g t o d o w i t h n u c l e a r e f f e c t s , b u t a c t u a l l y it is n o t . T h e p r o b l e m of

d e s c r i b i n g a n u c l e u s m a d e u p of m a n y i n t e r a c t i n g n u c l e o n s i s in m a n y

w a y s s imi l a r t o t h e p r o b l e m of d e s c r i b i n g a g a s m a d e u p of m a n y

i n t e r a c t i n g p a r t i c l e s . If o n e d o e s n o t w a n t t o g e t i n v o l v e d in t h e

i m p o s s i b l e p r o b l e m of d e s c r i b i n g t h e m o t i o n of e a c h p a r t i c l e in d e t a i l , o n e

t r e a t s t h e s y s t e m a s a n e n s e m b l e , a n d d i s c u s s e s o n l y t h e g r o s s p r o p e r t i e s ,

i g n o r i n g t h e d e t a i l e d s t r u c t u r e a s m u c h a s p o s s i b l e . I n t h e c a s e of t h e

l i qu id , o n e u s e s t h e r m o d y n a m i c s o r fluid m e c h a n i c s . S i n c e a fluid is t h e

s i m p l e s t s y s t e m in w h i c h t h i s a v e r a g i n g p r o c e s s i s d o n e , i t i s n a t u r a l t o t r y

t o a p p r o x i m a t e a n y s y s t e m w i t h a c o m p l e x i n t e r n a l s t r u c t u r e b y a fluid.

T h e l i q u i d d r o p m o d e l r e p r e s e n t s s u c h a z e r o - o r d e r a p p r o x i m a t i o n t o t h e

b e h a v i o r of l a r g e n u c l e i .

I n t h e d i s c u s s i o n of s u r f a c e t e n s i o n , w e s h o w e d h o w t h e e x i s t e n c e of

a n a t t r a c t i v e f o r c e b e t w e e n t h e c o n s t i t u e n t p a r t i c l e s of a l i q u i d g a v e r i s e

t o a s u r f a c e f o r c e . A n u c l e u s is m a d e u p of p r o t o n s a n d n e u t r o n s , s o t h a t

if t h e r e w e r e n o f o r c e s p r e s e n t o t h e r t h a n e l e c t r o m a g n e t i c o n e s , t h e

n u c l e u s w o u l d h a v e t o fly a p a r t b e c a u s e of t h e C o u l o m b r e p u l s i o n

b e t w e e n p r o t o n s . T h e e x i s t e n c e of n u c l e i is t h u s e v i d e n c e f o r t h e

e x i s t e n c e of s h o r t - r a n g e a t t r a c t i v e f o r c e s b e t w e e n t h e n u c l e o n s . ( T h e s e

a r e t h e " s t r o n g i n t e r a c t i o n s " w h i c h c o n s t i t u t e o n e of t h e m a j o r fields of

i n v e s t i g a t i o n in m o d e r n p h y s i c s . ) S u c h a f o r c e w o u l d , of c o u r s e , g i v e r i s e

t o a s u r f a c e t e n s i o n in t h e n u c l e a r " f l u i d . " T h e s t a b i l i t y of t h e n u c l e u s is

t h u s s e e n t o b e a r e s u l t of t h e c o m p e t i t i t o n b e t w e e n t h e C o u l o m b

e l e c t r o s t a t i c f o r c e s , w h i c h t e n d t o b l o w t h e n u c l e u s a p a r t , a n d t h e s t r o n g

i n t e r a c t i o n s g i v i n g r i s e t o a s u r f a c e t e n s i o n , w h i c h t e n d s t o h o l d t h e

n u c l e u s t o g e t h e r . ( T h e s e t w o f o r c e s p l a y s i m i l a r r o l e s t o g r a v i t y a n d

c e n t r i f u g a l f o r c e , w h o s e c o m p e t i t i o n w a s t h e m a i n p o i n t of i n v e s t i g a t i o n

in o u r s t u d y of s t a r s in C h a p t e r 2.)

I n o u r d i s c u s s i o n of s t a b i l i t y in C h a p t e r 3 , w e s a w t h a t o n e w a y t o

d e c i d e w h e t h e r a s y s t e m is s t a b l e a g a i n s t s o m e p e r t u r b a t i o n is t o s e e

w h e t h e r t h a t p e r t u r b a t i o n i n c r e a s e s o r d e c r e a s e s t h e e n e r g y of t h e

s y s t e m . T h e r e f o r e , l e t u s c o n s i d e r t h e s t ab i l i t y of n u c l e i b y c o n s i d e r i n g t h e

d e f o r m a t i o n of a n u c l e u s w h o s e r a d i u s w h e n u n d i s t u r b e d is a , a n d w h o s e

s t r o n g i n t e r a c t i o n s g i v e r i s e t o a s u r f a c e t e n s i o n T. L e t u s t a k e a n a r b i t r a r y

d e f o r m a t i o n of t h e s u r f a c e s u c h a s t h a t s h o w n in F i g . 7 . 1 , s o t h a t t h e

d i s t a n c e f r o m t h e c e n t e r t o t h e s u r f a c e is j u s t

(7 .B.1)


A s in t h e p r o b l e m in t h e p r e c e d i n g s e c t i o n in w h i c h t h e p o t e n t i a l of a

d e f o r m e d s p h e r e w a s c a l c u l a t e d , w e sha l l r e p l a c e t h e d e f o r m e d s p h e r e b y

a s p h e r e of r a d i u s a a n d a s p h e r i c a l she l l w h o s e t h i c k n e s s is s m a l l

c o m p a r e d t o t h e r a d i u s of t h e s p h e r e ( s e e F i g . 7 .2) . T h i s p r o b l e m i s s i m i l a r

in m a n y r e s p e c t s t o t h e c a l c u l a t i o n of t h e p o t e n t i a l of t h e d e f o r m e d e a r t h

in S e c t i o n 7 .A , b u t in c a l c u l a t i n g s t a b i l i t y in t h e w a y w e a r e d o i n g i t , w e

wi l l b e c o n c e r n e d w i t h t h e e n e r g y of a c h a r g e d i s t r i b u t i o n in t h e p o t e n t i a l ,

a n d n o t in t h e p o t e n t i a l i tself .

T h e C o u l o m b e n e r g y c a n b e w r i t t e n

Ec = j p f t d V , (7 .B .7)

w h e r e t h e i n t e g r a t i o n is u n d e r s t o o d t o e x t e n d o v e r t h e e n t i r e d e f o r m e d

s p h e r e . W e wil l find it e a s i e r t o t r e a t t h e s y s t e m a s if t h e t h i n she l l a n d t h e

T h i s d i s t o r t i o n h a s t w o c o m p e t i n g e f f ec t s . F i r s t , b y i n c r e a s i n g t h e

s u r f a c e a r e a , w e i n c r e a s e t h e s u r f a c e e n e r g y , w h i c h is g i v e n b y

Es = TS, (7 .B.2)

w h e r e S i s t h e t o t a l s u r f a c e a r e a , a n d , s e c o n d , w e m o v e t h e c h a r g e s

f a r t h e r a p a r t f r o m e a c h o t h e r , s o w e d e c r e a s e t h e C o u l o m b e f f ec t s . T h e

i n t e r p l a y b e t w e e n t h e s e t w o e f f ec t s wi l l d e t e r m i n e t h e s t a b i l i t y of t h e

s y s t e m .

I n P r o b l e m 7 . 3 , it is s h o w n t h a t t h e s u r f a c e a r e a of a s p h e r e d e f o r m e d

a c c o r d i n g t o E q . (7 .B .1) is

(7 .B.3)

If w e w r i t e t h e s u r f a c e e n e r g y of t h e u n d e f o r m e d s p h e r e a s

Es° = 4ira2T, (7 .B.4)

t h e n t h e c h a n g e in s u r f a c e e n e r g y a c c o m p a n y i n g d e f o r m a t i o n is j u s t

(7 .B.5)

F o r t h e p u r p o s e of c a l c u l a t i n g t h e C o u l o m b e n e r g y of t h e d e f o r m e d

s p h e r e , w e a s s u m e t h a t t h e t o t a l c h a r g e of t h e n u c l e u s ( w h i c h w e sha l l ca l l

Ze, w h e r e e i s t h e c h a r g e o n a s ing le p r o t o n ) is s p r e a d o u t u n i f o r m l y o v e r

t h e s p h e r e , s o t h a t t h e c h a r g e d e n s i t y is j u s t

(7 .B .6)


s p h e r e w e r e t w o s e p a r a t e e n t i t i e s . I n t h i s c a s e , t h e C o u l o m b e n e r g y w o u l d

b e m a d e u p of t h r e e t e r m s : t h e s e l f - e n e r g y of t h e s p h e r e , g i v e n b y

w h e r e w e h a v e u s e d t h e o r t h o g o n a l i t y p r o p e r t i e s of t h e s p h e r i c a l

h a r m o n i c s t o e l i m i n a t e t h e l i n e a r t e r m in a\m a n d t o c o l l a p s e t h e d o u b l e

s u m in t h e q u a d r a t i c t e r m ( s e e A p p e n d i x F ) .

(7 .B .8)

t h e s e l f - e n e r g y of t h e she l l , g i v e n b y

(7 .B.9)

a n d t h e i n t e r a c t i o n e n e r g y b e t w e e n t h e s p h e r e a n d t h e she l l , g i v e n b y

(7 .B .10)

w h e r e w e h a v e w r i t t e n t h e p o t e n t i a l of t h e s p h e r e a s f l , t h e p o t e n t i a l of

t h e she l l a s fls, a n d t h e d e n s i t y of t h e she l l ( s e e F i g . 7.3) a s p s .

T h e d i f f e r e n c e b e t w e e n t h e C o u l o m b e n e r g y in t h e u n d i s t o r t e d s t a t e

a n d t h e d i s t o r t e d s t a t e is t h e n

A E C = E 2 + E 3 . (7 .B .11)

T h e c a l c u l a t i o n of E3 is r e l a t i v e l y s i m p l e . T h e p o t e n t i a l a t t h e she l l d u e

t o t h e s p h e r e is j u s t

(7 .B .12)

(7 .B.13)

s o t h a t

w h e r e dco r e p r e s e n t s t h e i n t e g r a l o v e r t h e so l id a n g l e . N o t e t h a t in t h e

e v e n t R < a, t h e i n t e g r a l o v e r r' wi l l c h a n g e s ign , s o t h a t w e n e e d n o t

w o r r y a b o u t w h e t h e r t h e p e r t u r b a t i o n p u s h e s R o u t o r p u l l s it in . C a r r y i n g

o u t t h e i n t e g r a l o v e r t h e r a d i a l v a r i a b l e ,

(7 .B.14)


Fig. 7.6. Coordinates for calculating a Coulomb potential for deformed nuclei.

T h e c a l c u l a t i o n of E2, t h e s e l f - e n e r g y of t h e s h e l l , c a n b e sp l i t i n t o t w o

p a r t s — t h e c a l c u l a t i o n of H s , t h e p o t e n t i a l a t a p o i n t in t h e she l l d u e t o t h e

r e s t of t h e she l l , a n d t h e n t h e c a l c u l a t i o n of E2 i tself . F r o m F i g . 7 .6 , w e

s e e t h a t w e c a n w r i t e

(7 .B .15)

W e s e e t h a t fls wil l d e p e n d o n R-a, h e n c e wil l b e l i n e a r in t h e s m a l l

p a r a m e t e r a l m . S i n c e in t h e c a l c u l a t i o n of E 3 , n s wi l l a p p e a r i n s i d e a n o t h e r

i n t e g r a l w h i c h wi l l d e p e n d o n R-a, it wi l l b e suff ic ient t o k e e p o n l y

l o w e s t - o r d e r t e r m s in t h e a b o v e e x p r e s s i o n .

P r o c e e d i n g a s in t h e s t e p s l e a d i n g t o E q . (7 .A . 12), w e find

(7 .B.16)

s o t h a t

(7 .B .17)

C o m b i n i n g E q s . (7 .B .3 ) , (7 .B .14) , a n d (7 .B .17 ) , w e find t h a t t h e t o t a l

e n e r g y c h a n g e in t h e s y s t e m w h e n a n in f in i t e s ima l d e f o r m a t i o n t a k e s

p l a c e is

(7 .B.18)

T h i s wi l l b e p o s i t i v e o r n e g a t i v e , d e p e n d i n g o n w h e t h e r t h e s e c o n d t e r m

( r e p r e s e n t i n g t h e C o u l o m b e n e r g y ) i s g r e a t e r o r l e s s t h a n t h e first


W e k n o w , of c o u r s e , f r o m t h e d i s c u s s i o n of C h a p t e r 3 t h a t if t h e s y s t e m is

u n s t a b l e a g a i n s t o n e p e r t u r b a t i o n , t h e n it wi l l n o t b e a b l e t o s u r v i v e in

n a t u r e .

T h u s , t h e n u c l e u s wi l l b e u n s t a b l e if t h e r e l a t i v e a m o u n t of p r o t o n s , Z ,

b e c o m e s l a r g e c o m p a r e d t o t h e n u m b e r of p r o t o n s a n d n e u t r o n s , A , w h i c h

g i v e r i s e t o t h e s u r f a c e t e n s i o n . I n P r o b l e m 7.4, it is s h o w n t h a t t h i s l e a d s

t o a p r e d i c t i o n f o r t h e l a r g e s t s t a b l e n u c l e u s w h i c h is p o s s i b l e in n a t u r e . If

t h e s t a b i l i t y c r i t e r i o n is n o t m e t , t h e n w e e x p e c t t h a t t h e n u c l e u s wi l l

u n d e r g o l a r g e o s c i l l a t i o n s a n d e v e n t u a l l y b r e a k u p . T h i s i s k n o w n a s

spontaneous fission, a n d will b e d i s c u s s e d in t h e n e x t s e c t i o n . I t is o n e

p r o c e s s w h i c h g i v e s r i s e t o n a t u r a l r a d i o a c t i v i t y .

Of c o u r s e , s p o n t a n e o u s f i s s ion is o n l y o n e k i n d of i n s t a b i l i t y t h a t a

n u c l e u s c a n h a v e , a n d o n l y a f e w n u c l e i in n a t u r e a c t u a l l y e x h i b i t i t . O t h e r

k i n d s of i n s t a b i l i t i e s w h i c h w o u l d b r e a k u p a n u c l e u s a r e p r o c e s s e s in

w h i c h t h e n u c l e u s w o u l d e m i t a n y of a n u m b e r of p a r t i c l e s . S u c h

p r o c e s s e s m u s t b e t r e a t e d q u a n t u m m e c h a n i c a l l y , h o w e v e r , a n d a r e n o t

i n c l u d e d in t h e l i qu id d r o p m o d e l .

F i n a l l y , w e n o t e t h a t t h e s t a b i l i t y c r i t e r i o n in E q . (7 .B .20) c a n b e w r i t t e n

( r e p r e s e n t i n g t h e s u r f a c e t e n s i o n ) . T h i s i s w h a t w e e x p e c t e d w h e n w e

r e m a r k e d e a r l i e r t h a t t h e s t a b i l i t y of t h e s y s t e m w o u l d d e p e n d o n t h e

i n t e r p l a y b e t w e e n t h e s e t w o f o r c e s .

L e t u s e x a m i n e t h i s s t a b i l i t y c r i t e r i o n a s a f u n c t i o n of t h e t o t a l n u m b e r

of p r o t o n s a n d n e u t r o n s in t h e n u c l e u s . W e c a n w r i t e

w h e r e pA i s t h e d e n s i t y of n u c l e o n s in t h e n u c l e u s . T h i s l e a d s u s t o e x p e c t

t h a t t h e r a d i u s of t h e n u c l e u s , a, s h o u l d b e r e l a t e d t o t h e n u c l e a r n u m b e r

A ,

a = r 0 A v \ (7 .B .19)

T h i s i s , in f a c t , t h e o b s e r v e d l a w of n u c l e a r s i z e , a n d t h e c o n s t a n t r 0 is

g e n e r a l l y g i v e n a v a l u e of a b o u t 1.2 x 1 0 - 1 3 c m . F o r / = 2 d e f o r m a t i o n s ,

t h i s m e a n s t h a t t h e e x p r e s s i o n f o r A E in E q . (7 .B .18) wil l b e p o s i t i v e , a n d

h e n c e t h e n u c l e u s wi l l b e s t a b l e , o n l y if

(7 .B .20)

Nuclear Fission 117

w h e r e E i a n d Es° a r e g i v e n in E q s . (7 .B .8) a n d (7 .B .4 ) . T h e p a r a m e t e r x i s

c a l l e d t h e fissionability parameter, a n d i s s o m e t i m e s u s e d in d i s c u s s i o n s

of fission.

C. NUCLEAR FISSION

T h e p r o b l e m of t h e fission of h e a v y e l e m e n t s h a s , u n t i l v e r y r e c e n t l y ,

de f i ed t h e o r e t i c a l a n a l y s i s . Y e t t h e u s e of t h e fission p r o c e s s in r e a c t o r s

h a s b e e n w i d e s p r e a d . L e t u s u s e t h e l i q u i d d r o p m o d e l of t h e n u c l e u s

d e v e l o p e d in t h e p r e v i o u s s e c t i o n t o s e e if w e c a n c o m e t o s o m e

q u a l i t a t i v e u n d e r s t a n d i n g of h o w e n e r g y c a n b e d e r i v e d f r o m fission.

L e t u s c o n s i d e r w h a t h a p p e n s w h e n , f o r s o m e r e a s o n , a n u c l e u s i s sp l i t

u p . R e m e m b e r t h a t any n u c l e u s c a n b e sp l i t u p . T h e s t a b i l i t y c r i t e r i o n j u s t

t e l l s u s w h i c h n u c l e i wi l l n o t b r e a k u p s p o n t a n e o u s l y . W e c a n a s k first

w h a t k i n d s of b r e a k u p a r e e n e r g e t i c a l l y f a v o r e d ; i .e . w h i c h p o s s i b l e final

s t a t e h a s t h e l o w e s t e n e r g y . L e t u s a s s u m e f o r s i m p l i c i t y t h a t t h e n u c l e u s

b r e a k s u p i n t o t w o f r a g m e n t s , o n e w i t h N n u c l e o n s , a n d t h e o t h e r w i t h

A-N. L e t u s a s s u m e t h a t t h e final p r o d u c t i s t h e t w o s p h e r e s s e p a r a t e d b y

a g r e a t d i s t a n c e . L e t u s a l s o , a s a first a p p r o x i m a t i o n , i g n o r e t h e C o u l o m b

e n e r g y in t h e final s t a t e [ th i s wi l l b e a g o o d a p p r o x i m a t i o n u n l e s s w e a r e

c l o s e t o t h e s t a b i l i t y l imi t ( s e e P r o b l e m 7 .6) ] . T h e n t h e final e n e r g y of t h e

s y s t e m a f t e r t h e sp l i t u p wi l l b e

Ef = Airro2T[Nm + ( A - N ) 2 / 3 ] , ( 7 . C . 1)

s o t h a t t h e n e t e n e r g y c h a n g e is

A E = 4 W T A 2 / 3 - E / .

T h i s wil l b e a m i n i m u m w h e n

o r

(7 .C .2)

T h u s , t h e l i q u i d d r o p m o d e l p r e d i c t s t h a t w h e n a n u c l e u s b r e a k s u p , it

s h o u l d sp l i t i n t o t w o e q u a l - s i z e d f r a g m e n t s . T h i s i s a c t u a l l y n o t t h e c a s e

(e .g . , w h e n u r a n i u m u n d e r g o e s fission, t h e e n d p r o d u c t s a r e c l u s t e r e d s o

t h a t w h e n o n e f r a g m e n t is a r o u n d A = 90 t h e o t h e r is a r o u n d

A = 140). T h i s is o n e of t h e m a i n diff icul t ies of t h e l i qu id d r o p

m o d e l — o n e of i t s f a i l u r e s . H o w e v e r , t h e q u e s t i o n of w h y n u c l e i s h o u l d


g o t o u n e q u a l f r a g m e n t s h a s b e e n t h e s u b j e c t of a l o n g i n v e s t i g a t i o n in t h e

t h e o r y of h e a v y n u c l e i , a n d t h e s o l u t i o n s t o t h e p r o b l e m w h i c h h a v e b e e n

a d v a n c e d d e p e n d in a v e r y c r i t i c a l w a y o n d e t a i l s of t h e q u a n t u m

m e c h a n i c s of m a n y b o d y s y s t e m s .

W e wi l l stil l u s e t h e m o d e l , h o w e v e r , b e c a u s e a l t h o u g h it is w r o n g in

s o m e d e t a i l s , it n e v e r t h e l e s s r e p r o d u c e s t h e g e n e r a l f e a t u r e s of n u c l e a r

s t r u c t u r e q u i t e w e l l in a v e r y s i m p l e w a y .

L e t u s s u p p o s e t h a t a n u c l e u s sp l i t s u p , t h e n , i n t o t w o e q u a l f r a g m e n t s ,

e a c h w i t h ha l f t h e p r o t o n s a n d n e u t r o n s of t h e p a r e n t n u c l e u s . W h a t is t h e

final e n e r g y of t h e s y s t e m ?

T h u s , t h e e n e r g y a s s o c i a t e d w i t h t h e b r e a k u p of t h e s y s t e m c a n b e w r i t t e n

in t e r m s of t h e fissionability p a r a m e t e r a s

s o t h a t w h e n x > 0 .35 t h e s y s t e m c a n g o t o a final s t a t e of l o w e r e n e r g y

t h a n t h e o r ig ina l s t a t e . B u t t h i s i s c o n f u s i n g , b e c a u s e w e h a v e s h o w n

a b o v e t h a t t h e s y s t e m is s t a b l e a g a i n s t s m a l l p e r t u r b a t i o n s un t i l x > 1.

H o w c a n t h e s e t w o s e e m i n g l y c o n t r a d i c t o r y r e s u l t s b e r e c o n c i l e d ?

T h e a n s w e r , of c o u r s e , is t h a t t h e r e s u l t s o n s t ab i l i t y te l l u s w h a t

h a p p e n s w h e n small p e r t u r b a t i o n s a r e a p p l i e d t o t h e s y s t e m . H o w e v e r , in

o r d e r f o r b r e a k u p t o o c c u r , t h e p e r t u r b a t i o n s m u s t b e v e r y l a r g e i n d e e d .

S c h e m a t i c a l l y , w e c a n i m a g i n e t h e t o t a l e n e r g y of t h e s y s t e m a s a

f u n c t i o n of p e r t u r b a t i o n p a r a m e t e r t o l o o k l ike F i g . 7 .7 . F o r s m a l l

p e r t u r b a t i o n s , t h e s y s t e m is s t a b l e . If a n a m o u n t of e n e r g y E f i s s i on is a d d e d t o

t h e s y s t e m , h o w e v e r , it wi l l b e a b l e t o o v e r c o m e t h e p o t e n t i a l b a r r i e r , a n d

(7 .C.3)

= E s ° ( 0 . 2 6 - 0 . 7 4 x ) , (7 .C.4)

E

a,

Fig. 7.7. The energy surface for a fissionable nucleus.

Problems 119

fall t o t h e s t a t e d i s c u s s e d a b o v e , w h i c h h a s a n e n e r g y A E B b e l o w t h e in i t ia l

s t a t e . T h i s is c a l l e d induced fission.

M o s t of t h e c u r r e n t r e s e a r c h o n f i s s ion h a s t o d o w i t h m a p p i n g o u t t h e

c o m p l i c a t e d e n e r g y s u r f a c e s w h i c h c o r r e s p o n d t o v a r i o u s d e f o r m a t i o n s

of t h e n u c l e u s , a n d t h e n t r y i n g t o d e c i d e h o w fission wi l l p r o c e e d f o r r e a l

n u c l e i .

I n d u c e d fission is t h e b a s i c p r i n c i p l e b y w h i c h a fission r e a c t o r w o r k s .

If t h e n u c l e u s in q u e s t i o n is U 2 3 5 , t h e n a n e u t r o n s t r i k i n g t h e n u c l e u s c a n

s u p p l y t h e e n e r g y n e e d e d t o p u t t h e n u c l e u s o v e r t h e t o p of t h e p o t e n t i a l

b a r r i e r . T h e e n e r g y r e l e a s e d is t h e n A E B ( s e e P r o b l e m 7.5) . S i n c e s o m e of

t h i s e n e r g y is r e l e a s e d in t h e f o r m of n e u t r o n s , w h i c h c a n , in t u r n , i n i t i a t e

f u r t h e r r e a c t i o n s , it i s p o s s i b l e t o s u s t a i n a c o n t i n u o u s fission p r o c e s s

f r o m w h i c h e n e r g y c a n b e e x t r a c t e d .

T h i s d i s c u s s i o n i l l u s t r a t e s a n i m p o r t a n t p o i n t a b o u t h y d r o d y n a m i c s

( a n d , i n d e e d , a b o u t a n y s y s t e m d e s c r i b e d b y n o n l i n e a r e q u i l i b r i u m ) . T h e

b e h a v i o r of t h e s y s t e m c l o s e t o e q u i l i b r i u m n e e d n o t b e r e l a t e d s i m p l y t o

t h e b e h a v i o r of t h e s y s t e m f a r f r o m e q u i l i b r i u m . T h i s a s p e c t of t h e

p h y s i c a l w o r l d is o n l y n o w b e g i n n i n g t o b e e x p l o r e d , a n d v e r y l i t t le i s

k n o w n a b o u t it a t p r e s e n t .

SUMMARY

A p p l i c a t i o n of t h e p r i n c i p l e s of fluids w h i c h w e r e d e v e l o p e d in

p r e v i o u s c h a p t e r s t o s p h e r i c a l fluid s y s t e m s l e a d s t o t w o i n t e r e s t i n g

p r e d i c t i o n s . F i r s t , a fluid ( s u c h a s t h e e a r t h ) a c t i n g u n d e r t h e i n f l u e n c e of

i t s o w n g r a v i t y wil l e x e c u t e p e r i o d i c v i b r a t i o n s a b o u t e q u i l i b r i u m

if d e f o r m e d a n d t h e n r e l e a s e d . S e c o n d , a c h a r g e d fluid u n d e r t h e in -

fluence of s u r f a c e t e n s i o n ( s u c h a s a n u c l e u s ) wi l l fission s p o n t a n e o u s l y f o r

c e r t a i n v a l u e s of t h e c h a r g e . T h i s w a s u s e d t o d i s c u s s t h e p r o c e s s of

n u c l e a r fission, w h i c h is a n e x a m p l e of a p r o c e s s in w h i c h d e v i a t i o n s f r o m

t h e s m a l l p e r t u r b a t i o n , l i n e a r t h e o r y w h i c h w e h a v e b e e n p r e s e n t i n g a r e

i m p o r t a n t .

PROBLEMS

7 .1. L o o k up the addition theorem for spherical harmonics , and use it, together with the proper t ies of the spherical harmonics discussed in Appendix F , to show that

Y L M ( 0 ' , 0 ' ) P / ( c o s y ) d ( c o s 6) dcf> •


7.2. Given that the gravitat ional at t ract ion at the surface of the moon is approximately \ tha t at the ear th , es t imate the period of the I = 2 free oscillations of the moon. H o w do they compare with those of the ear th?

7.3. Le t us consider the surface of a deformed sphere ,

(a) F r o m Eq . (7.B.1), show that

H e n c e show that the direction cosines of the deformed surface are

where

(b) H e n c e show that the change in surface area of an infinitesimal volume

element is

(c) Integrate to obtain the surface area in the form

7 .4. Find a good value to the surface tension T of a nucleus , and calculate the largest value of Z 2 / A which a nucleus can have and still be stable. H o w does this compare to the actual stability of heavy e lements?

7.5. Calculate the energy which will be released if the nucleus U235 is made to

undergo fission, assuming that the liquid drop model is correct in stating that the

final s tate will b e t w o identical nuclei . H o w does this compare to the actual value

of this number?

7.6. Show that including the Coulomb effect in Eq . (7.C.1) will not affect the

conclusion of E q . (7.C.2) for heavy nuclei. (Hint: Wha t is the relation be tween A

and Z around uranium?)

7.7. Verify the express ion for Ex in Eq . (7.B.8).

7.8. Suppose that the ear th had a total charge Q spread uniformly through its volume. H o w would Eq . (7.A. 17) be al tered? Are there values of Q for which the frequencies of vibrat ion will be complex , and therefore represent an instability? Relate this to the resul ts of Section 7.B.

References 121

REFERENCES

K. E. Bullen, An Introduction to the Theory of Seismology, Cambridge, U.P., 1965. Chapter 14 gives the theory of oscillation for a solid earth, and a survey of observations.

Lawrence Willets, Theories of Nuclear Fission, Clarendon Press, Oxford, 1964 A survey of nuclear fission. This should give an overview of the field.

M. Brack, J. Damgaard, A. S. Jensen, H. C. Pauli, V. M. Strutinsky, C. Y. Wong, Reviews of Modern Physics 44, 320 (1972). A review of the latest ideas in the theory of fission.

I. Prigogine, G. Nicolis, and A. Babloyantz, Physics Today 25, numbers 11 and 12 (1972). They give a discussion of how a living system which is far from the equilibrium of its consituents might arise by processes similar to that considered in Section 7.C.

8

Viscosity in Fluids

Slow as molasses in January

Southern folk saying

A. THE IDEA OF VISCOSITY

U p t o t h i s p o i n t , w e h a v e i g n o r e d m a n y of t h e p r o p e r t i e s of r e a l f luids

w h i c h m i g h t s e r v e t o c o m p l i c a t e o u r c o n s i d e r a t i o n s of s i m p l e s y s -

t e m s . W e h a v e a r g u e d t h a t t h i s is a va l i d w a y t o p r o c e e d in m a n y

c a s e s . A s m i g h t b e e x p e c t e d , h o w e v e r , t h e r e a r e m a n y p h e n o m e n a fo r

w h i c h t h e " i d e a l f l u id" wi l l s i m p l y n o t p r o v i d e a n a d e q u a t e d e s c r i p t i o n .

I n a n i d e a l fluid, t h e o n l y w a y in w h i c h a f o r c e c a n b e g e n e r a t e d o r ,

e q u i v a l e n t l y , in w h i c h m o m e n t u m c a n b e t r a n s f e r r e d , is t h r o u g h t h e

p r e s s u r e g r a d i e n t . O n t h e a t o m i c l e v e l , t h i s c o r r e s p o n d s t o c o l l i s i o n s in

w h i c h t h e m o m e n t u m of a m o l e c u l e in t h e d i r e c t i o n of t h e f o r c e is

r e v e r s e d . C l e a r l y , a f o r c e of t h i s t y p e m u s t a l w a y s b e n o r m a l t o t h e

s u r f a c e o n w h i c h it is b e i n g e x e r t e d . I n a d d i t i o n , if w e w e r e s o m e h o w a b l e

t o r e a c h i n t o a n i d e a l fluid a n d a p p l y a f o r c e t o a s ing le fluid e l e m e n t , t h e r e

w o u l d b e n o t h i n g o t h e r t h a n p r e s s u r e g r a d i e n t s t o o p p o s e t h e m o t i o n of

t h e e l e m e n t , s o t h a t it c o u l d b e q u i c k l y a c c e l e r a t e d .

T o s e e t h e s h o r t c o m i n g s of t h i s d e s c r i p t i o n of a fluid, c o n s i d e r t h e

f o l l o w i n g e x a m p l e : L e t t h e r e b e a fluid of d e p t h h w h i c h is n o t m o v i n g .

L e t a n o t h e r l a y e r of i d e n t i c a l fluid b e f lowing a c r o s s t h e t o p of t h e

s t a t i o n a r y l a y e r a t a v e l o c i t y v. F o r a c l a s s i c a l i d e a l fluid, t h e fluid in t h e

u p p e r l a y e r wi l l k e e p m o v i n g inde f in i t e ly , e v e n if n o f o r c e s a r e a c t i n g o n

122

The Idea of Viscosity 123

i t . O u r i n t u i t i o n t e l l s u s , h o w e v e r , t h a t in a r e a l s i t u a t i o n , t h e t o p l a y e r

w o u l d e v e n t u a l l y s l o w d o w n a n d s t o p . T h i s m e a n s t h a t t h e r e m u s t b e

s o m e w a y of e x e r t i n g f o r c e s w h i c h a r e d i f f e r en t f r o m t h e p r e s s u r e , a n d

w h i c h a c t along a s u r f a c e , r a t h e r t h a n n o r m a l t o i t .

T h e t e r m u s u a l l y u s e d t o d e s c r i b e s u c h a s i t u a t i o n is t h a t t h e fluid is

c a p a b l e of e x e r t i n g a shear force, in a d d i t i o n t o t h e p r e s s u r e . T h e

p h e n o m e n o n a s s o c i a t e d w i t h t h i s f o r c e i s c a l l e d viscosity.

T o u n d e r s t a n d h o w v i s c o s i t y w o r k s a t t h e a t o m i c l e v e l , c o n s i d e r a

c o l l i s i o n b e t w e e n t w o a t o m s in t h e a b o v e e x a m p l e . If o n l y p r e s s u r e f o r c e s

c o u l d b e e x e r t e d , t h e n m o m e n t u m t r a n s f e r s c o u l d o c c u r o n l y in a

d i r e c t i o n n o r m a l t o t h e i n t e r f a c e b e t w e e n t h e fluids, a n d t h e m o m e n t u m

of e a c h a t o m a l o n g t h e i n t e r f a c e w o u l d h a v e t o r e m a i n c o n s t a n t

( e s s e n t i a l l y , t h e a t o m in t h e m o v i n g fluid w o u l d r e t a i n , o n t h e a v e r a g e , a

v e l o c i t y v). W h e n w e p u t t h i n g s t h i s w a y , it is c l e a r t h a t t h e a s s u m p t i o n s

a s s o c i a t e d w i t h i d e a l fluids a r e r a t h e r ar t i f ic ia l . S u p p o s e w e t h o u g h t a b o u t

a m o r e r e a l i s t i c a t o m i c p i c t u r e , in w h i c h m o m e n t u m c o u l d b e t r a n s f e r r e d

in a n y d i r e c t i o n . T h e n t h e a t o m s in t h e l o w e r l a y e r w o u l d , o n t h e a v e r a g e ,

b e s p e e d e d u p b y c o l l i s i o n s , w h i l e t h e a t o m s in t h e u p p e r l a y e r w o u l d , o n

t h e a v e r a g e , b e s l o w e d d o w n . T h e n e t r e s u l t w o u l d b e t h a t t h e r e l a t i v e

v e l o c i t y b e t w e e n t h e t w o l a y e r s w o u l d b e r e d u c e d ( e v e n t u a l l y ) t o z e r o .

T h i s m e c h a n i s m is s imi l a r t o t h e p h e n o m e n o n of f r i c t i o n in m e c h a n i c s .

I n o r d e r t o c o m e t o s o m e b a s i c u n d e r s t a n d i n g of v i s c o s i t y , l e t u s r e t u r n

t o t h e d e r i v a t i o n s of t h e E u l e r e q u a t i o n in C h a p t e r 1, in w h i c h N e w t o n ' s

s e c o n d l a w of m o t i o n w a s a p p l i e d t o a n in f in i t e s ima l e l e m e n t of t h e fluid

t o g i v e t h e e q u a t i o n

(8 .A.1)

F r o m t h e p o i n t of v i e w of t h e v o l u m e e l e m e n t o n w h i c h t h e v a r i o u s

f o r c e s ( p r e s s u r e , g r a v i t y , e t c . ) a r e a c t i n g , t h e e x i s t e n c e of v i s c o s i t y wi l l b e

a n a d d i t i o n a l w a y in w h i c h t h e m o m e n t u m of t h e e l e m e n t c a n b e c h a n g e d ,

o r , b y N e w t o n ' s s e c o n d l a w , a n a d d i t i o n a l f o r c e . T o s e e w h y t h i s

s h o u l d b e s o , c o n s i d e r a n e l e m e n t in t h e m o v i n g fluid w e d i s c u s s e d e a r l i e r .

B e c a u s e of t h e c o l l i s i o n s b e t w e e n m o v i n g a n d s t a t i o n a r y a t o m s , it w o u l d

e x p e r i e n c e a n e t d e c e l e r a t i o n . T o a n o b s e r v e r o n t h e e l e m e n t w h o k n e w

n o t h i n g of a t o m i c s t r u c t u r e , t h i s w o u l d a p p e a r t o b e d u e t o s o m e s o r t of

i n t e r n a l f o r c e g e n e r a t e d w i t h i n t h e fluid, j u s t a s t h e f r i c t i o n a l f o r c e

g e n e r a t e d w h e n a b l o c k of w o o d s l i d e s a c r o s s a t a b l e s l o w s d o w n t h e

b l o c k . ( A s a m a t t e r of h i s t o r i c a l i n t e r e s t , a c o m m o n w a y of t h i n k i n g a b o u t

v i s c o s i t y in c l a s s i c a l t e r m s is t o i m a g i n e t h e fluid flow a s b e i n g m a d e u p of

124 Viscosity in Fluids

(8 .A.5)

Fig. 8.1. A fluid element in a generalized volume enclosed in a surface S.

s o t h a t , if n o o u t s i d e f o r c e s a r e a c t i n g o n t h e fluid, t h e t o t a l f o r c e a c t i n g o n

t h e fluid is j u s t

(8 .A.4)

T h i s i s p u r e l y a f o r m a l o p e r a t i o n , b u t it t u r n s o u t t o b e e a s i e r t o d i s c u s s

t h e t e n s o r trlk t h a n t h e f o r c e i tself . I n a n y c a s e , if w e c a n d e t e r m i n e w h a t

t h e t e n s o r cr ik i s , t h e v i s c o u s f o r c e c a n b e d e r i v e d i m m e d i a t e l y . W e sha l l s e e

l a t e r ( C h a p t e r 12) t h a t o- ik is o n e e x a m p l e of a s t r e s s t e n s o r .

T o u n d e r s t a n d t h e p h y s i c a l s i gn i f i c ance of t h e t e n s o r cr ik, c o n s i d e r a

m a s s of f luid of v o l u m e V a n d s u r f a c e S ( s e e F i g . 8 .1) .

T h e t o t a l f o r c e p e r u n i t m a s s a c t i n g o n a v o l u m e e l e m e n t is j u s t

a s e r i e s of s h e e t s s l id ing o v e r e a c h o t h e r , a n d v i s c o s i t y a s b e i n g t h e

f r i c t i o n b e t w e e n t h e s h e e t s . )

T h e e x i s t e n c e of t h i s e x t r a f o r c e , o r m o m e n t u m t r a n s f e r , m e a n s t h a t

t h e r e m u s t b e a n a d d i t i o n a l t e r m in t h e E u l e r e q u a t i o n . F o r t h e s a k e of

d e f i n i t e n e s s , w e wi l l t r e a t v i s c o s i t y a s a f o r c e , a n d p u t it o n t h e r i g h t - h a n d

s i d e of E q . ( 8 .A .1 ) , b u t w e c o u l d j u s t a s w e l l t r e a t it a s a m o m e n t u m

c h a n g e , a n d p u t in o n t h e l e f t - h a n d s i d e . I s t h e r e a n y t h i n g w e c a n s a y

a b o u t t h e f o r m t h a t t h i s e x t r a t e r m in t h e E u l e r e q u a t i o n m u s t t a k e o n

g e n e r a l g r o u n d s ? I t t u r n s o u t t h a t t h e r e i s a g r e a t d e a l t h a t c a n b e s a i d .

T h e first t h i n g t h a t w e n o t e a b o u t t h e E u l e r e q u a t i o n in t h e f o r m

(8 .A.2)

is t h a t w e c a n a l w a y s w r i t e t h e e x t r a f o r c e a s

The Idea of Viscosity 125

w h e r e t h e l a s t s t e p , a s in E q . ( 1 .C .12 ) , f o l l o w s f r o m G a u s s ' l a w . T h e t e r m s

in t h e s u r f a c e i n t e g r a l s h o u l d l o o k f a m i l i a r . I n C h a p t e r 1, w e s a w t h a t t h e

t e r m

c o u l d b e i n t e r p r e t e d a s t h e s u m of t h e p r e s s u r e f o r c e s a c t i n g a c r o s s t h e

s u r f a c e of t h e fluid a n d t h e m o m e n t u m c a r r i e d a c r o s s t h e s u r f a c e b y t h e

fluid m o t i o n . T h e a d d i t i o n a l t e r m w h i c h w e n o w h a v e a d d e d ,

h a s a s i m i l a r i n t e r p r e t a t i o n . I t i s c l e a r l y j u s t t h e f o r c e e x e r t e d a c r o s s t h e

s u r f a c e S b y t h e v i s c o u s f o r c e s w h i c h a c t i n t h e fluid. I n m i c r o s c o p i c

t e r m s , it r e p r e s e n t s t h e m o m e n t u m t r a n s f e r r e d a c r o s s t h e s u r f a c e S b y

i n e l a s t i c c o l l i s i o n s of t h e a t o m s n e a r t h e s u r f a c e .

W e c a n l e a r n a g r e a t d e a l a b o u t t h e t e n s o r cr,k if w e a s k o u r s e l v e s t h e

q u e s t i o n " U n d e r w h a t c i r c u m s t a n c e s wi l l w e e x p e c t n o v i s c o u s f o r c e s t o

b e p r e s e n t ? " C l e a r l y , f r o m o u r p r e v i o u s d e s c r i p t i o n s , w e e x p e c t t h e

v i s c o u s f o r c e s t o b e a b s e n t w h e n e v e r t h e fluid i s m o v i n g in s u c h a w a y

t h a t t h e r e i s n o r e l a t i v e v e l o c i t y b e t w e e n d i f f e r en t p a r t s of t h e fluid, s i n c e

t h e n t h e r e w o u l d b e n o n e t g a i n o r l o s s of e n e r g y b y a n y p a r t of t h e fluid

d u e t o i n e l a s t i c a t o m i c c o l l i s i o n s . T h i s s i t u a t i o n c a n a r i s e in t w o w a y s :

(i) t h e fluid i s m o v i n g e v e r y w h e r e w i t h t h e s a m e v e l o c i t y u ;

(ii) t h e fluid is in a s t a t e of u n i f o r m r o t a t i o n , s o t h a t

w h e r e co is t h e r o t a t i o n a l f r e q u e n c y .

F r o m t h e a b s e n c e of v i s c o s i t y in t h e first c a s e , w e c o n c l u d e t h a t t h e

v i s c o u s f o r c e , a n d h e n c e t h e t e n s o r aik c a n n o t d e p e n d o n t h e v e l o c i t y

i tself , b u t m u s t d e p e n d o n t h e v e l o c i t y t h r o u g h t e r m s l i ke dUildxk a n d

d2UildxkdXj... w h i c h v a n i s h if t h e v e l o c i t y is a c o n s t a n t .

F r o m t h e s e c o n d c a s e , w e c o n c l u d e t h a t t h e t e n s o r m u s t v a n i s h if

u = to x r. T h e o n l y c o m b i n a t i o n s of d e r i v a t i v e s of t h e v e l o c i t y w h i c h

s a t i s f y t h e s e t w o c o n d i t i o n s a r e

(8 .A.6)

(8 .A.7)

u = w x r, (8 .A.8)

a n d


a n d , of c o u r s e , a l a r g e n u m b e r of t e r m s i n v o l v i n g s e c o n d a n d h i g h e r

d e r i v a t i v e s of t h e v e l o c i t y . W e h a v e n o r e a s o n t o e x p e c t t h a t s u c h t e r m s

wi l l n o t b e p r e s e n t in trik, b u t it is c l e a r t h a t o u r t h e o r y w o u l d b e m u c h

s i m p l e r if t h e v i s c o u s f o r c e s d e p e n d e d o n l y o n t h e first d e r i v a t i v e s of t h e

v e l o c i t y . T h e r e f o r e , f o l l o w i n g t h e l e a d of W i l l i a m of O c c a m , t w e wil l

a s s u m e t h a t w e a r e e n t i t l e d t o u s e t h e s i m p l e s t p o s s i b l e t h e o r y w e c a n

w r i t e d o w n ( c o n s i s t e n t w i t h t h e c o n d i t i o n s (i) a n d (ii) , of c o u r s e ) u n t i l w e

a r e f o r c e d t o d o o t h e r w i s e b y t h e d a t a . I n f a c t , it h a s b e e n f o u n d t h a t t h e

s i m p l e t h e o r y , in w h i c h t h e v i s c o u s f o r c e i s a s s u m e d t o d e p e n d o n l y o n

t h e first d e r i v a t i v e s of t h e v e l o c i t y , i s a p e r f e c t l y a d e q u a t e d e s c r i p t i o n of

t h e m o t i o n of f lu ids . A n a l t e r n a t e d e r i v a t i o n of t h i s r e s u l t is g i v e n in

P r o b l e m 12.7 in t e r m s of t h e s t r e s s t e n s o r .

T h i s m e a n s t h a t w e c a n w r i t e t h e m o s t g e n e r a l t e n s o r in t h e f o r m

(8 .A.9)

w h e r e t h e coef f i c i en t s TJ a n d £ a r e c a l l e d coef f i c i en t s of v i s c o s i t y . W e

h a v e w r i t t e n crlk in t h e s e c o n d f o r m b e c a u s e t h i s i s t h e w a y it is u s u a l l y

f o u n d d i s c u s s e d in t e x t b o o k s .

I t s h o u l d b e n o t e d in p a s s i n g t h a t b y w r i t i n g t h e m o s t g e n e r a l f o r m of o-ik

in E q . (8 .A .9 ) , w e h a v e , in f a c t , a s s u m e d t h a t t h e coef f i c i en t s of v i s c o s i t y

d o n o t d e p e n d o n p o s i t i o n i n t h e f luid, a n d h e n c e a r e r e a l l y n e g l e c t i n g

t h i n g s l i ke a p o s s i b l e d e p e n d e n c e of t h e coef f i c ien t s o n t e m p e r a t u r e o r

o t h e r p a r a m e t e r s in t h e fluid. T h i s wi l l b e a g o o d a p p r o x i m a t i o n f o r t h e

a p p l i c a t i o n s w h i c h w e w i s h t o m a k e , b u t it m u s t b e b o r n e in m i n d t h a t it

m a y n o t b e v a l i d in e v e r y p r o b l e m .

I n m o s t of t h e w o r k w h i c h w e h a v e d o n e u p t o t h i s p o i n t , w e h a v e

c o n f i n e d o u r a t t e n t i o n t o i n c o m p r e s s i b l e f lu ids ; i .e . f luids f o r w r r h t h e

e q u a t i o n (8 . A . 10)

i s va l i d . W e a r g u e d t h a t t h i s is a g o o d a p p r o x i m a t i o n f o r l i q u i d s , b u t

p e r h a p s n o t s o g o o d f o r g a s e s . F o r t h e c a s e of i n c o m p r e s s i b l e f lu ids , t h e

tWilliam of Occam (or Ockham), 1280-1349. He was an Oxford philosopher who had a rather exciting life, including a trial by the Pope at Avignon for heresy. He put forward the philosophical dictum "pluritas non est ponenda sine necessitate", or "multiplicity is not to be posited without necessity," which is usually known as Occam's razor. It is frequently cited in cases such as this when there is no inescapable reason to neglect complications.

Viscous Flow through a Pipe 127

v i s c o u s f o r c e b e c o m e s

(8 . A . 11)

s o t h a t t h e E u l e r e q u a t i o n i s

(8 . A . 12)

w h i c h , in a m o r e f a m i l i a r v e c t o r f o r m b e c o m e s

(8 . A . 13)

w h e r e v = r\lp i s u s u a l l y c a l l e d t h e kinematic viscosity coefficient.

T h i s e q u a t i o n is g e n e r a l l y c a l l e d t h e Navier-Stokes e q u a t i o n , b u t it wi l l

b e suff ic ient f o r u s t o r e m e m b e r t h a t it is s i m p l y N e w t o n ' s s e c o n d l a w

a p p l i e d t o a fluid in w h i c h i n t e r n a l f r i c t i o n , o r v i s c o s i t y , i s k n o w n t o e x i s t .

T h e a b o v e f o r m a p p l i e s only t o i n c o m p r e s s i b l e f lu ids . If t h e fluid i s

c o m p r e s s i b l e , s o t h a t E q . (8 .A . 10) i s n o t v a l i d , t h e n a m o r e c o m p l i c a t e d

f o r m of t h e e q u a t i o n c o u l d b e d e r i v e d ( s e e P r o b l e m 8.1).

B. VISCOUS FLOW THROUGH A PIPE (Poisieulle Flow)

A n e x a m p l e of v i s c o u s flow w h i c h o c c u r s o f t e n i n p r a c t i c a l a p p l i c a t i o n

is t h e flow of a fluid t h r o u g h a p i p e . L e t u s c o n s i d e r a v i s c o u s fluid flowing

t h r o u g h a p i p e of c i r c u l a r c r o s s s e c t i o n w h o s e w a l l s a r e p e r f e c t l y r ig id

( l a t e r , w h e n w e c o n s i d e r flow of t h e b l o o d in a r t e r i e s , w e sha l l c o n s i d e r

t h e r a m i f i c a t i o n s of a l l o w i n g t h e w a l l s t o b e e l a s t i c ) .

L e t u s f u r t h e r s u p p o s e t h a t t h e s y s t e m i s i n a s t e a d y s t a t e , a n d t h a t t h e

v e l o c i t y of t h e fluid i s e v e r y w h e r e in t h e z - d i r e c t i o n ( a l t h o u g h w e a l l o w

t h e p o s s i b i l i t y t h a t t h e z - v e l o c i t y m a y d e p e n d o n t h e c o o r d i n a t e r ) a n d

t h a t t h e r e is n o d e p e n d e n c e o n t h e a z i m u t h a l a n g l e ( t h i s f o l l o w s f r o m t h e

s y m m e t r y of t h e p r o b l e m ) .

T h e z - c o m p o n e n t of t h e N a v i e r - S t o k e s e q u a t i o n t h e n c a n b e w r i t t e n

U n d e r t h e c o n d i t i o n s o u t l i n e d f o r t h i s p r o b l e m ( s t e a d y s t a t e flow a n d

t h e v e l o c i t y b e i n g o n l y in t h e z - d i r e c t i o n a n d d e p e n d i n g o n l y o n t h e r a d i a l

c o o r d i n a t e ) , t h e t e r m s o n t h e l e f t - h a n d s i d e of t h e N a v i e r - S t o k e s

(8 .B .1)


e q u a t i o n v a n i s h , a n d w e h a v e f o r t h e z - c o m p o n e n t

w h e r e d a n d C 2 a r e c o n s t a n t s of i n t e g r a t i o n .

A s w i t h a n y d i f f e ren t i a l e q u a t i o n , i t i s n e c e s s a r y t o i m p o s e b o u n d a r y

c o n d i t i o n s t o d e t e r m i n e t h e s e c o n s t a n t s . I n t h i s c a s e , w e c a n r e q u i r e t h a t

t h e v e l o c i t y b e e v e r y w h e r e f in i te , i n c l u d i n g t h e p o i n t r = 0 . T h i s

r e q u i r e m e n t is m e t b y s e t t i n g Ci = 0 .

T o d e t e r m i n e t h e o t h e r c o n s t a n t , it i s n e c e s s a r y t o s p e c i f y t h e v e l o c i t y

s o m e w h e r e e l s e . W e s a w in t r e a t i n g n o n v i s c o u s fluids t h a t t h e b o u n d a r y

c o n d i t i o n a t a so l id s u r f a c e w a s t h a t t h e c o m p o n e n t of v e l o c i t y n o r m a l t o

t h e s u r f a c e h a d t o v a n i s h , b u t t h a t t h e c o m p o n e n t a l o n g t h e s u r f a c e c o u l d

b e a r b i t r a r y . I n t h e c a s e of v i s c o s i t y , h o w e v e r , t h i s b o u n d a r y c o n d i t i o n

d o e s n o t s e e m a d e q u a t e , s i n c e w e a r e d e a l i n g w i t h a fluid in w h i c h e n e r g y

t r a n s f e r c a n t a k e p l a c e b e c a u s e of t h e e x i s t e n c e of i n e l a s t i c c o l l i s i o n s a t

t h e a t o m i c l e v e l .

If w e t h i n k f o r a m o m e n t a b o u t t h e fluid n e a r t h e w a l l of t h e t u b e , w e

wi l l r e a l i z e t h a t t h e a t o m s in t h e fluid wi l l c o l l i d e w i t h t h e a t o m s in t h e

w a l l . I n t h e i d e a l i z e d c a s e w h e r e t h e a t o m s in t h e w a l l a r e p e r f e c t l y r ig id

(8 .B.2)

w h i l e t h e r - c o m p o n e n t of t h e e q u a t i o n y i e l d s

(8 .B .3)

E q u a t i o n (8 .B .3 ) , t o g e t h e r w i t h t h e r e q u i r e m e n t t h a t t h e p r e s s u r e n o t

d e p e n d o n t h e a n g l e <p, i m p l i e s t h a t e a c h p l a n e p e r p e n d i c u l a r t o t h e z - a x i s

is a p l a n e of c o n s t a n t p r e s s u r e . S i n c e w e a r e d e a l i n g w i t h a n inf in i te ly l o n g

p i p e , t h i s i m p l i e s t h a t t h e p r e s s u r e d r o p in t h e z - d i r e c t i o n m u s t b e

u n i f o r m , o r

(8 .B.4)

w h e r e A P is t h e p r e s s u r e d r o p in a l e n g t h A/. E q u a t i o n (8 .B.2) t h e n

b e c o m e s

(8 .B.5)

w h i c h c a n b e i n t e g r a t e d t o g i v e

(8 .B.6)

Viscous Flow through a Pipe 129

( i . e . , w h e r e t h e y c a n a b s o r b a n inf in i te a m o u n t of e n e r g y w i t h o u t r e c o i l i n g

o r m o v i n g ) w e w o u l d e x p e c t t h a t t h e a t o m s in t h e m o v i n g fluid w o u l d b e

r e d u c e d t o a s t a t e of r e s t a s w e l l . I n t h i s c a s e , t h e n , t h e c o r r e c t b o u n d a r y

c o n d i t i o n f o r t h e fluid w o u l d b e t h a t t h e v e l o c i t y a t t h e s u r f a c e v a n i s h

i d e n t i c a l l y ( a n d n o t j u s t in t h e n o r m a l d i r e c t i o n ) f o r p e r f e c t v i s c o s i t y .

Of c o u r s e , in a r e a l fluid w e w o u l d e x p e c t t h a t t h e fluid a t t h e s u r f a c e

w o u l d h a v e s o m e s m a l l v e l o c i t y . T h i s p h e n o m e n o n is k n o w n a s " s l i p , "

a n d w o u l d h a v e t o b e t a k e n i n t o a c c o u n t in d e t a i l e d c a l c u l a t i o n s . T h e

s i t u a t i o n is q u i t e s i m i l a r t o t h e m e c h a n i c a l p r o b l e m of a b a l l r o l l i n g a c r o s s

a s u r f a c e . I n t h e c a s e of " p e r f e c t f r i c t i o n , " w e a s s u m e t h a t t h e v e l o c i t y of

t h e s u r f a c e of t h e ba l l a t t h e p o i n t of c o n t a c t is e x a c t l y z e r o . W e r e a l i z e ,

h o w e v e r , t h a t in a r e a l s i t u a t i o n t h e v e l o c i t y a t t h a t p o i n t wi l l n o t b e z e r o ,

b u t t h a t s o m e s l i p p i n g wi l l o c c u r . N o n e t h e l e s s , in m o s t p r o b l e m s w e a r e

c o n t e n t t o i g n o r e t h i s s m a l l e f fec t in o r d e r t o e n j o y t h e g r e a t e r s i m p l i c i t y

of t h e i d e a l i z e d c a s e .

T h u s , in o u r p r o b l e m , w e wi l l a s s u m e t h a t t h e s e c o n d b o u n d a r y

c o n d i t i o n i s j u s t

s o t h a t t h e s o l u t i o n f o r t h e v e l o c i t y is j u s t

w h i c h m e a n s t h a t t h e v e l o c i t y prof i le l o o k s l ike t h e o n e s h o w n in F i g . 8.2,

i .e . t h e v e l o c i t y is z e r o a t t h e w a l l s , a n d a t t a i n s i t s m a x i m u m a t t h e c e n t e r of

t h e p i p e . S u c h a s i t u a t i o n is u s u a l l y r e f e r r e d t o a s P o i s i e u l l e flow.

F o r t h e s a k e of c o m p l e t e n e s s , w e n o t e t h a t s i n c e t h e t o t a l a m o u n t of

fluid p a s s i n g t h r o u g h a t u b e in t i m e A t is j u s t

vz(r = R) = 0, (8 .B.7)

(8 .B .9)

Fig. 8.2. Fully developed Poisieulle flow.


T h i s r e s u l t i s c a l l e d t h e P o i s i e u l l e f o r m u l a . T h e i m p o r t a n t f e a t u r e t o

w h i c h w e sha l l r e f e r w h e n d i s c u s s i n g b l o o d f low is t h e f a c t t h a t t h e f low

r a t e d e p e n d s o n t h e f o u r t h p o w e r of t h e r a d i u s , s o t h a t l a r g e c h a n g e s in

t h e p r e s s u r e a r e r e q u i r e d t o c o m p e n s a t e f o r s m a l l c o n s t r i c t i o n s in t h e

t u b e .

C. VISCOUS REBOUND—THE VISCOSITY OF THE EARTH

O n e of t h e p r o p e r t i e s of a v i s c o u s fluid is t h a t i t s r e s p o n s e t o e x t e r n a l

f o r c e s is n o t i n s t a n t a n e o u s . O n e n e e d o n l y t h i n k of m o l a s s e s flowing f r o m

a j a r t o r e a l i z e t h i s . O n t h e o t h e r h a n d , o n e w o u l d e x p e c t t h a t t h e r a t e a t

w h i c h t h e fluid r e s p o n s e d t o e x t e r n a l f o r c e s w o u l d d e p e n d r a t h e r s t r o n g l y

o n t h e v i s c o s i t y , s o t h a t , t u r n i n g t h e p r o b l e m a r o u n d , w e s h o u l d in

p r i n c i p l e b e a b l e t o d e t e r m i n e t h e v i s c o s i t y of a fluid b y m e a s u r i n g i t s

r e s p o n s e t o k n o w n f o r c e s .

O n e p a r t i c u l a r l y f a s c i n a t i n g a p p l i c a t i o n of t h i s i d e a is in m e a s u r i n g t h e

v i s c o s i t y of t h e e a r t h . W e s a w in S e c t i o n 2 . D t h a t in s o m e c a s e s , it is

p o s s i b l e t o t r e a t t h e e a r t h a s a u n i f o r m fluid. If t h i s is s o , t h e n it s h o u l d b e

p o s s i b l e t o m a k e m e a s u r e m e n t s w h i c h w o u l d a l l o w u s t o a s c r i b e a

v i s c o s i t y t o t h a t fluid. If t h e c o n j e c t u r e in t h e p r e v i o u s p a r a g r a p h is

c o r r e c t , t h e n w e s h o u l d b e a b l e t o d e t e r m i n e t h e v i s c o s i t y of t h e e a r t h b y

a p p l y i n g a k n o w n f o r c e t o t h e s u r f a c e , a n d t h e n m e a s u r i n g t h e t i m e

r e s p o n s e t o t h a t f o r c e .

Of c o u r s e , w e c a n n o t p r o d u c e m a n - m a d e f o r c e s of suff ic ient m a g n i t u d e

t o p r o d u c e a p p r e c i a b l e d e f o r m a t i o n s of t h e e a r t h ' s c r u s t o v e r l a r g e

d i s t a n c e s . H o w e v e r , n a t u r e h e r s e l f h a s p r o v i d e d t h e s e f o r c e s in m a n y

c a s e s . W e sha l l c o n s i d e r t w o c a s e s , w h i c h r e s u l t f r o m d i f f e r en t g e o l o g y ,

b u t o b e y t h e s a m e p h y s i c a l p r i n c i p l e s .

C o n s i d e r a c a s e w h e r e t h e r e i s a g r e a t w e i g h t i m p r e s s e d o n t h e s u r f a c e

of t h e e a r t h o v e r a l o n g p e r i o d of t i m e . E x a m p l e s of t h i s m i g h t b e t h e

e x i s t e n c e of a l a k e o r g l a c i e r . T h e s u r f a c e of t h e e a r t h wi l l t h e n b e

d e f o r m e d b y t h e p r e s e n c e of t h i s a d d e d w e i g h t ( s e e F i g . 8.3.). N o w s u p p o s e

t h a t f o r s o m e r e a s o n , t h e o v e r b u r d e n is r e m o v e d . T h i s m i g h t r e s u l t f r o m

t h e e v a p o r a t i o n o r d r a i n i n g of t h e l a k e , o r f r o m t h e m e l t i n g of t h e g l a c i e r .

t h e r a t e of flow t h r o u g h a c i r c u l a r p i p e is j u s t

(8 .B .10)

Viscous Rebound—The Viscosity of the Earth 131

Fig. 8.3. The deformation of the earth's surface due to a glacier.

T h e n a n i m b a l a n c e of f o r c e s wi l l e x i s t , a n d t h e s u r f a c e of t h e e a r t h wil l

s l o w l y r e b o u n d t o i t s o r i g i n a l s h a p e . T h e r a t e of r e b o u n d wi l l d e p e n d o n t h e

v i s c o s i t y . I t is t h i s s o r t of p r o c e s s t h a t w e w i s h t o c o n s i d e r in t h i s s e c t i o n .

T h e t w o e x a m p l e s w h i c h w e h a v e i n m i n d a r e t h e s o - c a l l e d F e n n o -

S c a n d i a n up l i f t a n d L a k e B o n n e v i l l e , U t a h . T h e f o r m e r is t h e r e s u l t of t h e

r e m o v a l of t h e g l a c i e r w h i c h c o v e r e d t h e S c a n d i n a v i a n p e n n i n s u l a d u r i n g

t h e l a s t i c e a g e . O v e r r e c o r d e d h i s t o r y , t h e l e v e l of l a n d in t h i s a r e a h a s

r i s e n b y h u n d r e d s of m e t e r s ! ( w e wi l l d i s c u s s a c t u a l n u m b e r s l a t e r ) . L a k e

B o n n e v i l l e w a s , d u r i n g t h e P l e i s t o c e n e e r a , a l a r g e b o d y of w a t e r w h i c h

w a s d r a i n e d a n d e v a p o r a t e d , a l l o w i n g t h e e a r t h t h e r e t o r e b o u n d a s w e l l .

I n o u r c o n s i d e r a t i o n s h e r e w e wi l l t r e a t t h e e a r t h a s if i t w e r e a n

o r d i n a r y fluid, a l t h o u g h it w o u l d b e h i g h l y v i s c o u s . H o w e v e r , it s h o u l d b e

n o t e d t h a t t h e c r u s t of t h e e a r t h ( a s o p p o s e d t o i t s i n t e r i o r ) i s n o t a n y t h i n g

l i k e t h e s u r f a c e of a fluid, b u t is a so l id a n d a s s u c h c a n e x e r t r e s t o r i n g

f o r c e s of i t s o w n . W e h a v e n o t y e t d i s c u s s e d t h e p r o b l e m of s o l i d s , b u t

w h e n w e d o w e sha l l s h o w t h a t a l t h o u g h t h e f o r c e s g e n e r a t e d b y t h e

e l a s t i c p r o p e r t i e s of t h e c r u s t a r e p r e s e n t in b o t h c a s e s of i n t e r e s t h e r e ,

t h e y a r e c o m p l e t e l y neg l ig ib l e c o m p a r e d t o t h e fluid f o r c e s w h i c h w e sha l l

a s s u m e c h a r a c t e r i z e t h e i n t e r i o r of t h e e a r t h .

T o a t t a c k t h i s p r o b l e m , le t u s c o n s i d e r t h e f o l l o w i n g c o n f i g u r a t i o n : L e t

t h e c r u s t of t h e e a r t h in e q u i l i b r i u m b e t h e p l a n e y = 0 , a n d s u p p o s e t h a t

t h e in i t ia l d e f o r m a t i o n of t h e c r u s t is of t h e f o r m

y c r ust ( t = 0 ) = & . (8 .C .1)

W e cal l t h e p o s i t i o n of t h e c r u s t a t a n y t i m e f (t) in o r d e r t o d i s t i n g u i s h it

f r o m t h e g e n e r a l c o o r d i n a t e y. T o d e t e r m i n e t h e p o s i t i o n a t s o m e l a t e r

t i m e , w e wi l l , in g e n e r a l , h a v e t o s o l v e t h e N a v i e r - S t o k e s e q u a t i o n f o r

t h i s s e t of in i t ia l c o n d i t i o n s . H o w e v e r , w e r e c a l l f r o m C h a p t e r 4 t h a t t h e

e a s i e s t m e t h o d of s o l v i n g t h e s e e q u a t i o n s is s i m p l y t o g u e s s a t t h e f o r m of a

s o l u t i o n , a n d t h e n v e r i f y t h a t t h e g u e s s d o e s i n d e e d w o r k . S i n c e w e a r e


d e a l i n g w i t h a h igh ly o v e r d a m p e d s y s t e m ( t h e v i s c o s i t y of t h e e a r t h i s , a f t e r

a l l , e x p e c t e d t o b e v e r y h i g h ) , a r e a s o n a b l e g u e s s f o r t h e s h a p e of t h e

d e f o r m a t i o n a t s o m e l a t e r t i m e t w o u l d b e

f ( 0 = (8 .C.2)

w h e r e k is a t i m e c o n s t a n t w h i c h m u s t b e d e t e r m i n e d , b u t w h i c h

p r e s u m a b l y d e p e n d s o n t h e v i s c o s i t y . I n o r d e r t o r e l a t e t h e v a r i a b l e £ (£ )

t o q u a n t i t i e s w h i c h o c c u r in t h e N a v i e r - S t o k e s e q u a t i o n , w e n o t e t h a t t h e

q u a n t i t y d£Idt, t h e r a t e a t w h i c h t h e s u r f a c e r i s e s , m u s t b e t h e s a m e a s t h e

y - c o m p o n e n t of t h e v e l o c i t y of a fluid p a r t i c l e in t h e s u r f a c e . W e h a v e

s e e n t h i s c o n d i t i o n b e f o r e in C h a p t e r 4 , w h e r e it w a s u s e d t o o b t a i n t h e

e q u a t i o n s g o v e r n i n g s u r f a c e w a v e s . If w e m a k e t h e u s u a l a p p r o x i m a t i o n

t h a t t h e d e f o r m a t i o n is s m a l l , s o t h a t w e c a n w r i t e vy(y) ~ vy(0) a s w e d i d

in C h a p t e r 4 , t h e n

B u t s i n c e

(8.C.4)

w e k n o w t h a t t o d e t e r m i n e t h e t i m e c o n s t a n t k ( w h i c h i s , of c o u r s e , a

m e a s u r a b l e q u a n t i t y ) , w e n e e d o n l y d e t e r m i n e t h e q u a n t i t y vy f r o m t h e

N a v i e r - S t o k e s e q u a t i o n .

S i n c e t h e v e l o c i t i e s in t h e p r o b l e m a r e s m a l l , w e c a n n e g l e c t t h e v • Vv

term, and write

(8 .C.3)

(8 .C .5)

L e t u s a l s o w r i t e , f o r s i m p l i c i t y ,

P' = P-Pgy, (8 .C.6)

s o t h a t t h e e q u a t i o n c a n b e p u t i n t o t h e f o r m

(8 .C.7)

Fig. 8.4. The coordinates for the viscous rebound problem.


a r e a s o n a b l e g u e s s a t a s o l u t i o n m i g h t b e

lA(y) s in Ix

B(y) c o s Ix

mA(y) c o s Ix

P ' ( y ) c o s Ix

(8 .C.9)

T h e a s s u m p t i o n t h a t t h e x- a n d z - c o m p o n e n t s of t h e v e l o c i t y h a v e

e s s e n t i a l l y t h e s a m e f o r m f o l l o w s f r o m t h e s y m m e t r y of t h e p r o b l e m , a n d

n e e d n o t b e c o n s i d e r e d a n e x t r a r e s t r i c t i o n o n t h e s o l u t i o n . W e n o w h a v e

t o d e t e r m i n e B(y). If w e p u t o u r a s s u m e d f o r m s f o r t h e v e l o c i t y i n t o t h e

e q u a t i o n of c o n t i n u i t y , w e find t h a t

T h i s t r i c k , of w r i t i n g t e r m s o n t h e r i g h t - h a n d s i d e of t h e N a v i e r - S t o k e s

e q u a t i o n a s g r a d i e n t s , a n d t h e n i n c o r p o r a t i n g t h e m i n t o a P' t e r m , is o n e

w h i c h w e wi l l u s e r e p e a t e d l y l a t e r .

F o l l o w i n g o u r s t a n d a r d p r o c e d u r e , w e wi l l n o w g u e s s a t t h e f o r m of t h e

s o l u t i o n f o r t h e q u a n t i t i e s w h i c h a p p e a r in t h e N a v i e r - S t o k e s e q u a t i o n ,

a n d in t h e e q u a t i o n of c o n t i n u i t y . W e h a v e a l r e a d y g u e s s e d a t t h e t i m e

d e p e n d e n c e of t h e t e r m s in E q . (8 .C .2 ) . S i n c e t h e v e l o c i t i e s a l s o h a v e t o

s a t i s fy t h e e q u a t i o n of c o n t i n u i t y fo r a n i n c o m p r e s s i b l e fluid

(8 .C.8)

(8 .C .10)

(8 .C .11)

w h e r e w e h a v e m a d e t h e u s e f u l de f in i t i on

a2=/2+m2.

W i t h t h i s r e s u l t , w e wil l g o b a c k t o t h e N a v i e r - S t o k e s e q u a t i o n s a n d

e l i m i n a t e t h e v a r i a b l e P' b e t w e e n t h e y - a n d x - c o m p o n e n t s of t h e

e q u a t i o n , l e a v i n g a n e q u a t i o n f o r B w h i c h w e c a n t h e n s o l v e .

T h e A : - c o m p o n e n t of t h e N a v i e r - S t o k e s e q u a t i o n i s , w h e n t h e a s s u m e d

f o r m s of t h e s o l u t i o n a r e i n s e r t e d a n d t h e o b v i o u s c a n c e l l a t i o n s m a d e ,

(8 .C.12)

wh ich , w i t h t h e a i d of E q . (8 .C .10) c a n b e w r i t t e n

(8 .C.13)


T h i s c o n c l u s i o n , of c o u r s e , d e p e n d s o n t h e v a l u e s of k a n d TJ w h i c h w e

will d e r i v e a s a r e s u l t of t h i s d i s c u s s i o n . T h e r e f o r e , w e wi l l r e g a r d t h i s

a p p r o x i m a t i o n a s a g u e s s w h i c h w e m a k e n o w , a n d wi l l v e r i f y a f t e r t h e

s o l u t i o n of t h e p r o b l e m h a s b e e n o b t a i n e d . If w e m a k e t h e a p p r o x i m a t i o n ,

t h e n t h e s o l u t i o n t o E q . (8 .C .15) i s j u s t

B(y) = - H(Gy + l ) * - ' + C ( D y + \)eay. (8 .C .17)

( T h i s c a n b e ver i f i ed b y s u b s t i t u t i n g t h e a s s u m e d f o r m of t h e s o l u t i o n

b a c k i n t o E q . (8 .C .15) . T h e s o l u t i o n t o t h e e q u a t i o n is d e r i v e d in s t a n d a r d

b o o k s o n o r d i n a r y d i f fe ren t i a l e q u a t i o n s . )

T h e r e a r e f o u r u n k n o w n c o n s t a n t s in t h i s s o l u t i o n , b e c a u s e w e s t a r t e d

w i t h a f o u r t h - o r d e r d i f f e ren t i a l e q u a t i o n . W e r e m a r k in p a s s i n g t h a t

f o u r t h - o r d e r e q u a t i o n s o f t e n o c c u r in p r o b l e m s i n v o l v i n g v i s c o s i t y b e -

c a u s e of t h e V 2 v t e r m in t h e N a v i e r - S t o k e s e q u a t i o n w h i c h w a s n o t

p r e s e n t in C h a p t e r 4 ( for e x a m p l e ) w h e n a p r o b l e m s imi l a r t o t h i s w a s

d i s c u s s e d fo r s u r f a c e w a v e s .

S i n c e t h e v e l o c i t y m u s t b e f ini te w h e n y a p p r o a c h e s inf in i ty , w e c a n

i m m e d i a t e l y w r i t e

C = 0. (8 .C .18)

T h e d e t e r m i n a t i o n of t h e o t h e r c o n s t a n t s i s s o m e w h a t m o r e c o m p l i -

c a t e d , a n d wil l r e q u i r e a l i t t le d i s c u s s i o n of w h a t b o u n d a r y c o n d i t i o n s a r e

a p p r o p r i a t e a t t h e f r e e s u r f a c e of a v i s c o u s f luid. I n C h a p t e r 2 , w h e n w e

d i s c u s s e d t h e s u r f a c e c o n d i t i o n f o r a n o n v i s c o u s s t a t i c fluid, w e s a w t h a t

in t h a t c a s e t h e s u r f a c e h a d t o b e a t a c o n s t a n t p r e s s u r e , s i n c e o t h e r w i s e

f o r c e s w o u l d e x i s t a t t h e s u r f a c e w h i c h w o u l d c a u s e e l e m e n t s of f luid a t

t h e s u r f a c e t o m o v e , d i s t o r t i n g t h e s u r f a c e .

T h e y - c o m p o n e n t of t h e N a v i e r - S t o k e s e q u a t i o n , w i t h s i m i l a r s u b s t i t u -

t i o n s a n d c a n c e l l a t i o n s is j u s t

(8 .C .14)

w h i c h , if w e s u b s t i t u t e P ' ( y ) f r o m E q . (8 .C .13 ) , b e c o m e s

(8 .C .15)

I n P r o b l e m 8.2, it is s h o w n t h a t fo r a l a r g e a r e a p h e n o m e n o n l ike t h e

F e n n o - S c a n d i a n up l i f t , t h e v a l u e s of t h e p h y s i c a l c o n s t a n t s in E q . (8 .C .15)

a r e s u c h t h a t

(8 .C .16)


T h e c o n d i t i o n t h a t a s u r f a c e b e f r e e , t h e n , is s i m p l y t h a t n o f o r c e s a c t

o n it in s u c h a w a y a s t o c a u s e it t o c h a n g e . I n t h e c a s e in w h i c h w e a r e

i n t e r e s t e d a t t h e m o m e n t , t h e r e is c l e a r l y a n i m b a l a n c e of f o r c e s in t h e

y - d i r e c t i o n , s i n c e t h e s u r f a c e is m o v i n g in t h a t d i r e c t i o n . H o w e v e r , t h e r e

s h o u l d b e n o f o r c e s a c t i n g a l o n g t h e s u r f a c e , in t h e J C - o r z - d i r e c t i o n . S u c h

f o r c e s a r e c a l l e d s h e a r f o r c e s . I n a n o n v i s c o u s fluid, n o s u c h f o r c e s e x i s t

f o r s m a l l v e l o c i t i e s , b u t w e s a w in E q . (8 .A.5) t h a t t h e p r e s e n c e of

v i s c o s i t y i n t r o d u c e s a n e w f o r c e , d e p e n d i n g o n cr i k. T h e c o n d i t i o n t h a t n o

s h e a r f o r c e s e x i s t a t t h e s u r f a c e y = 0 m u s t t h e n b e t h a t

cryx(y =0) = tryz(y = 0) = 0 . (8 .C .19)

B e c a u s e of t h e s y m m e t r y , w e wil l c o n s i d e r o n l y o n e of t h e s e c o n d i -

t i o n s . B y de f in i t ion

(8 .C .20)

(8 .C .21)

w h i c h , u s i n g o u r a s s u m e d s o l u t i o n s b e c o m e s

tryx = 7]l s in Ix c o s raz e

s o t h a t i m p o s i n g t h e b o u n d a r y c o n d i t i o n , w e find

(8 .C .22)

w h e r e t h e s e c o n d e q u a l i t y f o l l o w s f r o m E q . (8 .C .10) . T h i s i m p l i e s

G = a. (8 .C .23)

T h u s , w e find t h a t t h e y - c o m p o n e n t of t h e v e l o c i t y of t h e fluid is j u s t

v y = - H(ay + l ) < T a y c o s Ix c o s m z e~kt. (8 .C.24)

It r e m a i n s t o find a r e l a t i o n s h i p b e t w e e n vy a n d t h e d i s p l a c e m e n t £. O n e

s u c h r e l a t i o n h a s , of c o u r s e , b e e n o b t a i n e d in E q . (8 .C .4 ) . T h e r e i s a n o t h e r

w h i c h c a n b e o b t a i n e d if w e l o o k a t t h e f o r c e s in t h e y - d i r e c t i o n a t t h e

s u r f a c e . F r o m t h e a b o v e d i s c u s s i o n a b o u t t h e f o r c e s e x e r t e d b y v i s c o s i t y

a t t h e s u r f a c e , it is c l e a r t h a t t h e f o r c e in t h e y - d i r e c t i o n a t t h e s u r f a c e

m u s t b e

(8 .C .25 ~ j -»y =u

= 2T\OH c o s Ix c o s mze~kt,

w h e r e w e h a v e u s e d E q s . (8 .C .13) , (8 .C .17 ) , a n d (8 .C .23) t o e v a l u a t e P' a t y = 0 .


T h u s , b y m e a s u r i n g t h e r a t e a t w h i c h t h e r e b o u n d of t h e e a r t h ' s c r u s t is

p r o c e e d i n g w i t h t i m e , t h e v i s c o s i t y of t h e e a r t h c a n b e e s t i m a t e d .

A c t u a l l y , in a p o p u l a t e d a r e a l i ke S c a n d i n a v i a , t h i s is n o t a s c o m p l i c a t e d

a s it s o u n d s , s i n c e o n e c a n l o o k a t o ld w h a r v e s w h i c h a r e n o w fa r i n l a n d , o r

a t g e o l o g i c a l e v i d e n c e . A full d i s c u s s i o n of t h e m e a s u r e m e n t s in t h e c a s e

is g i v e n in t h e t e x t b y H e i s k a n e n a n d V e n i n g M e i n e s z (1958) . F o r o u r

e x a m p l e , w e n o t e t h a t in S c a n d i n a v i a , t h e d e f l e c t i o n in 8000 B . C . w a s

556 m , a n d is a b o u t 80 m t o d a y , s o t h a t t h e t i m e c o n s t a n t i s j u s t

k « 6 x 1 0 " 1 2 s e c 1 ,

w h i c h l e a d s t o a v i s c o s i t y e s t i m a t e of

n ~ 1 0 2 2 p o i s e .

F o r L a k e B o n n e v i l l e , h o w e v e r , t h e u p w a r d d e f l e c t i o n is e s t i m a t e d t o b e

a b o u t 64 m in 4 0 0 0 y e a r s . T h i s l e a d s t o a n e s t i m a t e d v i s c o s i t y of

r/ ~ 1 0 2 1 p o i s e .

T h e d i f f e r e n c e s b e t w e e n t h e s e t w o c o u l d b e d u e t o a n u m b e r of c a u s e s .

I n o u r d e v e l o p m e n t , w e h a v e a s s u m e d t h a t t h e o v e r b u r d e n w a s l i f ted

i n s t a n t a n e o u s l y , w h e r e a s in b o t h c a s e s w e c o n s i d e r e d — t h e m e l t i n g of a

g l a c i e r a n d t h e e m p t y i n g of a l a k e — t h e r e m o v a l of t h e o v e r b u r d e n w o u l d

t a k e p l a c e o v e r a t i m e s c a l e w h i c h is n o t t e r r i b l y s m a l l c o m p a r e d t o t h a t of

t h e r e b o u n d . W e h a v e a l s o n e g l e c t e d t h e f a c t t h a t t h e e a r t h is n o t a p e r f e c t

fluid, b u t in f a c t c h a n g e s d e n s i t y a p p r e c i a b l y o v e r d i s t a n c e s of t h e o r d e r

F r o m t h e p r i n c i p l e of A r c h i m e d e s , t h i s m u s t b e t h e b u o y a n t f o r c e , a n d

m u s t t h e r e f o r e b e e q u a l t o t h e w e i g h t of t h e d i s p l a c e d l i qu id . F r o m F i g .

(8 .4) , t h e w e i g h t of d i s p l a c e d l i qu id a t a n y p o i n t is j u s t g i v e n b y

FB = pgfe (8 .C .26)

s o t h a t e q u a t i n g E q . (8 .C .26) t o E q . (8 .C .25) g i v e s

(8 .C.27) -H c o s Ix c o s mz e k t .

T h i s e x p r e s s i o n , w h i c h g i v e s t h e d i s p l a c e m e n t of t h e s u r f a c e in t e r m s

of t h e v i s c o s i t y of t h e e a r t h , is p r e c i s e l y t h e e x p r e s s i o n w h i c h w e s e e k . If

w e u s e E q s . (8 .C .3 ) , (8 .C .4 ) , (8 .C .24) , a n d , (8 .C .27) , w e find t h a t t h e t i m e

c o n s t a n t f o r t h e r e b o u n d is g i v e n b y

(8 .C.28)

Problems 137

of m a g n i t u d e w h i c h w e a r e c o n s i d e r i n g h e r e . F i n a l l y , t h e d i f f e r e n c e s

m i g h t s i m p l y b e a r e s u l t of t h e f a c t t h a t t h e e a r t h is j u s t a l i t t le l e s s r ig id in

N o r t h A m e r i c a t h a n it is in N o r t h e r n E u r o p e .

B u t w h a t e v e r t h e o u t c o m e of t h e d i s c u s s i o n of t h e d e t a i l s of t h i s t y p e

of a n a l y s i s , t h e i m p o r t a n t p o i n t f o r o u r d i s c u s s i o n is t h a t it is p o s s i b l e ,

s t a r t i n g w i t h t h e s i m p l e N a v i e r - S t o k e s e q u a t i o n , t o l o o k a t t h e p r o c e s s of

e l a s t i c r e b o u n d in t h e c r u s t of t h e e a r t h a n d c o m e u p w i t h r e a s o n a b l e

e s t i m a t e s of t h e e a r t h ' s v i s c o s i t y . T h i s i l l u s t r a t e s a g a i n t h e p o i n t w h i c h

w a s m a d e in t h e first c h a p t e r — t h a t g i v e n a f e w s i m p l e p h y s i c a l p r i n c i p l e s

w h i c h g o v e r n t h e b e h a v i o r of fluids, t h e r e i s a l m o s t n o e n d t o t h e n u m b e r

of i n t e r e s t i n g e x a m p l e s w h i c h c a n b e d e s c r i b e d w i t h t h e m .

SUMMARY

W e h a v e s e e n t h a t t h e e f f ec t s of v i s c o s i t y c a n b e i n c l u d e d in o u r

d e s c r i p t i o n of fluids b y t h e a d d i t i o n of a t e r m t o t h e E u l e r e q u a t i o n . F o r

i n c o m p r e s s i b l e fluids, t h i s t e r m is of t h e f o r m 17 A 2 v , w h e r e 17 is c a l l e d t h e

coef f ic ien t of v i s c o s i t y . T h e e x a m p l e s of t h e flow of a fluid t h r o u g h a r ig id

p i p e a n d t h e v i s c o u s r e b o u n d of t h e e a r t h ' s s u r f a c e a f t e r t h e r e m o v a l of

a n o v e r b u r d e n l ike a g l a c i e r w e r e w o r k e d o u t .

PROBLEMS

8 . 1 . Der ive the form of the Nav i e r -S tokes equat ion for the case of a compress ible fluid whose coefficients of viscosity are cons tant .

8.2. Verify that for the Fenno-Scandian uplift area, which is approximately 1400 km on a side, the approximat ion

8.3. Show that the boundary condit ion in Eq . (8.C.10) and the subsequent determinat ion of the coefficient in Eq . (8.C.23) imply that there is no mot ion of the fluid in the x- or 2 -d i rec t ion in the case of viscous rebound. Is this consis tent with the boundary condi t ions we have imposed on the problem?

8.4. The introduct ion of viscosity means that there is a new mechanism for dissipating energy in a fluid sys tem. Let us repeat the energy ba lance analysis of Sect ion l .E for an incompressible v iscous fluid.

(a) Show that the Nav i e r -S tokes equat ion leads to the result

is valid.


(b) H e n c e show that

(c) Using the definition of cr,*, show that an appropr ia te choice of the surface S leads to

8.5. Consider a fluid of viscosity 17 flowing be tween two infinitive parallel plates a dis tance h apart . Le t there be a pressure gradient dPjdz exer ted by some outside agency, so that the fluid will flow in the 2 -d i rec t ion .

(a) Calculate the velocity profile of the fluid be tween the plates . (b) Hen ce calculate the tensor crik in the fluid. (c) Show that there will be a force in the z -direction per unit area on each plate

given by

This phenomenon , in which a viscous liquid exer ts a force on the material at its boundary , is called drag.

8.6. Repeat Problem 8.5 for the case where the upper plate is moving in the z-direction with velocity V.

8.7. The general method outlined in the above two problems can be applied to calculating the drag on any body moving through a fluid (or, equivalently, a stat ionary body around which a fluid flows). One case which can be solved explicitly is that of a sphere in a fluid. The result of this calculation, called Stoke's formula, says that the drag force on a sphere of radius a in a fluid which is moving with velocity V relative to the sphere , is given by

F = 6TTRVV.

Derive this result by calculating the velocity field around a sphere , deriving aik

from the field, and integrating over the sphere to find the force. (Hint: You may want to consult some of the texts cited in Chapter 1, since the derivation is somewhat complicated.)

8.8. S toke ' s formula tells us what the effect of air res is tance would be on a falling sphere . There is a common folktale involving Galileo which says that he discovered that the accelerat ion due to gravity was independent of the mass by dropping different weights off of the leaning tower of Pisa. Calculate the

(d) and hence

Problems 139

difference in arrival t imes be tween two spheres whose masses are a factor of q different, but whose radii are the same, if they are d ropped from rest from a height h. U s e this result to comment on the historical validity of Galileo's exper iment .

8.9. Consider the flow of a fluid in a two-dimensional plane. Le t us define

co = V x v)2. Show that the Nav ie r -S tokes equat ion and the equat ion of continuity imply that

Deo _2

The variable co is usually called the vorticity, and this equat ion is called the vorticity transport equation. Does it resemble any other equat ion you know of?

8.10. Show that the potential flow of an incompressible fluid will automatical ly satisfy the Nav ie r -S tokes equat ion provided that it satisfies the corresponding Euler equat ion.

8.11. Consider two cylinders of radii rx and r 2 , rotating at angular speed cox and co2, respect ively.

(a) Wri te down the Nav ie r -S tokes equat ion and the boundary condit ions which must apply in this case .

(b) Show that if the inner cylinder is held fixed, the to rque per unit length exer ted by the outer cylinder is

This result has been utilized as a means of measur ing the viscosity of fluids, (d) Show that in the case of a single cylinder rotating alone,

(Hint: This is a limit of the result in part (b).)

8.12. Consider a flat plate which is initially at rest in an infinite fluid, and which at t ime t = 0 is ins tantaneously accelerated to its final velocity V, which we will take to be along the plate.

(a) Show that the Nav ie r -S tokes equat ion reduces to

M — 47717

where x is the coordinate perpendicular to the plate, (b) If we assume a solution of the form

u = t>/(f), and define


(c) Solve this equat ion (Hint: Look up the incomplete error function), and sketch the velocity near the wall.

8.13. Work through Problem 8.12 for the case where the wall is oscillating, so that its velocity is

V = v0 cos cot.

Show that the fluid velocity is given by

u(x, t) = Vo*T v ^ cos (cot - x A / ^ ; ) .

These two examples are called Stoke's first and second problems.

8.14. In bo th of the above two problems, the fluid at large dis tances was essentially at rest , while the fluid near the moving plate was in motion. Show that the dis tance to the point at which the velocity has been reduced to about 1% of the velocity of the plate is given in both cases by

8 « Wt,

where t is a typical t ime in the problem. The significance of this result will become clear in the next chapter .

REFERENCES

In addition to the texts cited in Chapter 1, an excellent reference on the topic of viscosity

H. Schlichting, Boundary Layer Theory, McGraw-Hill, New York, 1968. This is a very thorough and surprisingly readable account of the theory of the flow of viscous fluids, leading to very good discussions of aerodynamics and turbulence.

For a discussion of the Fenno-Scandian problem, see

W. A. Heiskanen and F. A. Vening Meinesz, The Earth and Its Gravity Field, McGraw-Hill, New York, 1958.

show that / is de termined by

9

The Flow of Viscous Fluids

Things are seldom what they seem.

GILBERT AND SULLIVAN

HMS Pinafore

A. THE REYNOLDS NUMBER

I n t h e p r e v i o u s s e c t i o n , w e e x a m i n e d t h e p r o b l e m of t h e flow of a

v i s c o u s fluid in a p i p e , t h e P o i s i e u l l e p r o b l e m . T h i s i s o n e of t h e s i m p l e s t

e x a m p l e s of t h e s t e a d y - s t a t e flow of fluids w h o s e v i s c o s i t y c a n n o t b e

n e g l e c t e d . T h e r e a r e s e v e r a l i m p o r t a n t c o n c l u s i o n s w h i c h c a n b e d r a w n

f r o m t h i s c a l c u l a t i o n . I n t h e first p l a c e , t h e v i s c o u s b o u n d a r y c o n d i t i o n ,

w h i c h s t a t e s t h a t t h e fluid m u s t b e a t r e s t a t a r ig id b o u n d a r y , g i v e s r i s e t o

flow p a t t e r n s w h i c h a r e q u i t e d i f f e r en t f r o m w h a t w e w o u l d e x p e c t in a

n o n v i s c o u s fluid, w h e r e o n l y t h e n o r m a l c o m p o n e n t of t h e v e l o c i t y m u s t

v a n i s h .

I n t h e s e c o n d p l a c e , in fu l ly d e v e l o p e d v i s c o u s flow, t h e n o n l i n e a r

t e r m s in t h e N a v i e r - S t o k e s e q u a t i o n c a n n o t , in g e n e r a l , b e i g n o r e d . T h i s

m e a n s t h a t e x c e p t f o r v e r y s i m p l e g e o m e t r i e s , l i ke a c i r c u l a r p i p e , t h e

e q u a t i o n s t h e m s e l v e s wil l b e n o n l i n e a r , a n d t h e r e f o r e q u i t e difficult t o

s o l v e . W e sha l l s e e t h i s in t h e e x a m p l e in t h e n e x t s e c t i o n .

B e f o r e g o i n g o n , h o w e v e r , w e wi l l s t u d y o n e g e n e r a l p r o p e r t y of t h e

N a v i e r - S t o k e s e q u a t i o n w h i c h is e x t r e m e l y i m p o r t a n t in a p p l i c a t i o n s . F o r

s t e a d y - s t a t e flow, w e h a v e

141

(9 .A.1)

142 The Flow of Viscous Fluids

N o w s u p p o s e t h a t w e h a v e a s y s t e m in w h i c h V, L , a n d P a r e " t y p i c a l "

v e l o c i t i e s , l e n g t h s , a n d p r e s s u r e s . F o r e x a m p l e , in t h e c a s e of P o i s i e u l l e

f low, a t y p i c a l v e l o c i t y m i g h t b e

V = C(r = 0 ) , (9 .A.2)

t h e m a x i m u m of t h e v e l o c i t y p rof i l e , w h i l e a t y p i c a l l e n g t h m i g h t b e g i v e n

b y

L = a,

a n d a t y p i c a l p r e s s u r e b y

P i = P ( z = 0 ) .

I n a n y p r o b l e m , s u c h t y p i c a l v a l u e s of t h e p a r a m e t e r s c a n b e de f ined .

N o w le t u s c h a n g e v a r i a b l e s , s o t h a t

(9 .A.3)

t h e n R is c a l l e d t h e Reynolds number.

T h e p h y s i c a l s i gn i f i cance of t h e R e y n o l d s n u m b e r c a n b e s t b e u n d e r -

s t o o d b y c o n s i d e r i n g t h e f o r c e s a c t i n g o n a n in f in i t e s imal v o l u m e in a fluid

in s t e a d y - s t a t e f low ( s e e F i g . 9 .1) . N e g l e c t i n g t h e v i s c o u s t e r m is e q u i v a l e n t

t o n e g l e c t i n g TJ V 2 V w i t h r e s p e c t t o (V • V ) V in t h e E u l e r e q u a t i o n . T h i s

l a t t e r t e r m is s o m e t i m e s c a l l e d t h e " i n e r t i a l f o r c e " , s i n c e it r e p r e s e n t s t h e

m o m e n t u m c a r r i e d in t h e m o v e m e n t of t h e fluid. F r o m E q . (8 .A.9) fo r a n

i n c o m p r e s s i b l e fluid, t h e v i s c o u s f o r c e in t h e z - d i r e c t i o n a l o n g t h e b o t t o m

f a c e of t h e c u b e is j u s t

o~zydx dz

a n d

If w e i n s e r t t h e s e n e w v a r i a b l e s i n t o E q . (9 .A.1) a n d d i v i d e b y V 2 / L , w e

find a n e w e q u a t i o n

(9 .A.4)

w h i c h is n o w w r i t t e n e n t i r e l y in t e r m s of d i m e n s i o n l e s s q u a n t i t i e s . T h e

c o l l e c t i o n of v a r i a b l e s o n t h e r i g h t is g i v e n a s p e c i a l n a m e . If w e w r i t e

(9 .A.5)

The Reynolds Number 143

t dy\ U,

dz

(9 .A.6)

O n t h e o t h e r h a n d , t h e de f in i t i on of t h e c o n v e c t i v e d e r i v a t i v e g i v e s

(9 .A.7)

f o r t h e i ne r t i a l f o r c e a c t i n g o n t h e b o d y . If w e n o w a s s u m e t h a t vz v a r i e s

a p p r e c i a b l y o v e r a d i s t a n c e L , s o t h a t

(9 .A.8)

w h i c h is t h e R e y n o l d s n u m b e r . T h a t t h i s r a t i o s h o u l d c o m e u p a g a i n is n o t

s u r p r i s i n g — t h e R e y n o l d s n u m b e r i s t h e o n l y d i m e n s i o n l e s s p a r a m e t e r

w h i c h c a n b e f o r m e d f r o m t h e v a r i a b l e s in t h e s i m p l e flow p r o b l e m .

A w o r d of c a u t i o n m u s t a l s o b e i n s e r t e d a t t h i s p o i n t . T h e de f in i t i on of

t h e R e y n o l d s n u m b e r is s o m e w h a t a r b i t r a r y . F o r e x a m p l e , w e c o u l d h a v e

c h o s e n t h e d i a m e t e r of t h e p i p e i n s t e a d of t h e r a d i u s in c h o o s i n g L , t h e

t y p i c a l l e n g t h in E q . ( 9 .A .5 ) . T h i s w o u l d h a v e m a d e a d i f f e r e n c e of a

Fig. 9.1. Forces on an infinitesimal volume element.

z

w h i l e t h a t a l o n g t h e t o p f a c e is j u s t

s o t h a t t h e n e t f r i c t i o n a l f o r c e is g i v e n b y

a n d

t h e n t h e r a t i o of i n e r t i a l t o v i s c o u s f o r c e s is j u s t


B. BOUNDARY LAYERS

W e h a v e r e p e a t e d l y r e f e r r e d t o t h e f a c t t h a t t h e r e is a g r e a t d e a l of

m a t h e m a t i c a l c o m p l e x i t y in t h e N a v i e r - S t o k e s e q u a t i o n . O n e c o n s e -

q u e n c e of t h i s in t h e n i n e t e e n t h c e n t u r y w a s t h a t t w o p a r a l l e l a n d r a t h e r

u n c o n n e c t e d fields of s t u d y h a d d e v e l o p e d in fluid m e c h a n i c s . O n e w a s

c a l l e d t h e o r e t i c a l h y d r o d y n a m i c s a n d i n v o l v e d t h e w o r k i n g o u t of t h e

E u l e r e q u a t i o n f o r p e r f e c t fluids. W e h a v e s e e n in p r e v i o u s c h a p t e r s t h a t

t h i s s o r t of t h i n g is c a p a b l e of d e s c r i b i n g a l a r g e p o r t i o n of t h e w o r l d

a r o u n d u s . W h e n it c a m e t o d e a l i n g w i t h p r o b l e m s of flow a r o u n d o r

t h r o u g h m a t e r i a l o b j e c t s , h o w e v e r , it w a s a r a t h e r d i s m a l f a i l u r e . I n

C h a p t e r 8, w e s a w h o w e v e n t h e s i m p l e p r o b l e m of flow t h r o u g h a

c i r c u l a r p i p e d e m a n d e d t h e i n c l u s i o n of v i s c o s i t y in t h e e q u a t i o n s of

m o t i o n . C o n s e q u e n t l y , t h e s e p a r a t e d i s c i p l i n e of h y d r a u l i c s g r e w u p . T h i s

w a s l a r g e l y a n e x p e r i m e n t a l e n g i n e e r i n g v e n t u r e , a n d m a d e l i t t le c o n t a c t

w i t h t h e t h e o r y of fluids a s w e a r e d i s c u s s i n g it in t h i s t e x t . T h e t w o

d i s c i p l i n e s w e r e b r o u g h t t o g e t h e r in t h e e a r l y 1900s b y L u d w i g P r a n d t l ,

w h o d e v e l o p e d t h e t h e o r y of b o u n d a r y l a y e r s .

f a c t o r of t w o in t h e de f in i t ion of t h e R e y n o l d s n u m b e r . T h u s , w h e n a

R e y n o l d s n u m b e r is de f i ned , s o m e c a r e s h o u l d b e t a k e n in s p e c i f y i n g

e x a c t l y w h i c h l e n g t h s , p r e s s u r e s , a n d v e l o c i t i e s a r e b e i n g t a k e n a s

" t y p i c a l , " s o t h a t c o m p a r i s o n s w i t h o t h e r c a l c u l a t i o n s ( p e r h a p s u s i n g

d i f f e ren t de f in i t i ons ) c a n b e m a d e .

O n c e t h e N a v i e r - S t o k e s e q u a t i o n h a s b e e n p u t i n t o d i m e n s i o n l e s s

f o r m , a s in E q . ( 9 .A .4 ) , a v e r y i n t e r e s t i n g r e s u l t e m e r g e s . S u p p o s e t h a t w e

h a d t w o d i f f e ren t s i t u a t i o n s in w h i c h t w o d i f f e ren t f luids w e r e f lowing in

(o r a r o u n d ) m a t e r i a l s w h i c h h a d s i m i l a r s h a p e s , b u t w e r e of a d i f f e ren t

s i z e . F o r e x a m p l e , w e m i g h t b e c o n s i d e r i n g t h e flow of a i r a r o u n d a n

o b s t r u c t i o n in a l a r g e t u n n e l a n d t h e flow of b l o o d a r o u n d a n o b s t r u c t i o n

in a n a r t e r y . S u p p o s e f u r t h e r t h a t t h e flows w e r e a d j u s t e d s o t h a t t h e r a t i o

PjpV2 w e r e t h e s a m e in e a c h c a s e , a n d s o t h a t t h e t w o flows h a d t h e

s a m e R e y n o l d s n u m b e r . T h e n a g l a n c e a t E q . (9 .A.4) t e l l s u s t h a t t h e s e

t w o s i t u a t i o n s will b e g o v e r n e d b y e x a c t l y t h e s a m e e q u a t i o n of m o t i o n .

T h i s m e a n s t h a t e x c e p t f o r t h e d i f f e r e n c e in s c a l e , t h e t w o flows wil l b e

i d e n t i c a l . T h i s is c a l l e d t h e law of similarity, a n d is of o b v i o u s u s e f u l n e s s

in m a n y a p p l i c a t i o n s of h y d r o d y n a m i c s . T h e e x a m p l e m o s t f a m i l i a r t o t h e

r e a d e r w o u l d b e t h e w i n d t u n n e l , in w h i c h s m a l l - s c a l e m o d e l a i r p l a n e

c o m p o n e n t s c a n b e t e s t e d . ( S e e P r o b l e m 9.2.)

Boundary Layers 145

f o r t h e R e y n o l d s n u m b e r a s s o c i a t e d w i t h t h e l e n g t h L .

N o w in a t y p i c a l s i t u a t i o n , RL c a n a t t a i n v a l u e s in t h e h u n d r e d s o r e v e n

t h o u s a n d s . F r o m t h e r e a s o n i n g in t h e p r e v i o u s s e c t i o n , w e w o u l d c o n -

c l u d e t h a t fo r s u c h c a s e s , t h e v i s c o u s f o r c e s w o u l d b e c o m p l e t e l y

neg l i g ib l e , a n d w e c o u l d u s e t h e E u l e r e q u a t i o n t o d e s c r i b e t h e f low.

T h i s w o u l d i m m e d i a t e l y l e a d t o p r o b l e m s , h o w e v e r , a s c a n b e s e e n b y

c o n s i d e r i n g t h e c a s e of a p r e s s u r e w h i c h is u n i f o r m in t h e y - d i r e c t i o n . I n

t h i s c a s e , t h e f low of t h e fluid w o u l d h a v e t o b e u n i f o r m a s w e l l , a n d h e n c e

c o u l d n o t v a n i s h a t t h e p l a t e . T h i s is t h e b a s i c conf l ic t b e t w e e n t h e t w o

p o i n t s of v i e w d i s c u s s e d a b o v e .

P r a n d t l ' s s o l u t i o n w a s q u i t e s i m p l e . H e p o i n t e d o u t t h a t w h i l e t h e

R e y n o l d s n u m b e r d e f i n e d a s in E q . (9 .B .1) m a y b e u s e f u l t h r o u g h o u t m o s t

of t h e fluid, a n d m a y r e p r e s e n t t h e r a t i o of i ne r t i a l t o v i s c o u s f o r c e s t h e r e ,

it d o e s n o t d o s o n e a r t h e p l a t e . W e c a n s e e t h i s q u i c k l y b y n o t i n g t h a t t h e

i ne r t i a l t e r m g o e s a s V2 w h i l e t h e v i s c o u s t e r m g o e s a s V, s o t h a t a s V

a p p r o a c h e s z e r o , t h e r e m u s t b e s o m e p o i n t a t w h i c h t h e i ne r t i a l t e r m

b e c o m e s l e s s t h a n t h e v i s c o u s t e r m , e v e n t h o u g h in t h e m a i n b o d y of t h e

fluid it is m u c h l a r g e r . T h u s , t h e r e will b e s o m e s m a l l r e g i o n n e a r t h e p l a t e

w h e r e v i s c o u s f o r c e s wil l d o m i n a t e t h e m o t i o n , e v e n t h o u g h t h e y c a n b e

n e g l e c t e d e v e r y w h e r e e l s e . T h i s s m a l l r e g i o n is c a l l e d t h e boundary layer.

Fig. 9.2. Mow oi a viscous nuia near a plate.

P e r h a p s t h e b e s t w a y t o u n d e r s t a n d t h e i d e a b e h i n d b o u n d a r y - l a y e r

t h e o r y is t o c o n s i d e r t h e c a s e of f low p a s t a p l a t e ( s e e F i g . 9 .2) . L e t U

d e n o t e t h e v e l o c i t y f a r f r o m t h e p l a t e . F r o m o u r d i s c u s s i o n of v i s c o s i t y ,

w e k n o w t h a t t h e v e l o c i t y m u s t v a n i s h a t t h e p l a t e , s o t h a t t h e r e m u s t b e

s o m e v a r i a t i o n of t h e v e l o c i t y w i t h y a s s h o w n .

N o w if w e w e r e t o f o r m t h e R e y n o l d s n u m b e r f o r t h i s s y s t e m , o u r first

i m p u l s e w o u l d b e t o t a k e U a s t h e t y p i c a l v e l o c i t y a n d L t o b e t h e l e n g t h

of t h e p l a t e , t o g i v e

(9 .B.1)


(9 .B.6)

T h u s , w e s e e t h a t t h e r e i s a s m a l l r e g i o n , w h o s e e x t e n t v a r i e s i n v e r s e l y

w i t h t h e s q u a r e r o o t of t h e R e y n o l d s n u m b e r , in w h i c h t h e v i s c o u s f o r c e s

c a n n o t b e n e g l e c t e d . I t is p r e c i s e l y t h i s r e g i o n w h i c h is i m p o r t a n t

w h e n w e a r e d e a l i n g w i t h t h i n g s l ike t h e v i s c o u s d r a g o n a n o b j e c t

in a m o v i n g fluid ( s u c h a s a n a i r fo i l ) . T h i s e x p l a i n s w h y t h e s i m p l e

n o n v i s c o u s t h e o r y c o u l d n o t b e u s e d in s o m a n y i m p o r t a n t a p p l i c a t i o n s ,

a n d w h y t h e i n c l u s i o n of v i s c o s i t y w a s n e c e s s a r y in t h e d e s i g n of a i r fo i l s

a n d s i m i l a r t h i n g s .

A c t u a l l y , a s w e h a v e d e f i n e d t h e R e y n o l d s n u m b e r , it d e p e n d s o n t h e

l e n g t h L of t h e p l a t e . W e s e e f r o m E q . (9 .B.6) t h a t t h e a c t u a l s i z e of t h e

b o u n d a r y l a y e r c a n b e e x p e c t e d t o i n c r e a s e a s VZ. T O p r o c e e d f a r t h e r ,

L e t u s p u t t h e i n t u i t i v e r e a s o n i n g in t h e a b o v e p a r a g r a p h i n t o m o r e

p r e c i s e f o r m . T h e e q u a t i o n d e s c r i b i n g t h e f low in F i g . 9.2 is j u s t

(9 .B.2)

w h e r e U i s a v e l o c i t y in t h e y - d i r e c t i o n , b u t c a n , in g e n e r a l , d e p e n d o n

b o t h y a n d JC.

A s b e f o r e , w e c a n w r i t e

(9 .B.3)

I n a s imi l a r w a y , t h e d e r i v a t i v e s in t h e v i s c o u s t e r m n e a r t h e p l a t e c a n b e w r i t t e n

a n d

(9 .B.4)

w h e r e t h e s e c o n d e x p r e s s i o n is t r u e o n l y in t h e r e g i o n in w h i c h t h e

v e l o c i t y is m a k i n g i t s r a p i d t r a n s i t i o n f r o m z e r o t o U. T h i s , of c o u r s e , is

t h e r e g i o n n e a r t h e p l a t e , a n d i t s t h i c k n e s s w e d e n o t e b y 5. C l e a r l y , t h e

v i s c o u s a n d ine r t i a l t e r m s wi l l b e c o m p a r a b l e w h e n

(9 .B .5)

o r , u s i n g E q . (9 .B .1 ) , w h e n

Boundary Layers 147

U . ( X )

t y

X

T

x = 0 x = L

Fig. 9.3. The development of the boundary layer.

a n d , in p a r t i c u l a r , t o d e t e r m i n e t h e c o n s t a n t of p r o p o r t i o n a l i t y in E q .

(9 .B .5 ) , it wi l l b e n e c e s s a r y t o w o r k o u t t h e e q u a t i o n s m o r e e x a c t l y .

L e t u s c o n s i d e r t h e p r o b l e m of f low p a s t a p l a t e in m o r e d e t a i l . L e t t h e

p l a t e s t a r t a t x = 0 , a n d b e of l e n g t h L ( s e e F i g . 9 .3) . L e t t h e v e l o c i t y

prof i le a t l a r g e y b e g i v e n b y Ux(x), Uy = 0 , a n d l e t t h e v e l o c i t y in t h e

b o u n d a r y l a y e r h a v e c o m p o n e n t s vx a n d vy w h i c h a r e b o t h , in g e n e r a l ,

f u n c t i o n s of b o t h JC a n d y. L e t u s a s s u m e t h a t o u t s i d e of t h e b o u n d a r y

l a y e r it is r e a s o n a b l e t o t r e a t t h e flow a s f r i c t i o n l e s s , a n d t h a t 8, t h e

t h i c k n e s s of t h e b o u n d a r y l a y e r , is m u c h l e s s t h a n L .

T h e e q u a t i o n s of m o t i o n t h e n b e c o m e

a n d

(9 .B .9)

T h e first a p p r o x i m a t i o n w h i c h w e sha l l m a k e is t h a t t h e p r e s s u r e

e v e r y w h e r e c a n b e w r i t t e n a s t h e p r e s s u r e w h i c h w o u l d o b t a i n if t h e r e

w e r e n o v i s c o s i t y ( a n d w h i c h d o e s a c t u a l l y e x i s t o u t s i d e of t h e b o u n d a r y

l a y e r ) . O n e w a y of j u s t i f y i n g t h i s a p p r o x i m a t i o n is t o u s e t h e r e s u l t of

P r o b l e m 9.3 t h a t dP/dy is of o r d e r 8, s o t h a t t h e p r e s s u r e d i f f e r e n c e

a c r o s s t h e b o u n d a r y l a y e r m u s t b e

(9 .B.8)

(9 .B.7)

w h i l e c o n t i n u i t y te l l s u s t h a t

(9 .B .10)


s o t h a t t h e p r e s s u r e a t t h e s u r f a c e i s , t o o r d e r 82, t h e s a m e a s t h e p r e s s u r e

a t t h e b o u n d a r y l a y e r . T h i s p r e s s u r e , in t u r n , m u s t b e t h e s a m e a s t h e

p r e s s u r e a s s o c i a t e d w i t h n o n v i s c o u s f low.

O n e i m p o r t a n t r e s u l t f o l l o w s i m m e d i a t e l y . F r o m t h e B e r n o u l l i e q u a -

t i o n , t h e p r e s s u r e a s s o c i a t e d w i t h n o n v i s c o u s flow is g i v e n b y

T h i s e q u a t i o n a n d t h e e q u a t i o n of c o n t i n u i t y , t a k e n t o g e t h e r , a r e c a l l e d

t h e Prandtl equations, a n d t h e y d e s c r i b e t h e b o u n d a r y - l a y e r flow. T h e y

c o n s t i t u t e t w o e q u a t i o n s in t w o u n k n o w n s , a n d h e n c e c a n b e s o l v e d ( t h e

p r e s s u r e is n o t a n u n k n o w n h e r e , s i n c e it is g i v e n b y t h e flow a t l a r g e y ) .

C o n s i d e r n o w t h e c a s e w h e r e

pUx

2(x) = c o n s t .

S i n c e in o u r p r o b l e m , Ux is a f u n c t i o n of x o n l y , w e h a v e

(9 .B .11)

Ux(x) = Uo = c o n s t . ,

a n d h e n c e , f r o m t h e B e r n o u l l i e q u a t i o n ,

I n t h i s c a s e , t h e e q u a t i o n s w h i c h m u s t b e s o l v e d a r e

(9 .B.13)

a n d

s u b j e c t t o t h e b o u n d a r y c o n d i t i o n s t h a t

vx = 0 ,

Vy = 0 , (9 .B.14)

a t y = 0, a n d vx(y - » o o ) = Uo.

U s i n g t h i s r e s u l t , a n d t h e r e s u l t s of t h e d i m e n s i o n a l a n a l y s i s in P r o b l e m

9 .3 , w e find t h a t t h e E q . (9 .B.7) b e c o m e s

(9 .B.12)

Boundary Layers 149

A t e c h n i q u e w h i c h is o f t e n u s e f u l in s o l v i n g h y d r o d y n a m i c e q u a t i o n s

i n v o l v e s t h e i n t r o d u c t i o n of a stream function. S u p p o s e t h a t w e de f ine a

f u n c t i o n ip b y t h e r e l a t i o n s

a n d

(9 .B .15)

T h e n t h e e q u a t i o n of c o n t i n u i t y in t e r m s of t h e s t r e a m f u n c t i o n is j u s t

a n d is a u t o m a t i c a l l y sa t i s f ied . T h e N a v i e r - S t o k e s e q u a t i o n b e c o m e s

(9 .B .16)

w h i l e t h e b o u n d a r y c o n d i t i o n s a r e

(9 .B .17)

a n d

W e s e e , t h e n , t h a t w r i t i n g t h e N a v i e r - S t o k e s e q u a t i o n in t e r m s of

s t r e a m f u n c t i o n s , w h i l e it a u t o m a t i c a l l y sa t i s f ies c o n t i n u i t y , a l s o l e a d s t o

a n e x t r e m e l y c o m p l i c a t e d t h i r d - o r d e r n o n l i n e a r d i f f e ren t i a l e q u a t i o n

w h i c h m u s t b e s o l v e d . W e c a n m a k e s o m e p r o g r e s s t o w a r d a s o l u t i o n b y

m a k i n g a c h a n g e of v a r i a b l e s . If w e le t

(9 .B .18)

(9 .B.19)

a n d w r i t e

t h e n s u b s t i t u t i o n i n t o E q . (9 .B .13) g i v e s a n e w e q u a t i o n in t e r m s of t h e

f u n c t i o n / w h i c h is

(9 .B .20)


T h i s e q u a t i o n is s o l v a b l e in p r i n c i p l e , a n d c a n , in f a c t b e s o l v e d

n u m e r i c a l l y . A t a b u l a t i o n of t h e f u n c t i o n / i s g i v e n in t h e t e x t b y

S c h l i c h t i n g m e n t i o n e d in t h e b i b l i o g r a p h y . F o r o u r p u r p o s e s , w e s i m p l y

Fig. 9.4. A graph of the solutions to Eq. (9.B.20) as a function of 1 7 . The position of the boundary layer is indicated by an arrow.

(9 .B .21)

w i t h t h e b o u n d a r y c o n d i t i o n s

/ ( 0 ) = 0,

2 4 6 8 10

1.0

0.8

0.6-

0.4

0.2

Summary 151

w h i c h a g r e e s , of c o u r s e , w i t h t h e o r d e r of m a g n i t u d e e s t i m a t e g i v e n in E q .

(9 .B .6 ) .

A n i m p o r t a n t f e a t u r e of t h i s r e s u l t is t h a t t h e w i d t h of t h e b o u n d a r y

l a y e r g r o w s a s t h e s q u a r e r o o t of t h e d i s t a n c e a l o n g t h e p l a t e , a r e s u l t

w h i c h w a s a n t i c i p a t e d in t h e p r e v i o u s d i s c u s s i o n . T h u s , t h e f a r t h e r w e g o

d o w n s t r e a m , t h e w i d e r t h e r e g i o n o v e r w h i c h t h e r o l e of v i s c o s i t y is

i m p o r t a n t . T h i s m a k e s s e n s e in t e r m s of t h e c l a s s i c a l i d e a of v i s c o s i t y a s a

f r i c t i o n a l f o r c e b e t w e e n a d j a c e n t l a y e r s of f luid, s i n c e t h e l a y e r s n e a r t h e

p l a t e wil l f ee l t h e f o r c e s first, a n d p a s s t h e m a l o n g t o t h e n e x t l a y e r .

O n e p r o p e r t y of b o u n d a r y - l a y e r t h e o r y , w h i c h is q u i t e i m p o r t a n t in

a p p l i c a t i o n s b u t w h i c h w e sha l l n o t d i s c u s s in d e t a i l h e r e , o c c u r s w h e n t h e

flow of t h e fluid i n s i d e of t h e b o u n d a r y l a y e r i s in t h e r e v e r s e d i r e c t i o n

f r o m t h e flow o u t s i d e . T h i s is c a l l e d separation, a n d g i v e s r i s e t o e d d i e s in

t h e flow, a n d is o n e i m p o r t a n t a s p e c t of t h e t r a n s i t i o n f r o m l a m i n a r t o

t u r b u l e n t flow.

SUMMARY

If t h e N a v i e r - S t o k e s e q u a t i o n is p u t i n t o d i m e n s i o n l e s s f o r m , t h e

R e y n o l d s n u m b e r c a n b e d e f i n e d a n d r e p r e s e n t s a m e a s u r e of t h e r e l a t i v e

i m p o r t a n c e of f r i c t i o n a l a n d i n e r t i a l f o r c e s in t h e fluid. W h i l e v i s c o u s

f o r c e s m a y b e s m a l l t h r o u g h o u t t h e fluid t a k e n a s a w h o l e , t h e r e i s a s m a l l

r e g i o n n e a r a s t a t i o n a r y b o d y , c a l l e d t h e b o u n d a r y l a y e r , w h e r e t h e y

c a n n o t b e n e g l e c t e d . A n e x a m p l e of v i s c o u s flow p a s t a p l a n e s h e e t w a s

w o r k e d o u t t o i l l u s t r a t e t h e t r a n s i t i o n f r o m t h e b o u n d a r y l a y e r t o t h e m a i n

b o d y of t h e fluid.

p l o t in F i g . 9.4 t h e r a t i o

(9 .B .22)

v e r s u s £ T h i s s h o w s t h e a p p r o a c h t o a s y m p t o t i c v a l u e s of t h e v e l o c i t y a s

w e c o m e a w a y f r o m t h e p l a t e .

T h e r e i s , of c o u r s e , a c e r t a i n a m o u n t of a m b i g u i t y in de f in ing t h e w i d t h

of t h e b o u n d a r y l a y e r , s i n c e t h e t r a n s i t i o n b e t w e e n z e r o v e l o c i t y a n d U0 is

s m o o t h . H o w e v e r , it i s c u s t o m a r y t o de f ine t h e e d g e of t h e b o u n d a r y l a y e r

a s t h e p o i n t a t w h i c h t h e v e l o c i t y h a s a t t a i n e d 9 9 % of i t s a s y m p t o t i c

v a l u e . F o r t h e g r a p h , t h i s o c c u r s a t a b o u t £ = 5 , s o t h a t w e h a v e

(9 .B .23)


PROBLEMS

9 . 1 . Consider Poisieulle flow in a tube to be a model of the flow of the blood in an ar tery. A typical size for an ar tery would be 1 cm, and a typical p ressure would be 100 mm of mercury .

(a) Calculate the Reynolds number for this type of flow, given that a typical value of viscosity for blood is th ree or four t imes that of water .

(b) H o w fast must the velocity of the blood be in order to allow us to d rop the viscous te rm in the Nav i e r -S tokes equat ion?

(c) H o w fast must it be to allow us to d rop the nonlinear t e rm?

9.2. Suppose that we wanted to make measurements of blood flow, but for var ious reasons wanted to use the flow of air in a tube as a scale model of blood flow in an ar tery. H o w would you go about designing the scale model , and what pressures and flow ra tes would you use in the exper iment?

9.3. (a) Using the techniques of Sect ion 9.A, put Eqs . (9.B.7) and (9.B.8) into dimensionless form.

(b) Show by dimensional analysis of the y - component of the Nav ie r -S tokes equat ion that

(c) Show by dimensional analysis that

9.4. Consider the problem of flow past a plate. T h e drag force on the plate is given

[see Eq . (8.A.9)] by

(a) Using simple es t imates , show that

(b) If the plate has length L and width b, find an express ion for the total drag force.

(c) Wha t is the drag on a plate six feet long and three feet wide moving through the air at 30 mph?

9.5. Verify E q s . (9.B.20) and (9.B.22).

9.6. Why did we not have to wor ry about boundary layers when we solved the problem of Poisieulle flow?

9.7. T h e argument that the law of similarity should be expec ted to hold was based on the assumpt ion that we were dealing with an incompressible fluid. If this were not the case , another dimensionless number would enter the problem. F r o m the

Problems 153

definition of the bulk modulus of a material as

and the fact that the speed of sound is

show that a fluid may be regarded as incompressible provided that the Mack number, M, (defined as M = vie) satisfies the relation

9.8. There are many dimensionless numbers like the Reynolds number which play impor tant roles in var ious fields of hydrodynamics . L o o k up and define the following: Taylor , Prandt l , Ecker t , and Grashof numbers .

9.9. Consider one plane inclined at an angle a t o another , and moving with velocity v with respec t to it, and let the space be tween the wedges be filled with an incompressible v iscous fluid.

(a) Show that for small Reynolds number , the Nav ie r -S tokes equat ion is

where x is measured along the lower (stat ionary) plane, and y perpendicular to it. (b) Show that the pressure in the fluid is

where L is the length along the flat plane, hi the height of the gap at the small end, and h2 the height at the large end, and h the height at the point x.

(c) Calculate the total p ressure and the total shearing force along the bo t tom plane. Show that the coefficient of friction P IF is proport ional to h2/L. Since this can be made very small, the introduct ion of the fluid be tween the two moving planes greatly reduces the friction. This is the theory of lubrication. 9.10. Show that in the case of two-dimensional flow, the boundary- layer equat ions t ake the form

where U is the velocity outside the boundary layer. (a) Define a s t ream function and write down the equat ion describing it. (b) Consider now the two-dimensional flow of a v iscous fluid past two planes

inclined at an angle a to each other with a " s i n k " (a place for the fluid to flow out)


REFERENCES

H. Schlichting (see reference in Chapter 8) presents the best discussion of the topics in this chapter, and the reader is referred to that text for further references.

at the origin. Using the techniques in t roduced in Prob lems 4.10 and 4.11, show

that the flow in the boundary layer of the plane must be

where x is the dis tance measured along the plane from the sink, (c) Defining a new variable

and a s t ream function

show that the equat ion for / is

f - / ' 2 + 1 = 0.

(d) Solve this equat ion to get

where

(e) H e n c e show numerically that the width of the boundary layer in this case is approximately

Compare this with Eq . (9.B.6).

10

Heat, Thermal Convection, and the Circulation of the Atmosphere

For I had done a dreadful thing And it would work us woe For all averred I'd killed the bird That made the breeze to blow

SAMUEL TAYLOR COLERIDGE

Rime of the Ancient Mariner

A. THE HEAT EQUATION AND THE BOSSINESQ APPROXIMATION

U p t o t h i s p o i n t in o u r s t u d i e s , w e h a v e p a i d v e r y l i t t le a t t e n t i o n t o t h e

p r o p e r t i e s of f lu ids t h a t h a v e t o d o w i t h t e m p e r a t u r e a n d h e a t . W e k n o w

t h a t s u c h p r o p e r t i e s e x i s t , h o w e v e r , a n d w e wi l l s t u d y s o m e of t h e i r

c o n s e q u e n c e s in t h i s c h a p t e r .

O n t h e a t o m i c s c a l e , w e a r e u s e d t o t h i n k i n g of t e m p e r a t u r e a s b e i n g

a s s o c i a t e d w i t h t h e m o t i o n s of a t o m s . If t h e a t o m s h a v e a l a r g e k i n e t i c

e n e r g y , w e s p e a k of a h i g h t e m p e r a t u r e . S i m i l a r l y , w e de f ine a b s o l u t e

z e r o c l a s s i c a l l y a s t h e t e m p e r a t u r e a t w h i c h t h e k i n e t i c e n e r g y v a n i s h e s .

C o n s i d e r w h a t w o u l d h a p p e n if w e h a d a fluid in w h i c h o n e p a r t w a s

h e a t e d t o a t e m p e r a t u r e h i g h e r t h a n i t s n e i g h b o r s . I n t h e h e a t e d s e c t i o n ,

t h e m o l e c u l e s w o u l d b e m o v i n g f a s t e r . I n t h e c o u r s e of t h e i r c o l l i s i o n s

w i t h s u r r o u n d i n g m o l e c u l e s , s o m e of t h i s e n e r g y w o u l d , o n t h e a v e r a g e ,

b e t r a n s f e r r e d t o m o l e c u l e s w h i c h w e r e o r i g i n a l l y m o v i n g m o r e s l o w l y ,

t h e r e b y s p e e d i n g t h e m u p . O b s e r v i n g t h i s , w e w o u l d s a y t h a t h e a t w a s

b e i n g t r a n s f e r r e d f r o m t h e h o t t o t h e c o l d r e g i o n .

I n m a n y of t h e a p p l i c a t i o n s w h i c h w e h a v e t r e a t e d s o f a r , it w a s

r e a s o n a b l e t o n e g l e c t e f fec t s of t h i s t y p e . T h e r e a r e s o m e e f f e c t s , l i ke

t h e r m a l c o n v e c t i o n , in w h i c h t h e e f f ec t s of t e m p e r a t u r e d e p e n d e n c e s a r e

155

156 Heat, Thermal Convection, and the Circulation of the Atmosphere

t h e m o s t i m p o r t a n t c o n s i d e r a t i o n . W e b e g i n , t h e r e f o r e , b y d i s c u s s i n g t h e

c l a s s i c a l m e t h o d s of d e a l i n g w i t h h e a t t r a n s f e r a n d t h e r m a l e f f ec t s .

C o n s i d e r t w o p l a n e s in a fluid a n in f in i t e s ima l d i s t a n c e A J C a p a r t , w i t h a

t e m p e r a t u r e g r a d i e n t A 0 b e t w e e n t h e m ( i .e . t h e l e f t - h a n d p l a n e i s a t a

t e m p e r a t u r e 0, a n d t h e r i g h t - h a n d p l a n e a t a t e m p e r a t u r e 6 + A 0 ) . T h e n

t h e h e a t flux ( h e a t e n e r g y p e r u n i t a r e a p e r u n i t t i m e ) w h i c h will f low

b e t w e e n t h e p l a n e s is j u s t

w h e r e dO/dn is t h e t e m p e r a t u r e g r a d i e n t n o r m a l t o t h e s u r f a c e .

If t h e m a t e r i a l i n s i d e t h e s u r f a c e h a s d e n s i t y p , t h e n a c h a n g e in

t e m p e r a t u r e c o r r e s p o n d s t o a c h a n g e in i n t e r n a l e n e r g y g i v e n b y

dU = pcvde (10 .A.3)

w h e r e c v is t h e spec i f i c h e a t of t h e m a t e r i a l in t h e v o l u m e . T h u s , t h e r a t e

of c h a n g e of i n t e r n a l e n e r g y i n s i d e of t h e v o l u m e a s s o c i a t e d w i t h

t e m p e r a t u r e c h a n g e s is j u s t

i .e . it s t a t e s t h a t a n y c h a n g e in e n e r g y in t h e s y s t e m e n c l o s e d in t h e

s u r f a c e m u s t b e b a l a n c e d b y a t r a n s f e r of e n e r g y a c r o s s t h e b o u n d a r y .

If w e u s e t h i s e q u a l i t y , a n d a p p l y G a u s s ' t h e o r e m t o E q . (10 .A.5) t o

c o n v e r t t h e s u r f a c e i n t e g r a l t o a v o l u m e i n t e g r a l , w e find t h a t t h i s

e q u a t i o n r e q u i r e s t h a t

s o t h a t t h e e q u a t i o n w h i c h g o v e r n s t h e t e m p e r a t u r e in a n a r b i t r a r y b o d y is

i u s t _ .

N o w t h e c o n s e r v a t i o n of e n e r g y ( t h e first l a w of t h e r m o d y n a m i c s )

r e q u i r e s t h a t _ „

(10 .A.5)

(10 .A.6)

(10 .A.2)

(10 .A.4)

(10 .A.1)

w h e r e K i s c a l l e d t h e coef f ic ien t of t h e r m a l c o n d u c t i v i t y . F o r a n a r b i t r a r y

s u r f a c e , t h e n , t h e t o t a l h e a t o u t f l o w is g i v e n b y

The Heat Equation and the Bossinesq Approximation 157

w h e r e K = K/pcv is c a l l e d t h e coefficient of diffusivity. F o r o u r p u r p o s e s ,

w e sha l l a s s u m e t h a t K is a c o n s t a n t fo r a g i v e n m a t e r i a l .

T h e r e a d e r ' s a t t e n t i o n is c a l l e d t o t h e s i m i l a r i t y b e t w e e n t h e a r g u m e n t

p r e s e n t e d a b o v e a n d t h e d e r i v a t i o n of t h e e q u a t i o n of c o n t i n u i t y in

S e c t i o n l . C . W h y s h o u l d t h i s b e s o ?

T h e h e a t e q u a t i o n m u s t n o w t a k e i t s p l a c e , a l o n g w i t h t h e E u l e r

e q u a t i o n , c o n t i n u i t y , a n d t h e e q u a t i o n of s t a t e , a s o n e of t h e b a s i c

e q u a t i o n s w h i c h m u s t b e s o l v e d in d e s c r i b i n g t h e m o t i o n of a fluid. I t is

n a t u r a l t o a s k , t h e n , h o w t h e o t h e r e q u a t i o n s a r e a l t e r e d b y t h e p r e s e n c e

of t h e r m a l e f f e c t s .

C o n s i d e r t h e E u l e r e q u a t i o n a s a n e x a m p l e . I n i t s m o s t g e n e r a l f o r m , it

c a n b e w r i t t e n

( e x t e r n a l ) , (10 .A.7)

w h e r e a\k is d e f i n e d in E q . (8 .A .9 ) . I n g e n e r a l , b o t h of t h e coe f f i c i en t s of

v i s c o s i t y , £ a n d 17, c a n d e p e n d o n t h e t e m p e r a t u r e of t h e fluid. M o r e

i m p o r t a n t , t h e v i s c o u s t e r m in t h e E u l e r e q u a t i o n wil l r e d u c e t o t h e

f a m i l i a r i)V2v o n l y in t h e c a s e of a n i n c o m p r e s s i b l e fluid. If, f o r s o m e

r e a s o n , w e c a n n o t u s e t h e s impl i f i ed f o r m of t h e e q u a t i o n of c o n t i n u i t y

w h i c h a p p l i e s t o a n i n c o m p r e s s i b l e fluid, t h e E u l e r e q u a t i o n wi l l b e c o m e

v e r y difficult t o h a n d l e m a t h e m a t i c a l l y .

T h e g e n e r a l e q u a t i o n of c o n t i n u i t y t a k e s t h e f o r m

(10 .A.8)

If p i s c o n s t a n t , t h i s wil l s imp l i fy t o

V • v = 0. (10 .A.9)

F o r a fluid in w h i c h t h e r m a l e f f ec t s a r e i m p o r t a n t , h o w e v e r , s u c h a

s imp l i f i ca t i on is n o t p o s s i b l e , s i n c e t h e r e wil l b e a n e q u a t i o n of s t a t e

w h i c h will l i nk d e n s i t y t o t e m p e r a t u r e . F o r a n i d e a l g a s , f o r e x a m p l e , w e

w o u l d h a v e

P = pR6.

T h u s in i n t r o d u c i n g t h e r m a l e f f e c t s , w e a r e in e f fec t l o o s i n g E q . (10 .A .9 ) .

S i n c e t h e r e l a t i v e l y s i m p l e f o r m of t h i s e q u a t i o n w a s i n s t r u m e n t a l in

a l l o w i n g u s t o s o l v e p r o b l e m s u p t o t h i s p o i n t , t h i s is a r a t h e r s e r i o u s

m a t t e r .

S o l o n g a s w e w e r e d e a l i n g w i t h s y s t e m s l ike t h e t i d e s , t h e

a p p r o x i m a t i o n of i n c o m p r e s s i b i l i t y w a s g o o d , s i n c e f o r t h e t y p e of

t e m p e r a t u r e d i f f e r e n c e s w h i c h e x i s t in t h a t p r o b l e m , d e n s i t y c h a n g e s a r e


n o t t o o g r e a t . T h i s is a c t u a l l y a r e a s o n a b l e a p p r o x i m a t i o n fo r m o s t l i qu id s

( e x c e p t in t h e c a s e of t h e r m a l c o n v e c t i o n ) . T h e r e a r e s o m e s i t u a t i o n s

w h e r e it is n o t s u c h a g o o d a p p r o x i m a t i o n , a n d w e a r e f a c e d w i t h t h e

p r o b l e m of e i t h e r t r e a t i n g t h e e q u a t i o n s of m o t i o n in t h e i r full c o m p l e x i t y

(a f o r m i d a b l e t a s k ) , o r f inding a n o t h e r r e a s o n a b l e a p p r o x i m a t i o n s c h e m e

fo r d i s c u s s i n g t h e r m a l e f fec t s in g a s e s . T h i s s c h e m e w a s first a d v a n c e d b y

H . B o s s i n e s q , a n d b e a r s h i s n a m e . T h e a p p r o x i m a t i o n is b a s e d o n t w o

o b s e r v a t i o n s :

1. T h e coef f i c i en t s of v i s c o s i t y a n d d i f fus ion v a r y s l o w l y w i t h

t e m p e r a t u r e f o r m o s t m a t e r i a l s .

2 . I n t r e a t i n g c o n v e c t i o n , w e e x p e c t t h e m o s t i m p o r t a n t e f f ec t s t o a r i s e

f r o m t h e f a c t t h a t w a r m a i r i s l i g h t e r t h a n c o l d a i r — i . e . f r o m t h e w a y

in w h i c h t h e g a s a t d i f f e ren t t e m p e r a t u r e s is a f f e c t e d b y g r a v i t y .

T h e s e o b s e r v a t i o n s l e d B o s s i n e s q t o p r o p o s e t h e f o l l o w i n g a p p r o x i m a t i o n

s c h e m e : I g n o r e t h e v a r i a t i o n of all q u a n t i t i e s in t h e e q u a t i o n s of m o t i o n

w i t h t e m p e r a t u r e except i n s o f a r a s t h e y a r e c o n c e r n e d w i t h g r a v i t a t i o n a l

e f f ec t s . I n o t h e r w o r d s , w e sha l l i n c l u d e t h e v a r i a t i o n of d e n s i t y w i t h

t e m p e r a t u r e in t h e t e r m p • F*(ex t ) o n t h e r i g h t - h a n d s i d e of E q . (10 .A .7 ) ,

b u t sha l l t r e a t p a s a c o n s t a n t in all o t h e r e q u a t i o n s .

T h i s i m m e d i a t e l y r e s u l t s in e n o r m o u s s i m p l i f i c a t i o n s . T h e e q u a t i o n of

c o n t i n u i t y r e d u c e s t o t h e f a m i l i a r f o r m of E q . (10 .A .9 ) . If w e a r e d e a l i n g

w i t h a s y s t e m in w h i c h t h e o n l y e x t e r n a l f o r c e is g r a v i t y ( a s w o u l d b e t h e

c a s e , f o r e x a m p l e , in c o n s i d e r i n g t h e m o t i o n of t h e a t m o s p h e r e ) , t h e

e x t e r n a l f o r c e t e r m in t h e E u l e r e q u a t i o n b e c o m e s

w h e r e a i s t h e coef f ic ien t of e x p a n s i o n f o r t h e g a s , de f i ned b y

a n d do is t h e t e m p e r a t u r e a t w h i c h t h e d e n s i t y is p 0 .

T h u s , t h e E u l e r e q u a t i o n f o r a fluid in w h i c h t h e r m a l e f f ec t s a r e a l l o w e d

r e d u c e s t o

p F ( e x t ) = pg = gpo = gpo-gpoaO, (10 .A.10)

(10 .A.11)

(10 .A.12)

w h e r e y is a u n i t v e c t o r in t h e d i r e c t i o n of t h e g r a v i t a t i o n a l f o r c e , a n d w e

h a v e de f ined o u r t e m p e r a t u r e s c a l e s o t h a t 0 O = 0.

Stability of a Fluid between Two Plates 159

B. STABILITY OF A FLUID BETWEEN TWO PLATES

A s t h e first e x a m p l e of a f luid s y s t e m in w h i c h t h e r m a l e f f ec t s a r e

i m p o r t a n t , l e t u s c o n s i d e r t h e s i t u a t i o n s h o w n in F i g . 10 .1 , in w h i c h t h e r e

a r e t w o r ig id p l a t e s a d i s t a n c e h a p a r t , w i t h t h e l o w e r o n e m a i n t a i n e d a t

t e m p e r a t u r e 0i a n d t h e u p p e r o n e a t 0 2 . L e t u s b e g i n b y e x a m i n i n g t h e

s t a b i l i t y of s u c h a s y s t e m .

If t h e s y s t e m is l o c a t e d in a g r a v i t a t i o n a l field, s u c h a s t h a t a t t h e

s u r f a c e of t h e e a r t h , a n d if 0i > 0 2 , o u r i n t u i t i o n t e l l s u s t h a t it s h o u l d b e

u n s t a b l e . T h i s f o l l o w s f r o m t h e f a c t t h a t t h e w a r m fluid a t t h e b o t t o m wi l l

b e l e s s d e n s e t h a n t h e c o l d fluid a t t h e t o p , s o t h a t t h e g r a v i t a t i o n a l e n e r g y

of t h e s y s t e m c o u l d b e l o w e r e d b y l e t t i n g t h e w a r m a i r r i s e a n d t h e c o l d

a i r fa l l , a s i n d i c a t e d in F i g . 10 .1 . T h i s e x c h a n g e i s , of c o u r s e , w h a t w e

n o r m a l l y t h i n k of a s t h e r m a l c o n v e c t i o n .

L e t u s s e e if t h i s i n t u i t i v e r e s u l t c a n b e d e r i v e d f r o m t h e e q u a t i o n s of

m o t i o n d e r i v e d in t h e p r e v i o u s s e c t i o n . T h e first s t e p in d i s c u s s i n g

s t a b i l i t y i s , of c o u r s e , t o find t h e e q u i l i b r i u m p o i n t of t h e s y s t e m . If w e

de f ine © a s t h e t e m p e r a t u r e a t t h e p o i n t y f o r e q u i l i b r i u m , t h e n b y

i n s p e c t i o n w e s e e t h a t t h e e q u a t i o n s wi l l b e sa t i s f i ed if

v = 0 (10 .B . 1 )

a n d

Fig. 10.1. The geometry for the discussion of thermal convection.

(10 .B .2)

w h e r e t h e l a s t e q u a l i t y f o l l o w s f r o m t h e a s s u m p t i o n t h a t t h e s y s t e m h a s

inf in i te e x t e n t in t h e z - a n d x - d i r e c t i o n s . T h e t e m p e r a t u r e e q u a t i o n c a n b e

s o l v e d t o g i v e

0 = /3y + 0 1 ? (10 .B .3)

T h i s a p p r o x i m a t e f o r m of t h e N a v i e r - S t o k e s e q u a t i o n , t o g e t h e r w i t h

t h e c o n t i n u i t y c o n d i t i o n , e q u a t i o n of s t a t e , a n d h e a t e q u a t i o n , t h e n

b e c o m e s t h e m e a n s b y w h i c h w e sha l l d e s c r i b e t h e m o t i o n of h e a t e d

f lu ids .


w h e r e

(10 .B.4)

T h i s e q u i l i b r i u m is e a s y t o p i c t u r e — i t c o r r e s p o n d s t o h a v i n g t h e fluid

c o m p l e t e l y a t r e s t , w i t h p r e s s u r e f o r c e s b a l a n c e d b y g r a v i t y , a n d a

u n i f o r m t e m p e r a t u r e g r a d i e n t b e t w e e n t h e p l a t e s . I n w h a t f o l l o w s , w e

sha l l i n v e s t i g a t e t h e s t a b i l i t y of t h i s s y s t e m b y a s s u m i n g t h a t t h e r e a r e s m a l l

t i m e - d e p e n d e n t d e v i a t i o n s f r o m t h i s e q u i l i b r i u m , a n d s e e w h e t h e r t h e y

g r o w a s a f u n c t i o n of t i m e o r n o t .

If w e de f ine a n e w v a r i a b l e b y t h e r e l a t i o n

0' = 0 - e = 0 - [fiy + 0i], (10 .B.5)

w h e r e 0 is t h e a c t u a l t e m p e r a t u r e a t y, t h e n t h e E u l e r e q u a t i o n in t h e f o r m

(10 .A . 12) c a n b e w r i t t e n

V ( P + g p y ) + a ( 0 ' + @)y + vV2\

fp + gpy + yp 0 ( y ' ) dy'^j + yS'y + vV2v, (10 .B.6)

w h e r e w e h a v e w r i t t e n

y = ga. (10 .B.7)

T h e h e a t e q u a t i o n c a n a l s o b e w r i t t e n in t e r m s of 6 ( n o t 6'), a n d is j u s t

= K V 2 ( @ + 0 ' ) = = ' < V 2 0 \

( 0 + 0 ' ) + v - V ( 0 ' + ©) .

(10 .B.8)

N o w b y de f in i t ion , t h e t i m e d e r i v a t i v e of S v a n i s h e s . S i n c e w e wil l b e

u s i n g t h i s e q u a t i o n t o e x a m i n e d e p a r t u r e s f r o m a n e q u i l i b r i u m in w h i c h

t h e v e l o c i t y is z e r o , v r e p r e s e n t s d e p a r t u r e s f r o m e q u i l i b r i u m , a n d c a n

t h e r e f o r e b e r e g a r d e d a s a s m a l l q u a n t i t y . T h e s a m e is t r u e of 0 ' . T h u s ,

t h e t e r m (v • V ) 0 ' in t h e a b o v e c a n b e d r o p p e d . S i n c e & is a f u n c t i o n of y

o n l y , w e a r e lef t w i t h

(10.B.9)

w h e r e w e h a v e u s e d t h e s t a t e m e n t d @/dy = /3 f r o m E q . (10 .B .3 ) .

L e t u s n o w e x a m i n e t h e s t a b i l i t y of t h e e q u i l i b r i u m w h i c h w e h a v e

f o u n d b y l o o k i n g a t t h e b e h a v i o r of t h e s y s t e m w h e n sma l l p e r t u r b a t i o n s


f r o m e q u i l i b r i u m a r e i n t r o d u c e d . I n p a r t i c u l a r , l e t u s , g u i d e d b y t h e

s y m m e t r y of t h e p r o b l e m , a s s u m e t h a t

A(y)\

B(y) \ei>xe>mieM. (10 .B .10)

C ( y ) /

s e e if w e c a n s a t i s fy all of t h e e q u a t i o n s w h e n w e

Q = ( 7 ^ ( 1 0 B 1 1 )

P' = P + gpy + yp J y @ ( y ' ) dy'.

T h e log i c of t h i s a p p r o a c h is a s f o l l o w s : W e a s s u m e v e l o c i t i e s ,

p r e s s u r e s a n d t e m p e r a t u r e d e v i a t i o n a s a b o v e . W e sha l l s e e t h a t t h e s e

f o r m s c a n , i n d e e d , s a t i s fy t h e b a s i c e q u a t i o n s . W e sha l l t h e n l o o k f o r t h e

b e h a v i o r of t h e s y s t e m a s a f u n c t i o n of t i m e b y s o l v i n g f o r n. If t h e v a l u e s

w h i c h w e find a r e p o s i t i v e , t h e n t h e s y s t e m wil l b e u n s t a b l e a g a i n s t t h e

t y p e of p e r t u r b a t i o n s t h a t w e h a v e a s s u m e d . S i n c e a n a r b i t r a r y

p e r t u r b a t i o n wil l c o n t a i n s o m e c o m p o n e n t w h i c h c a n b e e x p r e s s e d a s t h e

a b o v e , t h i s m e a n s t h a t t h e s y s t e m will b e u n s t a b l e , a n d wi l l n o t s t a y in i t s

e q u i l i b r i u m c o n f i g u r a t i o n .

T h e d e t e r m i n a t i o n of t h e f u n c t i o n s A ( y ) , JB(y) , a n d C ( y ) in E q .

(10 .B .10) i s , in g e n e r a l , n o t s o m e t h i n g w h i c h c a n b e d o n e b y i n s p e c t i o n ,

s i n c e it i n v o l v e s t h e b o u n d a r y c o n d i t i o n s a t t h e t w o s u r f a c e s y = 0 a n d

y = h. C o n s e q u e n t l y , w e sha l l d i s c u s s t h e g e n e r a l t e c h n i q u e w h i c h c a n b e

u s e d t o s o l v e f o r t h e s e f u n c t i o n s , b u t w o r k o u t in d e t a i l o n l y t h e s i m p l e s t

p o s s i b l e b o u n d a r y c o n d i t i o n s — t h a t in w h i c h b o t h p l a n e s b o u n d i n g t h e

fluid a r e c o n s i d e r e d t o b e f r e e s u r f a c e s , s o t h a t t h e c o n d i t i o n

vy = 0 a t y = 0, y = h,

m u s t h o l d . C l e a r l y , if t h e s u r f a c e s w e r e r ig id , w e w o u l d h a v e in a d d i t i o n

t o t h e a b o v e t h e s t a t e m e n t t h a t n o t o n l y vy, b u t vx a n d vz w o u l d a l s o v a n i s h

a t t h e s e s u r f a c e s .

I n a d d i t i o n t o t h e b o u n d a r y c o n d i t i o n s a b o v e , w e k n o w t h a t t h e

v e l o c i t i e s m u s t s a t i s fy t h e e q u a t i o n of c o n t i n u i t y fo r a n i n c o m p r e s s i b l e

fluid, w h i c h is

I n a d d i t i o n , w e wil l

a s s u m e

w h e r e


a n d

If t h e s e c o n d i t i o n s d i d n o t h o l d , it w o u l d b e i m p o s s i b l e t o s a t i s fy t h e

e q u a t i o n of c o n t i n u i t y a t all p o i n t s in t h e fluid.

I n g e n e r a l , w e w o u l d a s s u m e a s o l u t i o n f o r B(y) of t h e f o r m

B(y) = esy,

a n d s o l v e f o r t h e v a l u e s of 5 w h i c h a r e c o n s i s t e n t w i t h t h e b o u n d a r y

c o n d i t i o n s . F o r t h e c a s e of t w o f r e e b o u n d a r i e s , h o w e v e r , w e c a n s e e b y

i n s p e c t i o n t h a t

J 5 ( y ) oc s in y^ = s in sy

( w h e r e q is a n i n t e g e r ) wi l l s a t i s f y t h e b o u n d a r y c o n d i t i o n s . T h u s , w e find

t h a t t h e e q u a t i o n of c o n t i n u i t y a n d t h e b o u n d a r y c o n d i t i o n s wi l l b e

sa t i s f ied p r o v i d e d t h a t

A ( y ) = A c o s sy,

B(y) = B s i n s y , (10 .B.12)

C ( y ) = C c o s sy,

w h e r e t h e c o n s t a n t s A, B, a n d C m u s t s a t i s fy t h e r e l a t i o n

UA + sB + imC = 0 . (10 .B.13)

If w e n o w d r o p t h e (v • V)v t e r m s a s b e i n g of s e c o n d o r d e r in s m a l l

q u a n t i t i e s , w e c a n i n s e r t t h e a s s u m e d f o r m s of t h e s o l u t i o n s in E q s .

(10 .B .10) a n d (10 .B .11) i n t o t h e t h r e e c o m p o n e n t s of t h e E u l e r e q u a t i o n t o

o b t a i n

(10 .B .14)

l2+m2+ s2 = a

If w e le t

a n d

(10 .B .15)

(10 .B.16) n1 = n + va,

s o t h a t a s i d e f r o m c o n s t a n t s of p r o p o r t i o n a l i t y , w e m u s t h a v e

A ( y ) o c C ( y )


t h e s e e q u a t i o n s s imp l i fy t o

T h e c o n t i n u i t y e q u a t i o n is j u s t

ilvx + imvz 4 - = 0 .

B y d i f f e r e n t i a t i n g t h e x- a n d z - c o m p o n e n t s of t h e E u l e r e q u a t i o n w i t h

r e s p e c t t o y, w e find

w h i c h , w h e n s u b s t i t u t e d i n t o t h e y - c o m p o n e n t of t h e E u l e r e q u a t i o n

[ t a k i n g a c c o u n t of E q . (10 .B .12 ) ] , y i e l d s

T h u s , t h e n e t r e s u l t of o u r m a n i p u l a t i o n s of t h e E u l e r e q u a t i o n a n d

c o n t i n u i t y is t o g i v e u s o n e e q u a t i o n r e l a t i n g vy a n d 0 ' . A n o t h e r s u c h

e q u a t i o n c a n b e o b t a i n e d b y i n s e r t i n g o u r a s s u m e d f o r m s of s o l u t i o n i n t o

t h e h e a t e q u a t i o n , g i v i n g

T h u s , t h e p r o b l e m of f ind ing a s o l u t i o n t o t h e E u l e r e q u a t i o n , t h e h e a t

e q u a t i o n a n d t h e c o n t i n u i t y c o n d i t i o n f o r p e r t u r b a t i o n s of t h e t y p e w h i c h

w e h a v e a s s u m e d r e d u c e s t o t h e p r o b l e m of s o l v i n g t h e a b o v e t w o l i n e a r

e q u a t i o n s . I t is w e l l k n o w n t h a t s o l u t i o n s f o r vy a n d 0 ' wi l l e x i s t p r o v i d e d

t h a t t h e W r o n s k i a n d e t e r m i n a n t v a n i s h e s — i . e .

n'avy-ye'(l2 + m2) = §. (10 .B .17)

fivy + [n + K a ] 0 ' = 0 . (10 .B .18)

o r

py(l2+m2) + n'na + n ' * a 2 = 0 . (10 .B.19)

T h i s e q u a t i o n d e t e r m i n e s t h e g r o w t h c o n s t a n t n. P r o v i d e d t h i s e q u a t i o n

is sa t i s f ied , s o l u t i o n s of t h e t y p e w h i c h w e h a v e a s s u m e d wil l e x i s t . T h e

q u e s t i o n of s t a b i l i t y o r i n s t a b i l i t y of t h e s y s t e m r e d u c e s , t h e n , t o f ind ing


o u t u n d e r w h a t c o n d i t i o n s t h e v a l u e of n f r o m t h e a b o v e e q u a t i o n will b e

p o s i t i v e .

I n o r d e r t o d i s c u s s t h e p h y s i c s of t h i s e q u a t i o n , le t u s c o n s i d e r a fluid

w h i c h h a s b o t h TJ a n d K s e t e q u a l t o z e r o . T h i s w o u l d c o r r e s p o n d t o a n

" i d e a l " fluid in t h e s e n s e t h a t it w o u l d b e n o n v i s c o u s , a n d t h e t e m p e r a t u r e

of a v o l u m e e l e m e n t w o u l d n o t c h a n g e , b u t t h e p r i m e f e a t u r e w h i c h w e

a r e c o n s i d e r i n g , w h i c h is t h e c h a n g e of d e n s i t y w i t h t e m p e r a t u r e , i s still in

t h e p r o b l e m . I n t h i s c a s e , t h e c o n d i t i o n in E q . (10 .B.19) r e d u c e s t o

N o w t h e r e a r e t w o p o s s i b i l i t i e s f o r n. If j8 > 0 ( i .e . if t h e t e m p e r a t u r e

g r a d i e n t is s u c h t h a t t h e h i g h e r t e m p e r a t u r e is a t t h e t o p , t h e n n = ± i\n\,

a n d all d e v i a t i o n s f r o m e q u i l i b r i u m b e h a v e l ike e±llnlt—i.e. t h e y d o

n o t g r o w a s a f u n c t i o n of t i m e , b u t o s c i l l a t e a b o u t t h e e q u i l i b r i u m

c o n f i g u r a t i o n . S u c h s i t u a t i o n s a r e s t a b l e .

O n t h e o t h e r h a n d , if p < 0 , ( i .e . if t h e l o w e r p l a t e i s m a i n t a i n e d a t a

h i g h e r t e m p e r a t u r e t h a n t h e u p p e r o n e ) , t h e n

a n d s m a l l d e v i a t i o n s f r o m e q u i l i b r i u m wil l g r o w e x p o n e n t i a l l y in t i m e .

T h i s , of c o u r s e , is t h e r e s u l t w h i c h w e e x p e c t e d i n t u i t i v e l y .

W e s e e t h e n t h a t in t h i s s i m p l e c a s e t h e fluid wi l l b e u n s t a b l e if t h e r e is

e v e n t h e s m a l l e s t a d v e r s e t e m p e r a t u r e g r a d i e n t . T h e q u e s t i o n of w h a t will

h a p p e n t o t h e s y s t e m a t l a r g e t i m e s wil l b e d i s c u s s e d in t h e n e x t s e c t i o n .

B u t E q . (10 .B .20) a l r e a d y h a s s o m e u n f o r t u n a t e c o n s e q u e n c e s , s i n c e it

p r e d i c t s t h a t if t h e r e is e v e r a s i t u a t i o n in w h i c h a i r a t a h i g h l eve l o v e r a

c i t y is w a r m e r t h a n t h e a i r n e a r t h e g r o u n d ( th i s is k n o w n a s a n inversion),

t h e r e will b e a s t a b l e s i t u a t i o n . W h e n t h i s h a p p e n s in L o s A n g e l e s , a s it

d o e s p e r i o d i c a l l y , t h e ef f luents in t h e a t m o s p h e r e c a n n o t b e r e m o v e d b y

t h e n o r m a l c i r c u l a t i o n of t h e a i r , a n d a s m o g c r i s i s r e s u l t s .

A n o t h e r q u e s t i o n w e c a n a s k a t t h i s s t a g e is w h a t t h e m a x i m u m v a l u e of

n i s , s i n c e t h i s c a n b e e x p e c t e d t o g o v e r n t h e g r o w t h r a t e of t h e

i n s t a b i l i t i e s . F r o m E q . (10 .B .21) , it is c l e a r t h a t n wil l b e a m a x i m u m w h e n

5 2 ( a n d t h e r e f o r e a ) is a m i n i m u m f o r a g i v e n / a n d m. T h u s , t h e f a s t e s t

g r o w i n g d i s t u r b a n c e will c o r r e s p o n d t o

(10 .B.20)

(10 .B.21)

(10 .B.22)


W e s h o u l d a l s o n o t e in p a s s i n g t h a t w h i l e w e h a v e b e e n a s s u m i n g t h a t

v e l o c i t i e s wil l g r o w e x p o n e n t i a l l y in t i m e , t h e y c l e a r l y c a n n o t d o s o

f o r e v e r . I n f a c t , s i n c e w e h a v e b e e n d r o p p i n g s e c o n d - o r d e r t e r m s in o u r

d e r i v a t i o n , w h i c h a m o u n t s t o i g n o r i n g v2 w i t h r e s p e c t t o v, o u r s o l u t i o n

wil l b e i n v a l i d f o r l a r g e v a l u e s of t in a n y c a s e .

H a v i n g s e e n h o w t h e e q u a t i o n s of m o t i o n c a n g i v e u s t h e r e s u l t w h i c h

w e e x p e c t e d in t h e " i d e a l " c a s e , le t u s n o w l o o k a t a m o r e r e a l i s t i c f luid, in

w h i c h n e i t h e r v n o r K a r e z e r o . I n t h i s c a s e , s o l v i n g E q . (10 .B .19) f o r n,

g i v e s

(10 .B .23)

O n c e m o r e , w e s e e t h a t t o h a v e i n s t a b i l i t y , t h e t e r m u n d e r t h e r a d i c a l

m u s t b e p o s i t i v e , w h i c h is o n l y p o s s i b l e if j3 < 0 . T h u s , a n a d v e r s e

t e m p e r a t u r e g r a d i e n t i s a g a i n a n e c e s s a r y ( b u t n o t suff ic ient ) , c o n d i t i o n

f o r s t a b i l i t y . A c t u a l l y , a m o r e p r e c i s e s t a t e m e n t of t h i s c o n d i t i o n is

\P\y(l2+m2)>Kva3.

W e s e e t h a t if e i t h e r K = 0 o r v = 0 , it will a l w a y s b e p o s s i b l e t o find s o m e

n > 0 , w h i c h is t h e r e s u l t t h a t w e h a d d e r i v e d p r e v i o u s l y . H o w e v e r , if

b o t h K a n d v a r e n o n z e r o , t h i s n e e d n o t n e c e s s a r i l y b e t h e c a s e .

D e f i n e

f(a) = \p\y(a - s2) - KVCL\ (10 .B .24)

If f(a) is p o s i t i v e , t h e n w e c a n h a v e i n s t a b i l i t y ( s e e E q . (10 .B .23) ) , s o t h e

p r d p e r t i e s of / ( a ) wil l d e t e r m i n e s t ab i l i t y . L e t u s e x a m i n e t h i s c u r v e fo r

fixed s a s a f u n c t i o n of I2 + m 2 . If / 2 + m 2 is v e r y l a r g e , f(a) b e c o m e s

n e g a t i v e . S i m i l a r l y , if / 2 + m 2 is v e r y s m a l l , f(a) b e c o m e s n e g a t i v e . S o m e

p o s s i b l e c u r v e s f o r / ( a ) a r e s h o w n in F i g . 10.2. W h i c h c u r v e r e p r e s e n t s t h e

Fig. 10.2. Some sample curves of f(a) as a function I2 + m2.


f u n c t i o n d e p e n d s o n t h e c h o i c e of o t h e r p a r a m e t e r s , i n c l u d i n g v, K, J8, a n d 5.

If w e h a d a s i t u a t i o n in w h i c h a c u r v e s u c h a s (a) w a s f o u n d t o h o l d , w e

n o t e t h a t t h e r e is a r a n g e of v a l u e s of / 2 + m 2 f o r w h i c h f(a) is p o s i t i v e ,

a n d f o r w h i c h n c o u l d t h e r e f o r e b e p o s i t i v e , a n d t h e s y s t e m c o u l d b e

u n s t a b l e . A c u r v e of t y p e ( c ) , h o w e v e r , w o u l d c o r r e s p o n d t o a s i t u a t i o n

w h e r e t h e r e w a s n o c h o i c e of p a r a m e t e r s w h i c h c o u l d s a t i s f y t h e

c o n d i t i o n n > 0 , s o t h a t n o v a l u e of n w o u l d b e p o s i t i v e . S u c h a c u r v e

w o u l d c o r r e s p o n d t o a s y s t e m w h i c h w a s c o m p l e t e l y s t a b l e a g a i n s t all

p e r t u r b a t i o n s of t h e t y p e w h i c h w e a r e c o n s i d e r i n g . T h e c u r v e (b)

c o r r e s p o n d s t o t h e " c r i t i c a l " c u r v e a t w h i c h t h e s y s t e m m o v e s f r o m

s t ab i l i t y t o i n s t a b i l i t y . T o find t h e v a l u e of I2 + m2 a t w h i c h t h e s y s t e m j u s t

b e c o m e s s t a b l e , w e s e t t h e d e r i v a t i v e of f(a) e q u a l t o z e r o ( th i s wil l

c o r r e s p o n d t o t h e m a x i m u m in t h e c u r v e a t t h e p o i n t P ) .

( s i n c e s = qir/h > ir/h) i .e . o n l y if t h e v a l u e of t h e f u n c t i o n a t i t s

(10 .B .25)

S i n c e t h e c u r v e (b) is spec i f i ed b y t h e e q u a t i o n

/ ( a ) = 0

a t P , w e c a n s o l v e t h i s f o r t h e p a r a m e t e r KV a n d p l u g t h e r e s u l t b a c k i n t o

E q . (10 .B .25) t o g i v e

w h i c h c o r r e s p o n d s t o t h e c o n d i t i o n

(10 .B.26)

W h e n t h i s c o n d i t i o n is sa t i s f i ed , t h e r e q u i r e m e n t t h a t f(a) b e z e r o a t t h e

p o i n t P b e c o m e s

(10 .B .27)

W h a t w e h a v e d e r i v e d , t h e n , is t h e f o l l o w i n g : U n l e s s f(a) c a n b e

p o s i t i v e f o r s o m e v a l u e of / 2 + m 2 , t h e s y s t e m wi l l b e s t a b l e a g a i n s t a n y

p e r t u r b a t i o n of t h e t y p e w e a r e c o n s i d e r i n g . B u t / ( a ) wil l b e p o s i t i v e o n l y

if

(10 .B .28)

Convection Cells 167

m a x i m u m is g r e a t e r t h a n z e r o . T h u s , w h e n w e i n c l u d e v i s c o s i t y in o u r

c o n s i d e r a t i o n s , t h e r e wi l l b e , f o r a g i v e n c h o i c e of m a t e r i a l f o r t h e fluid, a

r a n g e of a d v e r s e t e m p e r a t u r e g r a d i e n t s f o r w h i c h t h e s y s t e m wil l b e

c o m p l e t e l y s t a b l e . T h e a d d i t i o n of v i s c o s i t y t h u s m a k e s a q u a l i t a t i v e

d i f f e r e n c e in t h e s t ab i l i t y p r o b l e m . T h e a b o v e c o n d i t i o n i s c a l l e d t h e

R a y l e i g h c o n d i t i o n , a f t e r L o r d R a y l e i g h , w h o first d i s c o v e r e d i t s

s i gn i f i cance .

A w a y of v i s u a l i z i n g t h i s e f fec t is t o c o m p a r e v i s c o s i t y t o s t a t i c f r i c t i o n

in m e c h a n i c s . If a w e i g h t r e s t s o n a s u r f a c e , it i s n e c e s s a r y t o a p p l y s o m e

f o r c e in o r d e r t o g e t it t o m o v e a t al l . T h e r e i s , in a d d i t i o n , a c r i t i c a l f o r c e

b e l o w w h i c h n o m o t i o n wil l r e s u l t . V i s c o s i t y p l a y s t h e s a m e r o l e in t h e

p r o b l e m of t h e r m a l i n s t a b i l i t i e s . W h e n w e a p p l y a n a d v e r s e t h e r m a l

g r a d i e n t , it is n e c e s s a r y t o o v e r c o m e t h e i n t e r n a l f r i c t i o n in t h e fluid in

o r d e r t o g e t it t o m o v e , a n d t h e c r i t i c a l v a l u e of j8 in t h e R a y l e i g h c r i t e r i o n

c o r r e s p o n d s t o t h e c r i t i c a l f o r c e in t h e m e c h a n i c a l p r o b l e m .

I t is i n t e r e s t i n g t o n o t e in p a s s i n g t h a t e v e r y n e w ef fec t w h i c h w e a d d t o

t h e s y s t e m — t h e r m a l c o n d u c t i o n , v i s c o s i t y , e t c . , s e e m s t o w o r k in t h e

d i r e c t i o n of i n c r e a s i n g t h e s t a b i l i t y of t h e s y s t e m a g a i n s t p e r t u r b a t i o n s .

T h i s is a g e n e r a l r u l e , a n d is f o u n d t o h o l d t r u e f o r r o t a t i o n a n d m a g n e t i c

e f f ec t s a s w e l l a s f o r v i s c o s i t y .

C. CONVECTION CELLS

U p t o t h i s p o i n t , w e h a v e c o n c e r n e d o u r s e l v e s o n l y w i t h t h e q u e s t i o n of

s t a b i l i t y of fluids in w h i c h t e m p e r a t u r e g r a d i e n t s e x i s t . S i n c e s u c h

s y s t e m s a r e s e e n t o b e u n s t a b l e , w e c a n t h e n a s k t h e n e x t q u e s t i o n —

w h a t will t h e s t e a d y - s t a t e m o t i o n s of t h e s y s t e m b e ?

F o r t u n a t e l y , it is n o t n e c e s s a r y t o t r a c e t h r o u g h t h e d e v e l o p m e n t

of i n s t a b i l i t i e s a s t h e y g r o w in t i m e a n d a p p r o a c h t h e s t e a d y - s t a t e

m o t i o n . W e c a n g e t t h e s t e a d y - s t a t e m o t i o n d i r e c t l y f r o m t h e r e s u l t s of

t h e l a s t s e c t i o n b y r e c a l l i n g w e wil l h a v e s t e a d y - s t a t e c o n d i t i o n s if t h e

t i m e d e r i v a t i v e of t h e v e l o c i t y v a n i s h e s . F o r t h e t y p e of d i s t u r b a n c e s

w h i c h w e t r e a t e d in E q . (10 .B .10 ) , t h i s c o r r e s p o n d s t o g e t t i n g t h e

p a r a m e t e r n = 0 in all s u b s e q u e n t e q u a t i o n s .

T h u s , f o r t h e spec i f i c c a s e of t w o f r e e s u r f a c e s w h i c h w a s d i s c u s s e d in

t h e l a s t s e c t i o n , t h e s t e a d y - s t a t e v e l o c i t i e s wil l b e g i v e n b y

vx = A c o s sy ellxeimz,

vy= B s in sy eilxeimz, (10 .C.1)

vz= C c o s sy ellxeimz,


w h e r e t h e c o n s t a n t s A , B, a n d C still s a t i s fy t h e a u x i l i a r y c o n d i t i o n of E q .

(10 .B.13) w h i c h is i m p o s e d b y c o n t i n u i t y . F o r t h e s a k e of s i m p l i c i t y , le t u s

d i s c u s s t h e g e o m e t r i c a l l y s i m p l e c a s e w h e r e m = 0 a n d C = 0 ( m o r e

c o m p l i c a t e d g e o m e t r i e s will b e lef t t o t h e p r o b l e m s ) . T h i s c o r r e s p o n d s t o

a s s u m i n g a s y m m e t r y in t h e z - d i r e c t i o n , a n d r e d u c e s t h e p r o b l e m of

t r a c i n g o u t t h e m o t i o n of t h e fluid t o t w o d i m e n s i o n s . T h e c o n t i n u i t y

c o n d i t i o n t h e n te l l s u s t h a t

- UA = sB. (10 .C.2)

If w e u s e t h i s r e s u l t a n d t h e n f o l l o w t h e u s u a l p r o c e d u r e of t a k i n g t h e r ea l

p a r t s of t h e c o m p l e x q u a n t i t i e s in E q . (10 .C.1) t o g e t a c t u a l p h y s i c a l

v e l o c i t i e s , w e find

R e vx = A c o s sy c o s Ix, y (10 .C.3)

R e vy = — A s in sy s in Ix. s

T o p i c t u r e t h e m o t i o n a s s o c i a t e d w i t h t h i s v e l o c i t y field, le t u s p l o y vy

a s a f u n c t i o n of x a t fixed y. I t will l o o k l ike t h e field s h o w n in F i g . 10.3 . F o r

d i f f e ren t c h o i c e s of x, t h e m a g n i t u d e of t h e v e l o c i t i e s will b e d i f f e ren t t h a n

t h o s e p i c t u r e d , b u t t h e p a t t e r n of t h e v e l o c i t y field r e p e a t i n g e v e r y t i m e w e

g o t h r o u g h a d i s t a n c e L ( w h i c h is c l e a r l y g i v e n b y L =2TTII) wil l r e a p p e a r

a l o n g e v e r y l ine of c o n s t a n t x. T h u s , t h e fluid wil l n a t u r a l l y d i v i d e i tself i n t o

r e g i o n s in w h i c h p e r i o d i c v e l o c i t i e s wil l r e p e a t t h e m s e l v e s . W e wil l r e f e r t o

s u c h u n i t s of d i v i s i o n a s ce l l s .

W e n o t e t h a t a l o n g t h e l i n e s JC = 0 a n d x = L, w e h a v e f r o m E q . (10 .C.3)

t h a t

vx=0 a n d

H- L *

y = 0

Fig. 10.3. The velocity in the y -direction as a function of x.

Convection Cells 169

y = h

, y = 0

O L

Fig. 10.4. The velocity in the x -direction as a function of y.

T h u s , it f o l l o w s t h a t fluid in o n e cel l wil l n e v e r l e a v e t h a t p a r t i c u l a r cel l

a n d e n t e r a n o t h e r . T h e cel l b o u n d a r i e s t h u s h a v e a p h y s i c a l s i gn i f i c ance ,

in t h a t t h e y d e l i n e a t e r e a l b o u n d a r i e s in t h e fluid, t h r o u g h w h i c h t h e fluid

m a y n o t f low.

T o v i s u a l i z e t h e p a t t e r n of f low i n s i d e of a g i v e n ce l l , l e t u s p l o t vx a s a

f u n c t i o n of x f o r s e v e r a l d i f f e r en t v a l u e s of y ( s e e F i g . 10.4).

T h e o v e r a l l p a t t e r n of f low in t h e cel l wil l t h e n b e o n e in w h i c h t h e fluid

f lows u p f r o m t h e b o t t o m in t h e c e n t e r of t h e ce l l , a n d fa l l s d o w n a t t h e

s i d e s , a s in F i g . 10.5. T h i s c o r r e s p o n d s t o o u r g e n e r a l n o t i o n of c o n v e c -

t i o n , in w h i c h h e a t is t r a n s f e r r e d f r o m t h e w a r m e r t o t h e c o l d e r t o w a r m e r

s u r f a c e b y m o t i o n of t h e fluid. W e s h o u l d n o t e in p a s s i n g t h a t n o t o n l y

d o e s t h e fluid n o t c r o s s t h e b o u n d a r i e s a t x = 0 , L , 2L,..., b u t it a l s o d o e s

n o t c r o s s t h e b o u n d a r i e s a t x = L / 2 , 3 L / 2 , . . . T h e fluid in t h e s i m p l e c a s e

c o m e s in " r o l l s , " a n d a l t e r n a t e ro l l s i n v o l v e fluid r o t a t i o n in o p p o s i t e

d i r e c t i o n s . T w o ro l l s t o g e t h e r c o m p r i s e w h a t w e h a v e t e r m e d a ce l l .

T h e s e ce l l s a r e c a l l e d c o n v e c t i o n c e l l s , o r Benard cells, a f t e r t h e F r e n c h

p h y s i c i s t w h o first o b s e r v e d t h e m in t h e l a b o r a t o r y . T h e y p l a y a n

i m p o r t a n t p a r t in all c o n s i d e r a t i o n s of m o t i o n of f lu ids d r i v e n b y t h e r m a l

d i f f e r e n c e s .

Fig. 10.5. The development of convection, or Benard, cells.


B e f o r e l e a v i n g t h i s t o p i c , w e n o t e t h a t t h e c o n d i t i o n d e r i v e d in t h e l a s t

s e c t i o n c o n c e r n i n g t h e d i m e n s i o n s of t h e f a s t e s t g r o w i n g s i n g u l a r i t y a l s o

g i v e s u s s o m e c l u e a s t o t h e r e l a t i o n s h i p b e t w e e n h, t h e t h i c k n e s s of t h e

l a y e r of l i qu id , a n d L , t h e s i z e of t h e ce l l . W e h a d

D. THE GENERAL CIRCULATION OF THE ATMOSPHERE

P r o b a b l y t h e m o s t i m p o r t a n t a p p l i c a t i o n of t h e t h e o r y of c o n v e c t i o n is

t h e m o t i o n s of t h e a t m o s p h e r e d u e t o h e a t i n g e f f ec t s . B e n a r d ce l l s a p p e a r

in m a n y p l a c e s in a t m o s p h e r i c m o t i o n . F o r e x a m p l e , t h e s u r f a c e a i r a b o v e

c i t i e s is u s u a l l y w a r m e r t h a n t h e a i r in t h e s u r r o u n d i n g c o u n t r y s i d e . T h i s

g i v e s r i s e t o c o n v e c t i o n ce l l s w h o s e s c a l e is o n t h e o r d e r of m i l e s a c r o s s .

I n a s i m i l a r w a y , a p a v e d s h o p p i n g c e n t e r ( o r a n i s l a n d in t h e o c e a n ) c a n

c a u s e ce l l s of s o m e w h a t s m a l l e r s i z e . I n t h i s s e c t i o n , w e will g i v e a

q u a l i t a t i v e d e s c r i p t i o n of a n o t h e r t y p e of c e l l — t h a t a s s o c i a t e d w i t h t h e

l a r g e - s c a l e m o v e m e n t of a i r a r o u n d t h e e a r t h .

C o n s i d e r t h e t e m p e r a t u r e a t t h e s u r f a c e of t h e e a r t h . T h e s ing le m o s t

d o m i n a n t f e a t u r e of t h e t e m p e r a t u r e d i s t r i b u t i o n is t h a t , in s i m p l e s t t e r m s ,

it is w a r m e r a t t h e e q u a t o r t h a n a t t h e p o l e s . T h u s , f o l l o w i n g o u r

Fig. 10.6. The Hadley cell.

r e c a l l i n g t h a t m = 0 in o u r c a s e , a n d t h e de f in i t ion of L , w e find

L = iVl h.

T h i s m e a n s t h a t w e e x p e c t t h e d i m e n s i o n s of a c o n v e c t i o n ce l l t o b e

r o u g h l y t h e s a m e a s t h e d e p t h of t h e l i qu id . T h i s r e s u l t is f a i r ly g e n e r a l ,

a l t h o u g h w e h a v e p r o v e d it o n l y f o r t h e s i m p l e s t p o s s i b l e p l a n e g e o m e t r y .

The General Circulation of the Atmosphere 171

d i s c u s s i o n of B e n a r d c e l l s , o n e m i g h t e x p e c t a n e t m o t i o n a s p i c t u r e d in

F i g . 1 0 . 6 — w h e r e w a r m a i r r i s e s a t t h e e q u a t o r , a n d d e s c e n d s a t t h e p o l e s .

T h i s p i c t u r e , c a l l e d a H a d l e y ce l l , a f t e r G . H a d l e y , w a s first p r o p o s e d a s a

w a y of e x p l a i n i n g t h e o b s e r v e d w i n d p a t t e r n s in t h e t r o p i c s . F r o m a

s i m p l e a p p l i c a t i o n of t h e C o r i o l i s f o r c e , w e c a n s e e t h a t in t h i s m o d e l ,

w i n d s in t h e n o r t h e r n h e m i s p h e r e w o u l d t e n d t o b l o w f r o m e a s t t o w e s t .

(I t is o n e of t h e c h a r a c t e r i s t i c s of m e t e o r o l o g y t h a t w i n d s a r e n a m e d b y

t h e d i r e c t i o n f r o m w h i c h t h e y c o m e , r a t h e r t h a n t h e d i r e c t i o n t o w h i c h

t h e y g o . W i n d s of t h e t y p e p r e d i c t e d b y t h e H a d l e y m o d e l w o u l d t h u s b e

t e r m e d e a s t e r l y w i n d s . )

A c t u a l l y , t h e g e n e r a l p a t t e r n of w i n d s o n t h e e a r t h is m o r e c o m p l i c a t e d

t h a n t h i s . N e g l e c t i n g d e t a i l s of l o c a l m o t i o n , t h e g e n e r a l w i n d s p a t t e r n s

c a n b e p i c t u r e d a s in F i g . 10.7. I n t h e r e g i o n of t h e t r o p i c s , f r o m 0° t o 30°

n o r t h l a t i t u d e , t h e w i n d s a r e g e n e r a l l y e a s t e r l y . T h e s e a r e c a l l e d t h e t r a d e

w i n d s , a n d w e r e e x p l a i n e d b y H a d l e y ' s o r i g ina l m o d e l . F r o m 30° t o 60°

n o r t h l a t i t u d e , t h e w i n d s a r e g e n e r a l l y w e s t e r l y . T h i s r e g i o n i n c l u d e s m o s t

of t h e t e m p e r a t e z o n e of t h e e a r t h . F i n a l l y , f r o m 60° t o 90° n o r t h l a t i t u d e s ,

t h e w i n d s b e c o m e e a s t e r l y a g a i n .

A c t u a l l y , t h i s p i c t u r e i s g r e a t l y o v e r s i m p l i f i e d . T h e l a t i t u d e s a t w h i c h

t h e p r e v a i l i n g w i n d s c h a n g e d i r e c t i o n a r e n o t s h a r p d i v i d i n g l i n e s , b u t a r e

s m e a r e d o u t , a n d c h a n g e w i t h t h e s e a s o n . T h e s t r u c t u r e of t h e r e g i o n of

p r e v a i l i n g w e s t e r l i e s ( a s w e sha l l s e e l a t e r ) i s m u c h m o r e c o m p l i c a t e d

t h a n i n d i c a t e d in t h e figure. N e v e r t h e l e s s , f o r o u r p u r p o s e s , t h i s p i c t u r e of

t h e g e n e r a l c i r c u l a t i o n of t h e a t m o s p h e r e wil l suffice. W e n o t e t h a t a

Fig. 10.7. A simplified picture of the circulation of the atmosphere.


m o d e l in w h i c h t h e r e a r e t h r e e H a d l e y ce l l s ( s e e F i g . 10.7) w o u l d g i v e t h e

c o r r e c t d i r e c t i o n s f o r t h e p r e v a i l i n g w i n d s . ( L i k e o u r p i c t u r e of t h e a c t u a l

w i n d m o t i o n s , t h i s m o d e l is g r e a t l y o v e r s i m p l i f i e d . )

T h e t r a n s i t i o n l a t i t u d e s , w h e r e t h e m a i n m o t i o n of t h e a i r is in t h e

v e r t i c a l d i r e c t i o n , a r e r e g i o n s w h e r e t h e r e is v e r y l i t t le w i n d a t t h e

s u r f a c e . T h e s e r e g i o n s w e r e wel l k n o w n t o e a r l y o c e a n n a v i g a t o r s . T h e

r e g i o n a t t h e e q u a t o r is c a l l e d t h e d o l d r u m s , a n d t h e r e g i o n a t a b o u t 30°

n o r t h is c a l l e d t h e " h o r s e l a t i t u d e s . " T h e n a m e d e r i v e d f r o m t h e f a c t t h a t

s h i p s sa i l ing t o t h e N e w W o r l d w o u l d b e b e c a l m e d w h e n t h e y e n t e r e d t h i s

r e g i o n , s o t h a t it w a s n e c e s s a r y t o j e t t i s o n a n y c a r g o t h a t c o n s u m e d f o o d

o r w a t e r . S i n c e t h e s e s h i p s u s u a l l y c a r r i e d h o r s e s , t h e y w e r e t h e first t o

g o . T h e s igh t of h o r s e c a r c a s s e s f loa t ing in t h e o c e a n g a v e t h e r e g i o n i t s

n a m e .

T h e a c t u a l c a l c u l a t i o n of t h i s g e n e r a l c i r c u l a t i o n is q u i t e difficult, f o r

t w o r e a s o n s . F i r s t , t h e c i r c u l a t i o n t a k e s p l a c e in a s p h e r i c a l shel l r a t h e r

t h a n o n a p l a n e , s o t h e g e o m e t r y is c o m p l i c a t e d , a n d s e c o n d , t h e e f fec t s of

t h e e a r t h ' s r o t a t i o n , a s e x p r e s s e d in t h e C o r i o l i s f o r c e , a d d c o m p l i c a t i o n s

t o t h e e q u a t i o n s of m o t i o n . L e t u s c o n s i d e r t h e c a l c u l a t i o n of t h e s i m p l e

H a d l e y ce l l , w i t h o u t r o t a t i o n , t o g i v e s o m e f lavor of w h a t t h e full

c a l c u l a t i o n m i g h t l o o k l i ke .

W e b e g i n b y a s s u m i n g t h a t t h e t e m p e r a t u r e d i s t r i b u t i o n is a f u n c t i o n of

l a t i t u d e o n l y , a n d i g n o r e t h e t e m p e r a t u r e d i f f e r e n c e s b e t w e e n n i g h t a n d

d a y . T h u s , w h e n w e de f ine t h e t e m p e r a t u r e a t a p o i n t , it s h o u l d b e

r e g a r d e d a s t h e a v e r a g e d a i l y t e m p e r a t u r e a n d n o t t h e i n s t a n t a n e o u s o n e .

S i n c e w e a r e i n t e r e s t e d in s o l v i n g f o r l o n g - t e r m w i n d p a t t e r n s , t h i s i s n o t a

d r a s t i c a p p r o x i m a t i o n . T h e s h o r t - t e r m d i u r n a l e f f ec t s w h i c h w e a r e

n e g l e c t i n g c a n b e e x p e c t e d t o g i v e r i s e t o s m a l l - s c a l e e f f ec t s w h i c h , in t h e

first a p p r o x i m a t i o n , d o n o t af fect t h e l o n g - t e r m w i n d s a t al l . T h i s i s k n o w n

a s t h e h y p o t h e s i s of zonal heating, a n d w a s a l s o i n t r o d u c e d b y H a d l e y in

1735.

W e shal l o n c e a g a i n u s e t h e B o s s i n e s q a p p r o x i m a t i o n i n t r o d u c e d in

S e c t i o n 10.A in w h i c h t h e e f f ec t s of c h a n g e s in d e n s i t y d u e t o c h a n g e s in

t e m p e r a t u r e a r e n e g l e c t e d e x c e p t i n s o f a r a s t h e y af fec t t h e a c t i o n of t h e

g r a v i t a t i o n a l f o r c e . W e shal l a l s o a s s u m e t h a t t h e h e a t i n g of t h e e a r t h h a s

g o n e o n f o r a l o n g t i m e , s o t h a t t h e t e m p e r a t u r e d i s t r i b u t i o n h a s s t a b i l i z e d

a n d r e a c h e d i t s s t e a d y - s t a t e v a l u e . I n t h i s c a s e , t h e h e a t e q u a t i o n is s i m p l y

V 2 0 = O , (10 .D.1)

t h e e q u a t i o n of c o n t i n u i t y is

V - v = 0 , (10 .D.2)


I n t h e B o s s i n e s q a p p r o x i m a t i o n , t h e g r a v i t a t i o n a l f o r c e d u e t o t h e

p r e s e n c e of t h e e a r t h is j u s t

w h e r e Me i s t h e m a s s of t h e e a r t h ( a s s u m e d t o b e s p h e r i c a l ) a n d G is t h e

g r a v i t a t i o n a l c o n s t a n t .

A s in t h e d e v e l o p m e n t of t h e s i m p l e B e n a r d ce l l , it sha l l b e c o n v e n i e n t

t o r e f e r t h e t e m p e r a t u r e s a n d v e l o c i t i e s t o a n e q u i l i b r i u m s o l u t i o n of t h e

e q u a t i o n s of m o t i o n . I n t h e c a s e of t h e B e n a r d ce l l , w e s a w t h a t a s o l u t i o n

e x i s t e d w h e n v = 0 . C a n w e find s u c h a s o l u t i o n t o t h e a b o v e e q u a t i o n s ? If

w e le t ® b e t h e e q u i l i b r i u m t e m p e r a t u r e d i s t r i b u t i o n , w e s e e t h a t t h e

N a v i e r - S t o k e s e q u a t i o n r e d u c e s in t h i s c a s e t o

w h e r e px is t h e e q u i l i b r i u m p r e s s u r e d i s t r i b u t i o n . S i n c e G M e V ( l / r ) is a

f u n c t i o n of r o n l y , t h i s e q u a t i o n c a n o n l y b e sa t i s f ied if b o t h @ a n d px a r e

f u n c t i o n s of r a l o n e a s w e l l . G i v e n t h i s , h o w e v e r , a n e q u i l i b r i u m s o l u t i o n is

i n d e e d p o s s i b l e .

T h e p h y s i c a l m e a n i n g of t h i s e q u i l i b r i u m s o l u t i o n t o t h e e q u a t i o n s is

q u i t e s i m p l e . I t c o r r e s p o n d s t o a s i t u a t i o n in w h i c h t h e a t m o s p h e r e is

u n i f o r m l y h e a t e d ( i .e . t h e r e is t h e s a m e h e a t flow i n t o t h e a t m o s p h e r e a t

e a c h p o i n t ) , a n d l o o s e s h e a t o n l y t h r o u g h r a d i a t i o n a t i t s u p p e r e d g e . I n

t h i s c a s e , t h e p r e s s u r e a d j u s t s i t se l f t o b a l a n c e t h e g r a v i t a t i o n a l f o r c e .

W e c a n n o w de f ine a n e w t e m p e r a t u r e

If w e i n s e r t t h i s i n t o t h e g r a v i t a t i o n a l f o r c e t e r m in E q . (10 .D.4 ) a n d u s e

t h e t w o i d e n t i t i e s ,

(10 .D.4)

(10 .D.5 )

0 ' = 0 - ( H > . (10 .D.6)

a n d

(10 .D.7)

(10 .D.3)

a n d t h e N a v i e r - S t o k e s e q u a t i o n f o r t h e s t e a d y s t a t e is j u s t


t h e g r a v i t a t i o n a l t e r m b e c o m e s

(10 .D.8)

N o w w e s a w in t h e d e r i v a t i o n of t h e s i m p l e B e n a r d cel l t h a t in s o l v -

ing f o r t h e v e l o c i t i e s , t h e p r e s s u r e w a s e l i m i n a t e d b e t w e e n d i f f e ren t

c o m p o n e n t s of t h e N a v i e r - S t o k e s e q u a t i o n s . S i n c e w e sha l l f o l l o w t h e

s a m e p r o c e d u r e h e r e , w e c a n spl i t u p t h e p r e s s u r e in a n y w a y w h i c h will

b e m a t h e m a t i c a l l y c o n v e n i e n t . I n p a r t i c u l a r , w e c a n w r i t e

p = p 2 ( l + 8 ) , (10 .D.9)

w h e r e p 2 , w h i c h will b e d e f i n e d b e l o w , is c l o s e l y r e l a t e d t o t h e e q u i l i b r i u m

p r e s s u r e a n d 8 is a sma l l p a r a m e t e r . W r i t i n g t h e p r e s s u r e in t h i s w a y , w e

c a n s e e t h a t t h e g r a d i e n t of t h e p r e s s u r e w h i c h a p p e a r s in E q . (10 .D.3)

c a n b e w r i t t e n

(10.D.10)

w h e r e w e h a v e d r o p p e d h i g h e r - o r d e r t e r m s in 8 a n d m a d e u s e of t h e

i d e n t i t y

T h e N a v i e r - S t o k e s e q u a t i o n f o r t h e s t e a d y s t a t e c a n n o w b e w r i t t e n

u s i n g t h e r e s u l t s of E q s . (10 .D .10) a n d (10 .D.8) a s

(10 .D.11)

W e de f ine p 2 s u c h t h a t

(10 .D.12)

( w e s e e t h a t in t h e c a s e of e q u i l i b r i u m h e a t i n g ( 0 ' = O ) p 2 b e c o m e s

i d e n t i c a l t o pu t h e e q u i l i b r i u m p r e s s u r e ) . W i t h t h i s a s s i g n m e n t of p 2 , t h e


Fig. 10.8. Polar coodinates for atmospheric circulation.

N a v i e r - S t o k e s e q u a t i o n f inal ly t a k e s t h e f o r m

(10 .D .13)

T h e d e v i a t i o n s f r o m t h e e q u i l i b r i u m t e m p e r a t u r e 0 ' i s g o v e r n e d b y t h e

e q u a t i o n

V 2 0 ' = O, (10 .D.14)

a n d h e n c e c a n b e e x p a n d e d in a s e r i e s ( a g a i n n e g l e c t i n g d e p e n d e n c e o n

t h e l o n g i t u d i n a l a n g l e ) a s

(10 .D.15)

T h e l o w e s t - o r d e r t e r m in t h e s e r i e s w h i c h g i v e s a r e a s o n a b l e a p p r o x i -

m a t i o n t o t h e a c t u a l d i f f e r e n c e s in t e m p e r a t u r e a s a f u n c t i o n of l a t i t u d e is

t h e t e r m / = 2 , o r

= T ( r ) ( l - 3 c o s 2 0 ) (10 .D .16)

( r eca l l t h a t t h e p o l a r a n g l e 0 is m e a s u r e d f r o m t h e p o l e , w h i l e t h e l a t i t u d e

a n g l e is m e a s u r e d f r o m t h e e q u a t o r — s e e F i g . 10.8). T h i s e q u a t i o n , t o g e t h e r

w i t h t h e N a v i e r - S t o k e s e q u a t i o n a n d t h e e q u a t i o n of c o n t i n u i t y , t h e n

d e t e r m i n e s t h e m o t i o n of t h e a t m o s p h e r e .

If w e p i c k t h e u s u a l p o l a r c o o r d i n a t e s a s s h o w n in F i g . 10.8 a n d le t


a n d

t h e c o m p o n e n t s of t h e N a v i e r - S t o k e s e q u a t i o n a r e

(10 .D.17)

a n d

w h i l e t h e e q u a t i o n of c o n t i n u i t y is

(10 .D.18)

L e t u s f o l l o w o u r u s u a l l i ne of a t t a c k a n d g u e s s a t a s o l u t i o n f o r t h e s e

e q u a t i o n s . F r o m t h e f o r m of 0 ' , a n d t h e m a n n e r in w h i c h it a p p e a r s in

t h e s e e q u a t i o n s , a r e a s o n a b l e g u e s s m i g h t b e

u r = ( l - 3 c o s 2 0 ) / ( r ) ,

ve = 6 c o s 0 s in 0<Mr), (10 .D.19)

= 0,

w h e r e f(r) a n d </>(r) a r e t o b e d e t e r m i n e d .

T h e a c t u a l w o r k i n g o u t of t h e f o r m of t h e f u n c t i o n s f(r) a n d cfi(r) is

s t r a i g h t f o r w a r d , b u t t e d i o u s , a n d is left t o P r o b l e m 10.3 . W e s i m p l y n o t e

t h a t if w e w r i t e

r = a + cr a n d a s s u m e t h a t

( th i s c o r r e s p o n d s t o t a k i n g t h e t h i c k n e s s of t h e a t m o s p h e r e t o b e sma l l

c o m p a r e d t o t h e r a d i u s of t h e e a r t h ) , w e find t h a t

(10 .D.20)

w h e r e

(10 .D.21)

a n d h is t h e h e i g h t of t h e a t m o s p h e r e .


W h a t s o r t of w i n d s d o e s t h i s s o l u t i o n d e s c r i b e ? W e n o t e s e v e r a l t h i n g s .

F i r s t , w e n o t e t h a t <f)(cr) c h a n g e s s ign a s cr g o e s f r o m 0 t o h. T h i s m e a n s

t h a t a s w e g o u p in t h e a t m o s p h e r e , ve m u s t c h a n g e s ign . S i n c e t h e f a c t o r

c o s 6 s in 6 is a l w a y s p o s i t i v e in t h e first q u a d r a n t , t h i s m e a n s t h a t fo r

s m a l l o-, ve is p o s i t i v e ( i .e . d i r e c t e d t o w a r d t h e e q u a t o r ) , a n d f o r h i g h e r

a l t i t u d e s it is n e g a t i v e ( i .e . d i r e c t e d t o w a r d t h e p o l e ) .

T h e f u n c t i o n f(tr) i s p o s i t i v e de f in i t e a s a g o e s f r o m 0 t o h , w h i c h

m e a n s t h a t t h e r a d i a l v e l o c i t y d o e s n o t c h a n g e a s a f u n c t i o n of a l t i t u d e .

H o w e v e r , a t a c r i t i ca l v a l u e of 0 C ( ~ 5 5 ° ) , t h e f u n c t i o n 1-3 c o s 2 6 c h a n g e s

s ign . T h i s m e a n s t h a t f r o m t h e e q u a t o r t o a b o u t 35° n o r t h l a t i t u d e , vr is

p o s i t i v e , a n d t h e a i r is r i s i ng , w h i l e f r o m t h i s l a t i t u d e t o t h e p o l e , vr is

n e g a t i v e a n d t h e a i r i s fa l l ing .

I n t h e a b s e n c e of r o t a t i o n , t h e n , t h e o v e r a l l p i c t u r e of t h e c i r c u l a t i o n

w h i c h w e h a v e d e r i v e d i s s h o w n in F i g . 10.9 a n d c o r r e s p o n d s t o t h e

g e n e r a l p i c t u r e w h i c h H a d l e y s u g g e s t e d t w o c e n t u r i e s a g o .

Fig. 10.9. The circulation corresponding to Eq. (10.B.20).

I t s h o u l d b e n o t e d t h a t t h e c o n s t a n t s A a n d A' c a n e a s i l y b e e x p r e s s e d

in t e r m s of t h e t e m p e r a t u r e d i f f e r e n c e b e t w e e n t h e p o l e a n d t h e e q u a t o r

b y n o t i n g t h a t t h e l a t t e r is

w h i l e t h e f o r m e r is

s o t h a t

(10 .D.22)


Fig. 10.10. The circulation which would result from the inclusion of the Coriolis force.

H a d w e i n c l u d e d t h e C o r i o l i s f o r c e in t h e E u l e r e q u a t i o n , t h e e q u a t i o n s

c o u l d b e s o l v e d in t h e s a m e w a y , a l t h o u g h t h e y a r e , of c o u r s e , m u c h m o r e

c o m p l i c a t e d . T h e g e n e r a l f e a t u r e s of t h e s o l u t i o n a r e s h o w n in F i g . 10.10.

I t t u r n s o u t t h a t t h e i n c l u s i o n of r o t a t i o n b r i n g s in a v e l o c i t y in t h e

</>-direction w h i c h g r e a t l y e x c e e d s ve a n d vr. T h i s m e a n s t h a t t h e r e wi l l b e

e a s t e r l y a n d w e s t e r l y w i n d s , r a t h e r t h a n n o r t h e r l y a n d s o u t h e r l y . I n f a c t ,

t h e t r a d e w i n d s a n d t h e p r e v a i l i n g w e s t e r l i e s f o l l o w t h i s m o d e l , a l t h o u g h

it fa i ls t o p r e d i c t t h e w i n d s n e a r t h e p o l e . T o d o b e t t e r , it w o u l d b e

n e c e s s a r y t o i n c l u d e h i g h e r - o r d e r t e r m s in t h e t e m p e r a t u r e d i s t r i b u t i o n in

E q . (10 .D .15) .

T h e m o d e l of t h e g e n e r a l c i r c u l a t i o n of t h e a t m o s p h e r e b e a r s l e s s

r e l a t i o n t o t h e a c t u a l a t m o s p h e r e t h a n t h e s i m p l e t h e o r y of t h e t i d e s

p r e s e n t e d in C h a p t e r 6 d o e s t o a c t u a l t i d e s . W e h a v e i g n o r e d m a n y

i m p o r t a n t e f fec t s b e s i d e s r o t a t i o n . T h e s e i n c l u d e t h e a c t u a l s t ra t i f i ed

s t r u c t u r e of t h e a t m o s p h e r e , t h e i m p o r t a n t e f f ec t s of t h e p r e s e n c e of

w a t e r v a p o r , a n d t h e e f f ec t s of d a y - n i g h t t e m p e r a t u r e d i f f e r e n c e s .

N e v e r t h e l e s s , t h e r e a d e r s h o u l d c o m e a w a y w i t h t h e r e a l i z a t i o n t h a t m a n y

of t h e g e n e r a l f e a t u r e s of a t m o s p h e r i c c i r c u l a t i o n c a n b e u n d e r s t o o d in

t e r m s of t h e s i m p l e p h y s i c a l p r i n c i p l e s w h i c h w e h a v e i n t r o d u c e d in t h i s

c h a p t e r .

SUMMARY

W h e n a fluid is h e a t e d in t h e p r e s e n c e of a g r a v i t a t i o n a l field, it is

p o s s i b l e f o r a n i n s t a b i l i t y t o o c c u r , in w h i c h t h e w a r m fluid will r i s e a n d

Problems 179

t h e c o l d fluid fal l . T h i s is c a l l e d t h e r m a l c o n v e c t i o n . I t wil l o c c u r

w h e n e v e r a n a d v e r s e t e m p e r a t u r e g r a d i e n t e x i s t s in a n o n v i s c o u s fluid,

a n d w h e n e v e r t h e R a y l e i g h c r i t e r i o n is m e t in a v i s c o u s o n e .

I n t h e s t e a d y s t a t e , t h i s t y p e of m o t i o n , i n v o l v i n g fluid r i s i ng a t o n e

p o i n t a n d fa l l ing a t n e i g h b o r i n g p o i n t s , g i v e s r i s e t o t h e p h e n o m e n o n of

B e n a r d c e l l s . T h e g e n e r a l c i r c u l a t i o n of t h e a t m o s p h e r e c a n b e t h o u g h t of

a s b e i n g d u e t o t h e u n e q u a l h e a t i n g of t h e e a r t h a t t h e p o l e s a n d e q u a t o r .

A s i m p l e m o d e l of t h e a t m o s p h e r e w a s d i s c u s s e d .

PROBLEMS

10.1 . Find the stability condition for a fluid of viscosity TJ be tween two free

surfaces a dis tance h apart , with a tempera ture gradient ]3 = ( 0 2 — 0 i ) / h ,

f ^ h

^ 01

in the case that the lower surface is solid, but the upper surface is free.

10 .2. One of the current ideas about the s t ructure of the ear th is the theory of

continental drift. The major idea of this theory is that the continental land masses

are drifting around on top of convect ion cells in the mantle of the ear th. Referring

to Chapter 13 for typical sizes of the mant le , show that from what we have learned

about Benard cells that there should be roughly as many convect ion cells in the

ear th as there are cont inents observed.

10 .3. Consider the assumed form of the solution to the equat ions of motion

without rotation given in Eq . (10.D.17).

(a) Show that the equat ion of continuity in terms of the new function f(r) and

cf>(r) is just

(b) Eliminate 8 from the two Euler equat ions to get a second equat ion relating f(r) and cj>(r).

(c) Assuming that Eq. (10.D.19) is valid, and applying the s tandard boundary condit ions: namely

vr=0 at or = 0,/i

verify that Eq . (10.D.20) does indeed give the required solution to our problem.

10.4. For the case of an a tmosphere with no rotat ion, calculate the variation of the

a tmospher ic pressure with height at var ious lat i tudes. Make rough sketches of this

variation.


10.5. Show that if a material contains heat sources (or sinks) which supply a quanti ty of heat Q per unit t ime, the heat equat ion (10.A.6) must be replaced by

10.6. Consider a sphere of coefficient of diffusivity K immersed in a fluid of diffusivity K2. If the sphere is of radius a, and a cons tant tempera ture gradient is maintained in the fluid, determine the tempera ture everywhere in the fluid and in the sphere.

10.7. Show that a law of similarity similar to that discussed in Chapter 9 for viscous flow can be derived for the s teady-state flow of a viscous fluid of diffusivity K. T o do this:

(a) Write down the equat ions of motion for the fluid. (b) Show that if we define a new dimensionless number ,

called the Prandt l number , the tempera ture distr ibutions in the fluid can depend on both R and P , while the velocity distribution can depend only on JR (both, of course , can be functions of posit ion).

(c) Hence , show that two flows are similar if the Reynolds and Prandt l numbers are equal .

10.8. In a similar way , show that for the type of convect ive processes which are discussed in the text , we can define a Grashof number

and that two convect ive flows will be similar if their Prandt l and Grashof numbers are equal. Why doesn ' t the Reynolds number enter into such considerat ions?

10.9. (a) Show that for small values of G, the heat transfer in a fluid must take place primarily through conduct ion, while for large values it must take place primarily through convect ion.

(b) Write the Rayleigh criterion [Eq. (10.B.28)] in terms of dimensionless numbers . For what relation be tween G and P will it be possible to have convect ion?

10.10. Consider the problem of the s teady-state flow of a fluid which is confined to a vertical tube of radius R, with the upper end of the tube maintained at a tempera ture 62 and the lower end at a tempera ture 0\.

(a) Assuming a form of the per turbat ion in which the velocity is along the z-axis (taken to be the axis of the tube) , and vz, 6' and dP'/dz depend only on the coordinates r and cp (the angle in the x-y plane), show that

Problems 181

(b) H e n c e show that the velocities and tempera ture differences in this problem

must be

vz = v0 cos (f) [Jx(Kr)h(KR)- h{Kr)JX{KR)\,

O' = vo (—) cos <l}[Ji(Kr)Il(KR) + I , (Kr)J , (KJR)] . \<xg /

where K = (GPIR4)114.

10.11. For the geometry of Problem 10.10, find the stability criterion correspond-ing to Eq . (10.B.28).

10.12. In going from per turbat ion methods to s teady-state solutions in Section 10.C, we simply set n = 0. Discuss the validity of this step in te rms of your physical unders tanding of the meaning of the Reynolds number .

10.13. With the introduction of heat , we have still another form of energy which must be included in the type of energy balance carried out in Section I .E. Show that for a viscous conduct ing fluid, conservat ion of energy requires that

where U is the sum of the internal energy defined in Eq. (10.A.3) and the usual potential energy.

10.14. H e n c e show that for the fluid of Prob lem 10.13,

where S is the ent ropy density. Show that for an ideal fluid, this equat ion implies ent ropy conservat ion. It is called the general equation of heat transfer.

10.15. Consider a sys tem which is made up of a mixture of two types of fluid, a normal fluid of density p„ and velocity v„ which is viscous and can carry ent ropy, and a superfluid of densi ty ps and velocity \ s which is nonviscous and carries no ent ropy.

(a) Show that the conservat ion of mass and ent ropy in such a fluid require

(b) Show that the Euler equat ion in such a fluid is


These results imply that there are two waves in the superfluid-normal mixture—a density wave , like the sound waves discussed in Chapter 5, and a thermal , or ent ropy wave which we have not seen before. This is called second sound, and is an important proper ty of the type of superfluids which we have been discussing here .

10.18. H o w will winds on Venus differ from those on ear th?

(c) Show that energy conservat ion in such a fluid requires

10.16. For the fluid of Prob lem 10.15, show that if we treat vn and vs as small per turbat ions , and treat the derivat ives of densit ies and thermodynamic quantit ies in the same way, we get

and

10.17. For the fluid of Prob lem 10.15, show that using the thermodynamic identities

and

along with the results of Prob lem 10.16 yields

where

References 183

REFERENCES

C. Eckart, Hydrodynamics of Oceans and Atmospheres, Pergamon Press, New York, 1960.

A well-organized and detailed study of the general motions of oceans and the atmosphere. Contains an excellent discussion of the equations of motion including rotation.

B. Saltzman, Theory of Thermal Convection, Dover Publications, New York, 1962. A collection of the classic papers on thermal convection. The original paper of Lord Rayleigh is the best presentation of the basic theory that I have found in the literature. There is a section on the motion of atmospheres.

S. Chandrasekar (cited in Chapter 3). This book contains an exhaustive study of the effects of stability of rotation, magnetic fields, and viscosity, and is highly recommended for anyone wishing to read further in the field of thermal convection.

11

General Properties of Solids— Statics

I can be pushed just so far.

H. L. WILSON

Ruggles of Red Gap

A. BASIC EQUATIONS

U p t o t h i s p o i n t , w e h a v e c o n s i d e r e d o n l y o n e t y p e of c l a s s i c a l

m a t e r i a l — f l u i d s . F l u i d s a r e c h a r a c t e r i z e d b y t h e f a c t t h a t o n t h e m i c r o s -

c o p i c l e v e l , t h e a t o m s i n t e r a c t m a i n l y b y c o l l i s i o n s . T h e o n l y f o r c e s w h i c h

a r e g e n e r a t e d i n s i d e of a fluid m a s s a r e t h o s e h a v i n g t o d o w i t h t h e

m o m e n t u m t r a n s f e r r e d t h r o u g h t h e s e c o l l i s i o n s . W e c u s t o m a r i l y r e f e r t o

s u c h f o r c e s a s p r e s s u r e s . If w e w i s h e d t o a p p l y a n e x t e r n a l f o r c e t o a

p a r t i c u l a r e l e m e n t in t h e fluid, h o w e v e r , it is c l e a r t h a t , a s i d e f r o m

p o s s i b l e v i s c o u s d r a g , t h e r e is n o w a y t o g e n e r a t e f o r c e s i n s i d e t h e fluid

w h i c h w o u l d o p p o s e t h e a p p l i e d f o r c e . C o n s e q u e n t l y , t h e fluid e l e m e n t

w o u l d b e in m o t i o n f o r a s l o n g a s t h e f o r c e w e r e a p p l i e d .

If w e t h i n k a b o u t a so l id , h o w e v e r , w e k n o w t h a t t h i s i s n o t t r u e . If I

p u s h o n a t a b l e t o p , t h e m a t e r i a l i m m e d i a t e l y u n d e r m y h a n d d o e s n o t

m o v e ( e x c e p t , p e r h a p s , f o r s o m e s m a l l ini t ia l d e f o r m a t i o n w h i c h w e wil l

c o n s i d e r l a t e r ) . T h i s m e a n s t h a t t h e so l id , u n l i k e t h e fluid, is c a p a b l e of

g e n e r a t i n g i n t e r n a l f o r c e s w h i c h c a n o p p o s e f o r c e s a p p l i e d f r o m t h e

o u t s i d e . T h e r e a s o n f o r t h i s b e c o m e s c l e a r if w e t h i n k a b o u t t h e

c r y s t a l l i n e s t r u c t u r e of t h e a t o m s in m o s t s o l i d s . T h e a t o m s a r e l o c k e d

i n t o t h e i r p l a c e s in a c r y s t a l l a t t i c e b y e l e c t r o m a g n e t i c i n t e r a c t i o n s w i t h

o t h e r a t o m s , s o t h a t in o r d e r t o m o v e o n e a t o m , it is n e c e s s a r y t o

184

Basic Equations 185

o v e r c o m e t h e s t r o n g f o r c e s w h i c h b i n d it t o o t h e r a t o m s ( w h i c h , in t u r n ,

a r e b o u n d t o o t h e r a t o m s , a n d s o f o r t h ) . I t is t h e s e a t o m i c f o r c e s w h i c h w e

d e s c r i b e c l a s s i c a l l y a s " i n t e r n a l l y g e n e r a t e d f o r c e s " in d i s c u s s i n g s o l i d s ,

a n d w h i c h a r e a b s e n t in t h e c a s e of f lu ids .

I n o u r d e v e l o p m e n t of fluid m e c h a n i c s , w e f o u n d it s i m p l e s t t o d i s c u s s

h y d r o s t a t i c s b e f o r e h y d r o d y n a m i c s . W e sha l l f o l l o w t h e s a m e l ine h e r e ,

a n d r e s t r i c t o u r a t t e n t i o n t o t h e s i m p l e c a s e in w h i c h a so l id f inds i t se l f in

s t a t i c e q u i l i b r i u m w i t h e x t e r n a l l y a p p l i e d f o r c e s , l e a v i n g t h e p r o b l e m of

t i m e - d e p e n d e n t e f fec t s f o r l a t e r . If e q u i l i b r i u m is i n d e e d e s t a b l i s h e d , t h e n

t h e i n t e r n a l l y g e n e r a t e d f o r c e s in t h e so l id m u s t e x a c t l y c a n c e l t h e

e x t e r n a l l y a p p l i e d f o r c e s . L e t u s s e e h o w t h i s i d e a l e a d s u s t o t h e b a s i c

e q u a t i o n s w h i c h d e s c r i b e t h e b e h a v i o r of s t a t i c s o l i d s .

C o n s i d e r a so l id ( s e e F i g . 11.1) in w h i c h a n e x t e r n a l l o a d q(x) p e r u n i t

l e n g t h is a p p l i e d e x t e r n a l l y . T h i s e x t e r n a l f o r c e m i g h t b e t h e w e i g h t of t h e

so l id i tself , o r a n y c o m b i n a t i o n of f o r c e s g e n e r a t e d b y t h e p h y s i c a l

s y s t e m . If w e c o n s i d e r o n e in f in i t e s ima l v o l u m e e l e m e n t s o m e w h e r e in

t h e so l id , t h e n t h e f o r c e s a c t i n g o n it in t h e y - d i r e c t i o n a r e

(i) t h e l o a d i n g , q(x) dx a c t i n g a t t h e c e n t e r of t h e e l e m e n t ,

(ii) a n i n t e r n a l l y g e n e r a t e d f o r c e F a c t i n g o n t h e l e f t - h a n d e d g e , w h i c h

w e t a k e t o b e a c t i n g in t h e p o s i t i v e d i r e c t i o n ( th i s f o r c e is w r i t t e n

<— d x —>

Fig. 11.1(b). Vertical forces on an element in a loaded solid.

Fig. 11.1(a). Loading of a solid.

186 General Properties of Solids—Statics

I n a d d i t i o n t o a b a l a n c i n g of f o r c e s in t h e y - d i r e c t i o n , t h e r e q u i r e m e n t

t h a t t h e so l id b e in s t a t i c e q u i l i b r i u m a l s o d e m a n d s t h a t t h e r e b e n o n e t

t o r q u e o n t h e v o l u m e e l e m e n t . I t f o l l o w s f r o m o u r d i s c u s s i o n of t h e

p r o p e r t i e s of a so l id t h a t a so l id wil l b e c a p a b l e of g e n e r a t i n g i n t e r n a l

t o r q u e s , a s we l l a s i n t e r n a l f o r c e s . T h i s , of c o u r s e , is a n o t h e r d i f f e r e n c e

b e t w e e n a so l id a n d a l iqu id .

W e c a n u n d e r s t a n d h o w t o r q u e s m i g h t b e g e n e r a t e d b y a s k i n g w h a t

h a p p e n s t o t h e l o a d e d so l id w e c o n s i d e r e d a b o v e w h e n a l o a d is a p p l i e d .

C l e a r l y , t h e so l id wil l b e n d u n d e r t h e w e i g h t a n d d e f o r m , s o t h a t a v o l u m e

e l e m e n t w h i c h s t a r t e d o u t a s a c u b e , fo r e x a m p l e , w o u l d e n d u p d e f o r m e d

a s we l l ( s e e F i g . 11.2). T h i s d e f o r m a t i o n of t h e v o l u m e e l e m e n t m u s t b e

q=Q

Fig. 11.2(a). The deformation of a loaded solid.

Fig. 11.2(b). Horizontal forces on an element in a loaded solid.

a s F in o r d e r t o e m p h a s i z e t h a t it i s a n i n t e r n a l l y g e n e r a t e d f o r c e ,

a n d not t h e f o r c e a p p l i e d t o t h e so l id b y a n o u t s i d e a g e n c y ) ,

(iii) s im i l a r i n t e r n a l l y g e n e r a t e d f o r c e a c t i n g o n t h e r i g h t - h a n d e d g e ,

w h i c h w e sha l l ca l l F + d F , a n d a s s u m e a c t s in t h e n e g a t i v e

y - d i r e c t i o n .

T h e n b a l a n c i n g f o r c e s in t h e y - d i r e c t i o n g i v e s

F-qdx-(F + dF) = 0,

s o t h a t t h e r a t e of c h a n g e of t h e i n t e r n a l l y g e n e r a t e d f o r c e is r e l a t e d t o t h e

e x t e r n a l l y a p p l i e d l o a d in a so l id b y t h e e q u a t i o n

(ll.A.l)

Basic Equations 187

a c c o m p l i s h e d t h r o u g h t h e g e n e r a t i o n of f o r c e s in t h e x - d i r e c t i o n s u c h a s

t h o s e p i c t u r e d , w h i c h a c t t o c o m p r e s s t h e t o p of t h e c u b e a n d s t r e t c h t h e

b o t t o m . If w e c o n s i d e r t h e n e t e f fec t of t h e f o r c e s a c t i n g o n o n e s i d e of

t h e v o l u m e e l e m e n t , it is t o a p p l y a t o r q u e t o t h e e l e m e n t , a t t e m p t i n g t o

c a u s e t h e e l e m e n t t o r o t a t e . T h u s , t h e i n t e r n a l f o r c e s g e n e r a t e d in a

d i r e c t i o n p e r p e n d i c u l a r t o t h e l o a d c a n b e r e p r e s e n t e d b y i n t e r n a l l y

g e n e r a t e d t o r q u e s .

If w e n o w c o n s i d e r a v o l u m e e l e m e n t , w e s e e t h a t t h e r e a r e t h r e e k i n d s

of t o r q u e s ( s e e F i g . 11.3):

(i) t h o s e g e n e r a t e d b y i n t e r n a l f o r c e s in t h e x - d i r e c t i o n ,

(ii) t h o s e g e n e r a t e d b y i n t e r n a l f o r c e s in t h e y - d i r e c t i o n ,

(Hi) t h o s e g e n e r a t e d b y t h e e x t e r n a l l o a d q(x).

B a l a n c i n g t h e s e t o r q u e s a b o u t t h e p o i n t P l e a d s t o t h e r e s u l t

w h i c h b e c o m e s , w h e n w e d r o p t e r m s of s e c o n d o r d e r in i n f i n i t e s i m a l s ,

( l l . A . :

o r , u s i n g E q . ( l l . A . l )

O n c e a g a i n , w e w r i t e t h e i n t e r n a l l y g e n e r a t e d t o r q u e s a s f t o d i s t i n g u i s h

t h e m f r o m e x t e r n a l l y a p p l i e d t o r q u e s .

T h e s e e q u a t i o n s , w h i c h r e l a t e t h e i n t e r n a l l y g e n e r a t e d f o r c e s t o t h e

e x t e r n a l l y a p p l i e d l o a d f o r a so l id in s t a t i c e q u i l i b r i u m , p l a y t h e r o l e of t h e

(11 .A.3)

qdx

1 F + dF

Fig. 11.3. Torques on a volume element in a deformed solid.


" e q u a t i o n s of m o t i o n " f o r s t a t i c s o l i d s . T h e i n t e r n a l f o r c e s , h o w e v e r , a r e

n o t t h e t y p e of t h i n g s w h i c h o n e u s u a l l y t r i e s t o c a l c u l a t e o r m e a s u r e . I n

C h a p t e r 2 , w e w e r e n o t i n t e r e s t e d in t h e i n t e r n a l f o r c e s o p e r a t i n g in a s t a r ,

b u t in t h e final s h a p e of t h e s t a r . S i m i l a r l y , it is m u c h m o r e u s u a l t o a s k

h o w a g i v e n so l id wil l d e f o r m w h e n a l o a d is a p p l i e d t h a n t o a s k a b o u t

i n t e r n a l f o r c e s in t h e so l id . T h e r e f o r e , it is n e c e s s a r y t o find s o m e r e l a t i o n

b e t w e e n t h e i n t e r n a l f o r c e s w h i c h w e h a v e c a l c u l a t e d a b o v e t h e d e f o r m a -

t i o n in t h e so l id .

B. HOOKE'S LAW AND THE ELASTIC CONSTANTS

T h e q u e s t i o n of h o w m u c h a n d in w h a t m a n n e r a so l id wil l d e f o r m

u n d e r a n a p p l i e d f o r c e is a n e x p e r i m e n t a l o n e . T h e r e is n o r e a s o n t o

e x p e c t , a priori, a n y p a r t i c u l a r k i n d of b e h a v i o r . F o r e x a m p l e , if w e

i m a g i n e d t h a t t h e i n t e r n a l f o r c e s b e t w e e n t h e a t o m s in a so l id c o u l d b e

r e p r e s e n t e d b y s p r i n g s , t h e n w e m i g h t e x p e c t t h a t t h e d e f o r m a t i o n w o u l d

b e p r o p o r t i o n a l t o t h e f o r c e a p p l i e d . S u c h a so l id is c a l l e d a n elastic solid,

a n d wil l o c c u p y m o s t of o u r a t t e n t i o n . W e c o u l d a l s o i m a g i n e t h a t t h e

f o r c e s b e t w e e n t h e a t o m s w e r e s u c h t h a t t h e y a l l o w e d n o m o t i o n of t h e

a t o m s u n l e s s t h e e x t e r n a l f o r c e w e r e s t r o n g e n o u g h t o o v e r c o m e t h e m . I n

t h i s c a s e , t h e r e w o u l d b e n o d e f o r m a t i o n f o r s m a l l f o r c e s , a n d l a r g e

d e f o r m a t i o n s f o r l a r g e f o r c e s w h e n , p r e s u m a b l y , t h e m a t e r i a l w o u l d

f r a c t u r e . B e t w e e n t h e s e t w o e x t r e m e s , o n e c o u l d i m a g i n e m a n y d i f f e ren t

k i n d s of s o l i d s , a n d , i n d e e d , t h e r e is a n e n t i r e field of s t u d y c a l l e d

rheology, w h i c h is d e v o t e d t o t h e s t u d y of t h e w a y in w h i c h m a t e r i a l s

r e a c t t o f o r c e s a p p l i e d t o t h e m .

F o r o u r p u r p o s e s , h o w e v e r , w e sha l l c o n s i d e r o n l y t h e s i m p l e c a s e of

a n e l a s t i c so l id . T o fix in o u r m i n d e x a c t l y w h a t is m e a n t b y s u c h a so l id ,

i m a g i n e a t h i n w i r e of l e n g t h / f r o m w h i c h w e i g h t s c a n b e h u n g . F o r a

g i v e n w e i g h t W, t h e w i r e wi l l s t r e t c h a d i s t a n c e A/. I t is a n e x p e r i m e n t a l

f a c t t h a t f o r m o s t m a t e r i a l s , t h e a m o u n t of s t r e t c h i n g is p r o p o r t i o n a l t o

t h e f o r c e , s o t h a t

E y = W. ( l l . B . l )

T h i s e x p e r i m e n t a l finding is c a l l e d Hooke's law, a n d t h e c o n s t a n t of

p r o p o r t i o n a l i t y E is c a l l e d Young's modulus.

T h e r e is a n i n t e r e s t i n g a n a l o g y b e t w e e n t h i s l a w a n d a r e s u l t w h i c h w e

f o u n d t r u e fo r f luids in C h a p t e r 1. T h e r e a d e r wil l r e c a l l t h a t in o r d e r t o

s p e c i f y t h e p h y s i c a l s i t u a t i o n i n v o l v i n g a fluid, it w a s n e c e s s a r y t o s a y

Bending of Beams and Sheets 189

T l + Al

Fig. 11.4. The deformation of a wire under a stress.

w h a t k i n d of fluid w e w e r e c o n s i d e r i n g — i . e . t o s p e c i f y a n e q u a t i o n of

s t a t e . S i m i l a r l y , in t h i s c a s e , it is n e c e s s a r y t o s p e c i f y t h e t y p e of so l id

b e i n g c o n s i d e r e d . T o d o t h i s , it is n e c e s s a r y t o g i v e a r e l a t i o n s h i p b e t w e e n

t h e f o r c e a p p l i e d t o a so l id a n d t h e a m o u n t of d e f o r m a t i o n su f f e r ed .

E q u a t i o n ( l l . B . l ) i s s u c h a r e l a t i o n s h i p , a n d h e n c e p l a y s t h e s a m e r o l e a s

t h e e q u a t i o n of s t a t e . E a c h e l a s t i c so l id wi l l b e c h a r a c t e r i z e d b y a

d i f f e r en t c o n s t a n t E , of c o u r s e , j u s t a s d i f f e r en t t y p e s of i n c o m p r e s s i b l e

f luids a r e c h a r a c t e r i z e d b y d i f f e r en t d e n s i t i e s .

I t is a l s o c l e a r t h a t if a so l id is s t r e t c h e d in l e n g t h , t h e m a t e r i a l w h i c h

g o e s t o m a k e u p t h e e x t r a l e n g t h m u s t c o m e f r o m s o m e w h e r e . I n g e n e r a l ,

if a so l i d is s t r e t c h e d in o n e d i m e n s i o n , it wi l l t h i n d o w n in t h e o t h e r

d i m e n s i o n ( s e e F i g . 11.4).

T h e r a t i o of t h e d e c r e a s e in l a t e r a l d i m e n s i o n t o t h e i n c r e a s e in l e n g t h is

c a l l e d Poisson's ratio, a n d is d e f i n e d b y

C. BENDING OF BEAMS AND SHEETS

A s a first e x a m p l e of t h e a p p l i c a t i o n of t h e l a w s d e r i v e d in t h e

p r e c e d i n g t w o s e c t i o n s , c o n s i d e r a t h i n b e a m of e l a s t i c m a t e r i a l of

Y o u n g ' s m o d u l u s E, o r i g ina l l y s t r a i g h t , b u t b e n t b y e x t e r n a l f o r c e s i n t o a n

a r c of r a d i u s p ( s e e F i g . 11.5).

(11 .B.2)

T h e q u a n t i t i e s E a n d a a r e c a l l e d elastic constants, b e c a u s e t h e t w o

t a k e n t o g e t h e r c o m p l e t e l y s p e c i f y t h e b e h a v i o r of a n e l a s t i c so l id . I n t h e

n e x t c h a p t e r , w e sha l l d i s c u s s o t h e r s e t s of e l a s t i c c o n s t a n t s ( w h i c h c a n

b e r e l a t e d t o E a n d a) w h i c h a r e s o m e t i m e s u s e d f o r t h e s a m e p u r p o s e .

F o r t h e p r e s e n t , h o w e v e r , w e sha l l w o r k o n l y w i t h t h e s e t w o , a n d sha l l

c o n s i d e r t h a t w e h a v e c o m p l e t e l y spec i f i ed t h e so l id w i t h w h i c h w e a r e

d e a l i n g if w e h a v e t h e s e t w o n u m b e r s .


I n t h e p r o c e s s of b e n d i n g , filaments n e a r t h e t o p of t h e b e a m , s u c h a s

P'Q', wil l b e s t r e t c h e d b e y o n d t h e i r n o r m a l l e n g t h , w h i l e t h o s e n e a r t h e

b o t t o m wil l b e c o m p r e s s e d . T h e r e wi l l b e o n e filament, d e n o t e d b y PQ,

w h i c h is n e i t h e r s t r e t c h e d n o r c o m p r e s s e d in t h e b e n d i n g , b u t r e t a i n s i t s

n o r m a l l e n g t h . T h i s is c a l l e d t h e n e u t r a l filament, a n d w e wil l t a k e p , t h e

r a d i u s of c u r v a t u r e , t o b e m e a s u r e d f r o m t h e c e n t e r of c u r v a t u r e t o t h e

l ine PQ. T h e s t r e t c h e d l e n g t h of P'Q' is j u s t

P'Q' = (p + z)<f>,

s o t h a t t h e f r a c t i o n a l c h a n g e in l e n g t h of t h e filament P'Q' is

( l l . C . l )

B u t f r o m t h e p r e v i o u s s e c t i o n , w e k n o w t h a t t h i s m e a n s t h a t t h e f o r c e

e x e r t e d o n t h e filament P'Q' m u s t j u s t b e ( b y H o o k e ' s l a w )

( 1 1 . C . 2 )

S u p p o s e w e n o w l o o k a t t h e b e a m e n d o n . T h e e n d of t h e filament P'Q'

will b e a n in f in i t e s ima l a r e a e l e m e n t a d i s t a n c e z a b o v e t h e p l a n e m a d e u p

of t h e e n d p o i n t s of t h e n e u t r a l filaments ( s e e F i g . 11.6). T h e r e f o r e , t h e

t o r q u e b e i n g a p p l i e d a t t h i s p a r t i c u l a r p o i n t is Fz dA, s o t h a t t h e t o t a l

t o r q u e b e i n g a p p l i e d t o t h e e n d of t h e b e a m is

(11 .C.3)

w h e r e I = / y 2 dA is t h e m o m e n t of i n e r t i a of t h e c r o s s s e c t i o n of t h e

b e a m , a n d wil l d e p e n d o n t h e s h a p e of t h e c r o s s s e c t i o n .

Fig. 11.5. The bending of a filament.

Bending of Beams and Sheets 191

•end of P'Q'

Fig. 11.6. End view of a deformed filament.

W e a r e n o w r e a d y t o r e l a t e t h e a p p l i e d t o r q u e t o t h e b e n d i n g of t h e

b e a m . S u p p o s e t h a t w h e n t h e r e is n o t o r q u e p r e s e n t , t h e b e a m is s t r a i g h t ,

a n d l i e s a l o n g t h e l i ne y = 0 . W h e n t h e t o r q u e is a p p l i e d , t h e b e a m wi l l b e

d e f o r m e d , a n d wi l l b e d e s c r i b e d b y s o m e c u r v e y ( x ) ( s e e F i g . 11.7). F r o m

e l e m e n t a r y c a l c u l u s , t h e r a d i u s of c u r v a t u r e a t e a c h p o i n t a l o n g t h e b e a m

wil l t h e n b e

(11 .C .4)

w h e r e t h e s e c o n d a p p r o x i m a t e e q u a l i t y h o l d s w h e n t h e d e f o r m a t i o n of

t h e b e a m is s m a l l ( t h i s is t h e o n l y c a s e w h i c h w e sha l l c o n s i d e r ) . F r o m E q .

(11 .C .3 ) , t h e a p p l i e d t o r q u e a t e a c h p o i n t m u s t t h e n b e

(11 .C.5)

I n o r d e r f o r t h e b e a m t o b e in s t a t i c e q u i l i b r i u m , t h i s e x t e r n a l l y a p p l i e d

t o r q u e T m u s t b e c a n c e l e d b y t h e i n t e r n a l l y g e n e r a t e d t o r q u e f ( i .e . w e

m u s t h a v e T = - T ) , s o t h a t t h e e x t e r n a l l o a d i n g of a b e a m ( w h i c h is

p r e s u m a b l y w h a t c a u s e s t h e t o r q u e in t h e first p l a c e ) is r e l a t e d t o t h e

d e f o r m a t i o n b y E q . (11 .A .2 ) ,

1

(11 .C .7)

y(x)

Fig. 11.7. Side view of a deformed filament.

a n d , f r o m E q . ( 11 .A .2 ) , t h e e x t e r n a l f o r c e is r e l a t e d t o t h e d e f o r m a t i o n b y

(11 .C .6)

y

X


i .e . t h e r e s u l t is e q u i v a l e n t t o t h e o n e d e r i v e d a b o v e e x c e p t t h a t t h e

s u b s t i t u t i o n E - * E / ( l - c r 2 ) is m a d e . C o n s e q u e n t l y , w e wil l u s e E q .

(11 .C.6) in all t h a t f o l l o w s , w i t h o u t l o s i n g a n y g e n e r a l i t y f o r t h e r e s u l t s .

D. THE FORMATION OF LACOLITHS

A s a n e x a m p l e of a p h y s i c a l s i t u a t i o n in w h i c h t h e p r i n c i p l e s d e r i v e d in

t h e p r e v i o u s s e c t i o n s o p e r a t e , c o n s i d e r t h e g e o l o g i c a l f o r m a t i o n k n o w n a s

a l a c o l i t h . T h e s e o c c u r w h e n a fissure d e v e l o p s in a l a y e r of r o c k b e l o w

t h e s u r f a c e of t h e e a r t h , a n d m o l t e n m a g m a u n d e r h i g h p r e s s u r e f lows

u p w a r d t h r o u g h t h i s fissure, f o r c i n g t h e o v e r l y i n g l a y e r s of r o c k u p w a r d

( s e e F i g . 11.8).

(11 .C.8)

magma

Fig. 11.8. Schematic diagram of the formation of a lacolith.

T h i s is t h e g e n e r a l s o l u t i o n w h i c h w e h a v e b e e n s e e k i n g . If w e a r e t o l d

h o w m u c h e x t e r n a l f o r c e is a p p l i e d t o a so l id ( i .e . if w e k n o w q(x)), w e

c a n c a l c u l a t e t h e d e f o r m a t i o n a t a n y p o i n t , y(x) s i m p l y b y s o l v i n g E q .

(11 .C .6) . T h e s o l u t i o n of t h i s e q u a t i o n wil l i n v o l v e f o u r i n t e g r a t i o n

c o n s t a n t s , a n d t h e s e m u s t b e s u p p l i e d b y t h e b o u n d a r y c o n d i t i o n s . F o r

e x a m p l e , if t h e p r o b l e m w e r e s e t u p s o t h a t t h e e n d of t h e b e a m a t x = 0

w e r e f r e e , t h e n t h e r e w o u l d b e n o t o r q u e s o r f o r c e s a t t h a t e n d . T h i s

w o u l d g i v e t w o c o n d i t i o n s o n t h e f o u r c o n s t a n t s . W e wil l c o n s i d e r

e x a m p l e s of o t h e r b o u n d a r y c o n d i t i o n s in s u b s e q u e n t s e c t i o n s , a n d in t h e

p r o b l e m s .

F i n a l l y , w e n o t e t h a t in all of o u r c o n s i d e r a t i o n s s o f a r , w e h a v e

c o n s i d e r e d t h e b e a m t o b e m a d e u p of inf in i te ly t h i n filaments w h i c h

w o u l d s t r e t c h , b u t w h i c h h a d n o l a t e r a l e x t e n t a t al l . T h i s m e a n s t h a t w e

h a v e n e g l e c t e d t h e k i n d of e f fec t s w h i c h l ed u p t o t h e def in i t ion of

P o i s s o n ' s r a t i o in E q . (11 .B .2 ) . A m o r e r e a l i s t i c b e a m w o u l d b o t h s t r e t c h

a n d t h i n o u t a s it w a s b e n t . I n P r o b l e m 11.2, t h e r e a d e r wil l s h o w t h a t

t a k i n g t h i s e f fec t i n t o a c c o u n t l e a d s t o t h e e q u a t i o n fo r t h e d e f l e c t i o n

The Formation of Lacoliths 193

T h e r e is a w e a l t h of i n f o r m a t i o n o n s u c h f o r m a t i o n s , s i n c e t h e y o c c u r

f r e q u e n t l y . T h e y a r e t y p i c a l l y a m i l e o r t w o a c r o s s . T h e r e is o n e o b s e r v e d

r e g u l a r i t y t o w h i c h w e will t u r n o u r a t t e n t i o n a n d t h a t is t h e f a c t t h e

h i g h e r t h e a l t i t u d e of t h e l a c o l i t h , t h e s m a l l e r it wi l l b e . L e t u s s e e h o w t h e

e q u a t i o n s d e r i v e d in t h e p r e v i o u s s e c t i o n c a n b e a p p l i e d t o t h i s p r o b l e m .

W e wil l c o n s i d e r t h e c a s e w h e r e t h e f i s su re is a s t r a i g h t l i ne w h i c h is

v e r y l o n g c o m p a r e d t o t h e w i d t h of t h e l a c o l i t h , s o t h a t w e c a n i g n o r e

w h a t h a p p e n s a t t h e e n d s . T h e n t h e f o r c e s o n a s t r i p of l e n g t h b a n d w i d t h

dx in t h e o v e r b u r d e n ( s e e F i g . 11.9) a r e

(i) T h e w e i g h t of t h e r o c k p r e s s i n g d o w n w a r d . If t h e d e n s i t y of t h e

r o c k is y, t h i s wil l b e yab dx.

(ii) T h e f o r c e of t h e m a g m a u p w a r d . T h i s wil l b e Pb dx, w h e r e P is t h e

p r e s s u r e of t h e fluid.

T h u s , t h e n e t e x t e r n a l f o r c e o n t h e o v e r b u r d e n p e r u n i t l e n g t h in t h e

x - d i r e c t i o n is j u s t

q(x) = b(P-ya), ( l l . D . l )

s o t h a t t h e e q u a t i o n of d e f o r m a t i o n is

Fig. 11.9. A fully formed lacolith.

(11 .D.2)

(11 .D.3)

w h i c h c a n b e i n t e g r a t e d t o g i v e

w h e r e d . . . , C 4 a r e c o n s t a n t s of i n t e g r a t i o n .

A s w e p o i n t e d o u t in t h e p r e v i o u s s e c t i o n , t h e s e c o n s t a n t s m u s t b e

d e t e r m i n e d b y t h e a p p l i c a t i o n of b o u n d a r y c o n d i t i o n s . A t t h e p o i n t x = 0 ,

w e h a v e y = 0 a n d dyldx = 0 ( th i s f o l l o w s f r o m t h e d e m a n d t h a t t h e r e b e


n o d i s c o n t i n u i t y in t h e r o c k o v e r b u r d e n ) . T h e s e c o n d i t i o n s g i v e

T h i s c u r v e a p p r o x i m a t e s t h o s e w h i c h a r e o b s e r v e d .

T h e h e i g h t of t h e l a c o l i t h is g i v e n b y t h e v a l u e of t h i s f u n c t i o n a t i t s

h i g h e s t p o i n t , w h i c h f r o m i n s p e c t i o n is t h e p o i n t x = L / 2 . W e find

(11 .D.7)

W e c a n n o w e x p l a i n t h e o b s e r v e d c o r r e l a t i o n of l a c o l i t h h e i g h t w i t h

a l t i t u d e w h i c h w e c i t e d e a r l i e r . C o n s i d e r t w o l a c o l i t h s a t d i f fe ren t

a l t i t u d e s b u t f e d f r o m t h e s a m e p o o l of m a g m a ( s e e F i g . 11.10). T h e

d i f f e r e n c e in h e i g h t A H b e t w e e n t h e m wil l r e s u l t in a d i f f e r e n c e in t h e

A H

Fig. 11.10. Two lacoliths at different altitudes.

(11 .D.4) C3 = C4 = 0.

S i m i l a r l y , a n d x = L , y = 0, a n d dyldx = 0 , s o t h a t

w h i c h g i v e s

(11 .D.5)

s o t h a t t h e e q u a t i o n d e s c r i b i n g t h e s h a p e of t h e l a c o l i t h is

(11 .D.6)

The Formation of Mountain Chains 195

p r e s s u r e of t h e m a g m a . S i n c e t h e h e i g h t of t h e l a c o l i t h is d i r e c t l y

p r o p o r t i o n a l t o t h e m a g m a p r e s s u r e , w e w o u l d e x p e c t t h a t t h e h i g h e r

a l t i t u d e l a c o l i t h w o u l d h a v e t h e s m a l l e r h e i g h t , a s o b s e r v e d .

E. THE FORMATION OF MOUNTAIN CHAINS

A n o t h e r g e o l o g i c a l p h e n o m e n o n w h i c h w e c a n u n d e r s t a n d o n t h e b a s i s

of t h e p h y s i c a l p r i n c i p l e s p r e s e n t e d in S e c t i o n l l . C is t h e f o r m a t i o n of

m o u n t a i n c h a i n s . I n g e n e r a l , w e c a n t h i n k of t h i s p r o c e s s a s a f o l d i n g of

t h e c r u s t w h e n a f o r c e is a p p l i e d a l o n g t h e s u r f a c e of t h e e a r t h . T h i s f o r c e

m i g h t a r i s e w h e n t h e l e a d i n g e d g e of a c o n t i n e n t is p u s h e d b y c o n t i n e n t a l

dr i f t a g a i n s t t h e u n d e r l y i n g m a n t l e . I t is t h o u g h t , f o r e x a m p l e , t h a t t h e

m o u n t a i n c h a i n o n t h e w e s t c o a s t of N o r t h a n d S o u t h A m e r i c a w a s

f o r m e d in t h i s w a y . I n g e n e r a l , a m o u n t a i n c h a i n wi l l h a v e t h e g e n e r a l

s h a p e s h o w n in F i g . 11 .11 , w h e r e t h e l a r g e s t m o u n t a i n s a r e c l o s e s t t o t h e

a p p l i e d f o r c e P , a n d t h e h e i g h t of t h e m o u n t a i n s v a r i e s i n v e r s e l y w i t h t h e

d i s t a n c e f r o m t h e f o r c e . T h e r e a r e , of c o u r s e , e x c e p t i o n s t o t h i s g e n e r a l

r u l e in n a t u r e , c a u s e d e i t h e r b y a n o n u n i f o r m i t y in t h e c r u s t o r b y

d e f l e c t i o n s of t h e s u r f a c e w h i c h e x i s t b e f o r e t h e f o r c e is a p p l i e d .

A s a m o d e l f o r t h i s p r o c e s s , l e t u s c o n s i d e r t h e c r u s t t o b e a t h i n ,

s emi - in f in i t e s h e e t of m a t e r i a l of Y o u n g ' s m o d u l u s E a n d m o m e n t of

i n e r t i a I [ s e e E q . (11 .C.6) a n d P r o b l e m 2.6] r e s t i n g o n t o p of a n inf in i te

e l a s t i c m e d i u m ( s e e F i g . 11.12). T h e e f fec t of t h i s m e d i u m wil l b e t o e x e r t

P

Fig. 11.11. Deformation of a plate due to a horizontal force.

Po X

elastic medium

Fig. 11.12. Model for the formation of a mountain chain.


a f o r c e w h i c h wil l o p p o s e t h e d e f l e c t i o n of t h e p l a t e . F o r o u r p u r p o s e s , w e

will t r e a t t h i s a s a s p r i n g , s o t h a t t h e f o r c e e x e r t e d o n a n y e l e m e n t of t h e

s h e e t i s p r o p o r t i o n a l t o t h e d e f l e c t i o n of t h a t e l e m e n t f r o m e q u i l i b r i u m . I t

is c l e a r t h a t t h i s is a n ef fec t w h i c h w o u l d b e p r e s e n t in t h e c a s e s of

p h y s i c a l i n t e r e s t , w h e r e a n y a t t e m p t t o p u s h t h e c r u s t d o w n i n t o t h e

m a n t l e w o u l d h a v e t o o v e r c o m e t h e f o r c e s e x e r t e d b y t h e m a n t l e i tself .

W e sha l l s e e t h a t t h i s t y p e of r e s t o r i n g f o r c e , w h i c h w e h a v e n o t

c o n s i d e r e d u p t o t h i s p o i n t , is r e s p o n s i b l e f o r t h e i n v e r s e v a r i a t i o n of

m o u n t a i n h e i g h t w i t h d i s t a n c e f r o m t h e a p p l i e d f o r c e .

F i n a l l y , w e a s s u m e t h a t s o m e e x t e r n a l f o r c e is a p p l i e d a t t h e e n d of t h e

s h e e t . W e l abe l t h e f o r c e a l o n g t h e c r u s t P 0 a n d t h e c o m p o n e n t of t h e

f o r c e p e r p e n d i c u l a r t o t h e c r u s t b y F 0 . W e i n c l u d e b o t h of t h e s e f o r c e s

b e c a u s e it is e x t r e m e l y u n l i k e l y t h a t n a t u r e w o u l d e v e r p r o v i d e a f o r c e

exactly a l o n g t h e p l a n e of t h e c r u s t .

If t h e l o a d p e r u n i t l e n g t h o n t h e c r u s t is q, t h e n t h e f o r c e s a n d t o r q u e s

a c t i n g o n a s e c t i o n of t h e c r u s t a f t e r e q u i l i b r i u m h a s b e e n e s t a b l i s h e d a r e

s h o w n in F i g . 11 .13.

I n t h i s d i a g r a m , P 0 is t h e a p p l i e d e x t e r n a l " a x i a l " f o r c e , F t h e i n t e r n a l l y

g e n e r a t e d s h e a r f o r c e , a n d ky dx a n d q dx t h e f o r c e s a p p l i e d b y t h e e l a s t i c

m e d i u m a n d t h e l o a d i n g , r e s p e c t i v e l y . B a l a n c i n g f o r c e s in t h e y - d i r e c t i o n

y i e l d s t h e e q u a t i o n

F - qdx - ( F + dF)-ky dx = 0 ,

o r

( l l . E . l )

Fig. 11.13. Forces and torques on a volume element in a mountain chain.

The Formation of Mountain Chains 197

w h i l e b a l a n c i n g t o r q u e s y i e l d s

o r

(11 .E .2 )

U s i n g t h e p r e v i o u s l y d e r i v e d r e l a t i o n s h i p b e t w e e n d e f l e c t i o n a n d

t o r q u e [ E q . (11 .C .3 ) ] , t h i s b e c o m e s

( 1 1 . E . 3

w h i c h is t h e w o r k i n g e q u a t i o n fo r o u r m o u n t a i n c h a i n m o d e l e q u i v a l e n t t o

E q . (11 .D.2) f o r t h e l a c o l i t h .

T h e g e n e r a l m e t h o d of s o l v i n g a n i n h o m o g e n e o u s d i f f e ren t i a l e q u a t i o n

of t h i s t y p e is t o n o t e t h a t t o a n y p a r t i c u l a r s o l u t i o n , y = yp of E q . (1 I . E . 3 )

w e c a n a d d y h , a s o l u t i o n of t h e h o m o g e n e o u s e q u a t i o n

(11 .E .4 )

(11 .E .5 )

s o t h a t t h e m o s t g e n e r a l s o l u t i o n is j u s t

y = yP + y h .

[ T h e r e a d e r c a n v e r i f y t h a t t h i s is i n d e e d a s o l u t i o n of E q . (11 .E .3 ) b y

d i r e c t s u b s t i t u t i o n . ]

I t is e a s y t o s e e t h a t t h e c h o i c e

(11 .E .6 )

sa t i s f ies E q . ( 1 1 . E . 3 ) . T h i s i s t h e p a r t i c u l a r s o l u t i o n d i s c u s s e d a b o v e , a n d

is a c t u a l l y of l i t t le i n t e r e s t . I t r e p r e s e n t s t h e a m o u n t t h e c r u s t s i n k s i n t o

t h e m a n t l e b e c a u s e of i t s o w n w e i g h t . W e will i g n o r e it in w h a t f o l l o w s .

T h e s t a n d a r d w a y t o find yh is t o a s s u m e a s o l u t i o n of t h e f o r m

y = e m x ( 11 .E .7 )

a n d d e t e r m i n e t h e p e r m i s s i b l e v a l u e s of m b y d i r e c t s u b s t i t u t i o n i n t o E q .

( 1 1 . E . 4 ) . If w e d o t h i s , a n d s o l v e t h e r e s u l t i n g e q u a t i o n in m, w e find

(11 .E .8 )


w h i c h , if P < 2 V E l k ( w e wil l s h o w in t h e n e x t s e c t i o n t h a t t h i s is t h e

o n l y p h y s i c a l l y i n t e r e s t i n g c a s e ) , g i v e s c o m p l e x v a l u e s f o r m ,

m = a + jj3, w h e r e

(11 .E .9 )

a n d

T h e m o s t g e n e r a l e x p r e s s i o n f o r yh is t h u s

yh = ( C i < T p * + C2e^) c o s ax

HCse-e* + C4epx)sinax. (11 .E .10)

A s in t h e p r e v i o u s s e c t i o n , t h e r e a r e f o u r c o n s t a n t s w h i c h m u s t b e

d e t e r m i n e d f r o m t h e b o u n d a r y c o n d i t i o n s . T w o c o n s t a n t s c a n b e d e t e r -

m i n e d d i r e c t l y f r o m t h e r e q u i r e m e n t t h a t t h e d e f l e c t i o n b e f ini te a s x

a p p r o a c h e s + 0 0 [x is m e a s u r e d p o s i t i v e t o t h e lef t in t h e d i a g r a m a b o v e

E q . ( l l . E . l ) ] . T h i s g i v e s

c2 = c4 = 0.

S i m i l a r l y , a t x = 0, t h e r e is n o e x t e r n a l t o r q u e b e i n g a p p l i e d , s o t h a t

T = - f = 0, w h i l e t h e a p p l i e d f o r c e F 0 m u s t b e e q u a l a n d o p p o s i t e t o t h e

i n t e r n a l l y g e n e r a t e d f o r c e . S i n c e

t h e t o r q u e c o n d i t i o n b e c o m e s

w h i c h , s u b s t i t u t i n g y h f r o m E q . (11 .E .10) l e a d s t o t h e r e s u l t

( l l . E . l l )

I n t h e s a m e w a y , u s i n g t h e e x p r e s s i o n f o r t h e i n t e r n a l l y g e n e r a t e d f o r c e in

E q . ( 11 .E .2 ) , t o g e t h e r w i t h E q . (11 .E .10) f o r yh, g i v e s

(11 .E .12)

(11 .E .13)

s o t h a t

Some Special Cases: Buckling and the Euler Theory of Struts 199

Fig. 11.14. A typical shape for a mountain chain.

T h i s i s , of c o u r s e , t h e e q u a t i o n of a d a m p e d o s c i l l a t i o n , h a v i n g a s h a p e

l i ke t h a t s h o w n in F i g . 11.14. T h i s is q u a l i t a t i v e l y t h e s h a p e w h i c h w e

d i s c u s s e d f o r m o u n t a i n t r a i n s in t h e b e g i n n i n g of t h i s s e c t i o n . O n c e a g a i n ,

w e s e e t h a t b y a p p l y i n g t h e s i m p l e i d e a s d e v e l o p e d in t h e i n t r o d u c t o r y

s e c t i o n s of t h i s c h a p t e r , t h e g e n e r a l f e a t u r e s of a r a t h e r c o m p l i c a t e d

s y s t e m c a n b e d e r i v e d . A c t u a l l y , t h e g e n e r a l e q u a t i o n (11 .E .13) d e s c r i b e s

t h e d i s t o r t i o n of a n y so l id s h e e t w h i c h is s u b j e c t e d t o a n a x i a l f o r c e a n d

e m b e d d e d in a n e l a s t i c m e d i u m . T h u s , in a d d i t i o n t o d e s c r i b i n g a

m o u n t a i n t r a i n , it w o u l d a l s o d e s c r i b e t h e f o l d i n g of a v e i n of m a t e r i a l

e m b e d d e d in o t h e r t y p e s of m a t e r i a l — e . g . , t h e b e n d i n g of q u a r t z v e i n s

e m b e d d e d in h a r d e r r o c k .

Of c o u r s e , t h e r e a r e m a n y e f f ec t s w h i c h w e h a v e i g n o r e d , s o t h a t E q .

(1 I . E . 13) s h o u l d b e r e g a r d e d a s a first a p p r o x i m a t i o n t o a c o r r e c t

d e s c r i p t i o n of a r e a l m o u n t a i n c h a i n . I n t h e p r o b l e m s , o n e s u c h e f f e c t —

t h e e x i s t e n c e of in i t ia l d e f l e c t i o n s — i s c o n s i d e r e d . E f f e c t s d u e t o

n o n h o m o g e n e i t y in t h e c r u s t o r m a n t l e , f r a c t u r i n g of t h e r o c k o r o t h e r

n o n e l a s t i c b e h a v i o r , a n d n o n u n i f o r m a p p l i e d f o r c e s wil l n o t b e d i s c u s s e d .

T h e r e a r e , h o w e v e r , s e v e r a l s p e c i a l c a s e s of E q . (11 .E .13) w h i c h a r e of

c o n s i d e r a b l e i n t e r e s t , a n d it is t o t h e s e t h a t w e wil l t u r n in t h e n e x t

s e c t i o n .

F. SOME SPECIAL CASES: BUCKLING AND THE EULER THEORY OF STRUTS

T h e r e is a g o o d d e a l of p h y s i c s c o n t a i n e d in E q . (1 I . E . 13). F o r e x a m p l e ,

it w o u l d s e e m f r o m e x a m i n i n g t h e e q u a t i o n if a f o r c e w e r e a p p l i e d t o t h e

so l id s h e e t d i r e c t l y a l o n g t h e p l a n e of t h e s h e e t ( i .e . if F 0 w e r e t o v a n i s h ) ,

t h e r e w o u l d b e n o d e f l e c t i o n . O u r i n t u i t i o n te l l s u s in t h i s c a s e t h a t t h e r e

w o u l d b e a f o r c e t e n d i n g t o c o m p r e s s t h e so l id , b u t n o t h i n g t o m a k e it

b u c k l e . O u r i n t u i t i o n a l s o t e l l s u s t h a t t h i s w o u l d b e a h i g h l y u n s t a b l e

s i t u a t i o n , s i n c e t h e s m a l l e s t f o r c e p e r p e n d i c u l a r t o t h e s h e e t w o u l d p r o d u c e

a f ini te d e f l e c t i o n . T h i s is s imi l a r t o t h e p r o b l e m of t h e s t a b i l i t y of s t a r s


s o t h a t t h e c r i t i ca l l o a d a t w h i c h b u c k l i n g o c c u r s i s j u s t

P c rit = P = VkEL (11 .F .2 )

F o r l o a d s b e l o w t h i s , t h e d e f l e c t i o n will r e m a i n f in i te , b u t a s P

a p p r o a c h e s P c r i t , t h e d e f l e c t i o n s wil l b e c o m e a r b i t r a r i l y l a r g e . T h u s ,

m a t e r i a l wil l s u p p o r t l o a d s l e s s t h a n P c r i t , b u t n o t g r e a t e r . T h i s w a s t h e

o r ig in of t h e s t a t e m e n t f o l l o w i n g E q . (11 .E .8 ) t h a t o n l y v a l u e s of P l e s s

t h a n iVEIk w e r e p h y s i c a l l y i n t e r e s t i n g — h i g h e r l o a d s w o u l d l e a d t o

b u c k l i n g .

Of c o u r s e , if F 0 w e r e z e r o a n d P w e r e a t i t s c r i t i ca l v a l u e , E q . (1 I . E . 13)

w o u l d b e of a n i n d e t e r m i n a t e f o r m ( z e r o d i v i d e d b y z e r o ) . A g a i n , s u c h a

c o n f i g u r a t i o n w o u l d b e h i g h l y u n s t a b l e , a n d n e e d n o t c o n c e r n u s f u r t h e r .

H o w e v e r , t h e a s t u t e r e a d e r wi l l a l r e a d y h a v e r e m a r k e d t h a t t h e c r i t i ca l

l o a d i n g g i v e n b y E q . (11 .F .2 ) c o u l d n o t a p p l y t o a s t r e s s e d m a t e r i a l w h i c h

w a s u n c o n f i n e d , b e c a u s e f o r s u c h a s y s t e m , t h e s p r i n g c o n s t a n t k w o u l d

b e z e r o . U n c o n f i n e d b e a m s w h i c h a r e r e q u i r e d t o c a r r y a n a x i a l l o a d a r e

c a l l e d " s t r u t s , " a n d t h e s t u d y of t h e i r p r o p e r t i e s i s , of c o u r s e , of i m m e n s e

p r a c t i c a l u s e f u l n e s s in c o n s t r u c t i o n of b u i l d i n g s , b r i d g e s , a n d o t h e r

s t r u c t u r e s .

T h e t h e o r y of s t r u t s , first d e v e l o p e d b y E u l e r , is a s p e c i a l c a s e of t h e

p r o b l e m t r e a t e d in t h e p r e v i o u s s e c t i o n , b u t w e s e e f r o m E q . (11 .E .9 ) t h a t if

k = 0 , ]8 2 = - a 2 s o t h a t d i v i d i n g b y a2 + j 8 2 , a s w e h a d t o d o t o d e r i v e E q .

(1 I . E . 13) wi l l n o l o n g e r b e v a l i d . I n f a c t , it is p r o b a b l y e a s i e r t o d e r i v e t h e

w h i c h w e c o n s i d e r e d in C h a p t e r 3 , w h e r e t h e s m a l l e s t d e v i a t i o n f r o m

e q u i l i b r i u m c o u l d d r i v e a s y s t e m a l o n g w a y w h e n t h e e q u i l i b r i u m

h a p p e n e d t o b e u n s t a b l e .

T o i n v e s t i g a t e t h e i n t e r e s t i n g a s p e c t s of t h i s p r o b l e m , l e t u s b e g i n b y

a s k i n g h o w l a r g e t h e ax ia l l o a d P c a n b e f o r a g i v e n m a t e r i a l . I m a g i n e t h a t

t h e l o a d P is a p p l i e d in t h e p r e s e n c e of a sma l l b u t f ini te F 0 , a n d t h e n

g r a d u a l l y i n c r e a s e d . W h a t will h a p p e n ?

E x a m i n i n g E q . (1 I . E . 13), w e s e e t h a t t h e d e f l e c t i o n wil l b e we l l b e h a v e d

e x c e p t w h e n w e a p p r o a c h t h e v a l u e of P w h i c h m a k e s

3 j 3 2 - a 2 = 0 . ( l l . F . l )

F o r t h i s v a l u e of P , t h e d e f l e c t i o n will b e c o m e inf in i te f o r a n y n o n z e r o F 0 .

T h i s p h e n o m e n o n is k n o w n a s " b u c k l i n g " of t h e m a t e r i a l . F r o m E q .

( 1 1 . E . 9 ) , it wil l o c c u r f o r a l o a d P w h i c h sa t i s f ies

Some Special Cases: Buckling and the Euler Theory of Struts 201

d e f l e c t i o n of a s t r u t b y s t a r t i n g f r o m E q . (11 .E .8 ) d i r e c t l y t h a n b y f ind ing

s o m e s u i t a b l e l imi t of E q . (1 I . E . 13).

If t h e s p r i n g c o n s t a n t is z e r o , t h e n t h e e q u a t i o n f o r m 2 is j u s t

s o t h a t m 2 ( E I m 2 + P ) = 0, (11 .F .3 )

(11 .F .4 )

w h i c h m e a n s t h a t y will b e g i v e n b y a n undamped o s c i l l a t i o n , r a t h e r t h a n

a d a m p e d o n e . F o r t h e s a k e of d e f i n i t e n e s s , l e t u s c o n s i d e r a s t r u t of l e n g t h

L l o a d e d w i t h a n a x i a l l o a d P a t b o t h e n d s ( s e e F i g . 11.15) .

T h e m o s t g e n e r a l s o l u t i o n f o r y wil l t h e n b e

y = A c o s yx + B s in yx.

T o d e t e r m i n e A a n d B it wi l l b e n e c e s s a r y t o de f ine b o u n d a r y c o n d i t i o n s .

T h e m o s t u s u a l a p p l i c a t i o n of t h e t h e o r y of s t r u t s i s in t h e c a s e w h e r e b o t h

e n d s a r e h e l d f ixed , a n d t h e s t r u t is c o m p r e s s e d . T h i s is t h e c a s e w e wi l l

c o n s i d e r h e r e , a n d t h e c a s e o f a s t r u t w i t h f r e e e n d s wi l l b e le f t t o t h e

p r o b l e m s .

If w e r e q u i r e t h a t y = 0 a t x = 0 a n d x = L , w e h a v e

a n d

w h i c h m e a n s t h a t e i t h e r

o r

A = 0

B s in yL = 0 ,

n = 0

I n t h e first c a s e , t h e d e f l e c t i o n is i d e n t i c a l l y z e r o w h i c h m e a n s t h a t t h e

b e a m wil l n o t b e n d a t al l . If P a p p r o a c h e s o n e of t h e c r i t i ca l v a l u e s g i v e n

—(f)"

x = 0 x = L

Fig. 11.15. A loaded strut.


h o w e v e r , t h e n a s o l u t i o n of t h e f o r m

y = B s in (11 .F .5 )

is p o s s i b l e . T h u s , t h e s t r u t m u s t e i t h e r n o t b e n d a t al l , o r b e b e n t i n t o a n

h a r m o n i c of a s i n e w a v e w h e n t h e c r i t i ca l l o a d is a p p l i e d . I n s u c h a c a s e ,

t h e v a l u e of B, t h e m a x i m u m d e f l e c t i o n of t h e b e a m is u n d e t e r m i n e d ,

a l t h o u g h in t h e o r y it i s r e l a t e d t o t h e f o r c e a p p l i e d a t t h e e n d s of t h e s t r u t

( s e e P r o b l e m 11.5) .

T h i s s u d d e n t r a n s i t i o n t o a d e f o r m e d s h a p e a s P is i n c r e a s e d is t h e

a n a l o g u e t o t h e b u c k l i n g of a m e m b e r e m b e d d e d in a n e l a s t i c m e d i u m

w h i c h w e d i s c u s s e d a b o v e .

G. FENNO-SCANDIA REVISITED

I n C h a p t e r 8, w e d i s c u s s e d t h e p r o b l e m of v i s c o u s r e b o u n d in t h e

c o n t e x t of t h e g e o l o g i c a l p h e n o m e n o n of t h e F e n n o - S c a n d i a n u p l i f t — t h e

r i s i ng of t h e c r u s t of t h e e a r t h a f t e r t h e m e l t i n g of t h e g l a c i e r s . I t w a s

c r u c i a l t o t h a t d i s c u s s i o n t h a t t h e f o r c e s a s s o c i a t e d w i t h t h e c r u s t i tself b e

neg l ig ib l e c o m p a r e d t o t h e b u o y a n t f o r c e s g e n e r a t e d b y t h e m a n t l e u n d e r

t h e c r u s t . W e a r e n o w in a p o s i t i o n t o s h o w t h a t t h i s w a s a v a l i d

a s s u m p t i o n .

F o r t h e s a k e of s i m p l i c i t y , c o n s i d e r a n ini t ial d e f o r m a t i o n of t h e c r u s t

g i v e n b y ( s e e F i g . 11.16)

W h e n t h e l o a d i n g is l i f ted (e .g . , w h e n t h e g l a c i e r m e l t s ) , t h e u n d e r l y i n g

fluid wil l e x e r t a r e s t o r i n g f o r c e t e n d i n g t o lift t h e d e f o r m e d p a r t of t h e

c r u s t . T h e d e v e l o p m e n t of t h i s p r o c e s s w a s t r e a t e d in S e c t i o n 8 .C. F o r

o u r p u r p o s e s , w e s i m p l y n o t e t h a t t h e r e s t o r i n g p r e s s u r e a t a p o i n t x is

(ll.G.l)

x =0

£(x)

Fig. 11.16. Elastic forces in the deformed crust.

Fenno-Scandia Revisited 203

s i m p l y t h e w e i g h t of t h e d i s p l a c e d f lu id—i .e .

P = P g & (11 .G.2)

w h e r e £ is t h e y - c o o r d i n a t e of t h e s u r f a c e , a n d a t t = 0 is e q u a l t o £ 0 .

T h e q u e s t i o n w h i c h w e w i s h t o a n s w e r c o n c e r n s t h e r e l a t i v e i m p o r -

t a n c e of t h e f o r c e a s s o c i a t e d w i t h t h i s p r e s s u r e a n d t h e f o r c e g e n e r a t e d

b y t h e c r u s t s n a p p i n g b a c k f r o m i t s d e f o r m e d p o s i t i o n . W e c a n a t t a c k t h i s

p r o b l e m in t h e f o l l o w i n g w a y : L e t u s c o n s i d e r a s e c t i o n of c r u s t w h i c h is

in i t ia l ly flat, b u t l o a d e d b y a f o r c e

T h i s f o r c e is p r o p o r t i o n a l , w i t h p r o p o r t i o n a l i t y c o n s t a n t A , t o t h e f o r c e

e x e r t e d b y t h e p r e s s u r e of t h e m a n t l e of t h e d e f o r m e d c r u s t .

If w e c o n s i d e r t h e c r u s t b e i n g d e f l e c t e d f r o m a flat c o n f i g u r a t i o n b y t h e

a p p l i e d f o r c e (11 .G .3 ) , t h e n t h e a m o u n t of d e f o r m a t i o n f o r a g i v e n a p p l i e d

f o r c e will tel l u s h o w m u c h f o r c e is r e q u i r e d t o p r o d u c e a g i v e n d e f l e c t i o n .

I n p a r t i c u l a r , w e c a n a s k w h a t v a l u e of A is n e e d e d t o p r o d u c e a d e -

f l ec t ion e q u a l t o t h a t p r o d u c e d b y t h e g l a c i e r , a n d g i v e n b y E q . (1 l . G . l ) . If

A t u r n s o u t t o b e v e r y s m a l l , w e will h a v e a s i t u a t i o n in w h i c h f o r c e s v e r y

sma l l c o m p a r e d t o t h o s e of t h e a c t u a l p r e s s u r e [ E q . (11 .G.2) ] wil l suffice t o

p r o d u c e l a r g e d e f l e c t i o n s of t h e c r u s t , w h i l e if A is l a r g e , it wil l t a k e

f o r c e s m u c h g r e a t e r t h a n t h o s e a s s o c i a t e d w i t h t h e p r e s s u r e t o de f l ec t t h e

c r u s t . S i n c e in t h e F e n n o - S c a n d i a n up l i f t t h e c r u s t is d e f l e c t e d o n l y a n

a m o u n t £ 0 , o u r p r e v i o u s a s s u m p t i o n t h a t t h e c r u s t a l f o r c e s c o u l d b e

n e g l e c t e d w o u l d a m o u n t t o a n a s s u m p t i o n t h a t A b e v e r y sma l l s i n c e in

t h a t c a s e m o s t of t h e b u o y a n t f o r c e s m u s t g o i n t o o v e r c o m i n g o t h e r t h i n g s

t h a n f o r c e s g e n e r a t e d in t h e c r u s t .

F r o m E q . (11 .C .6 ) , w e h a v e

s o t h a t , a s s u m i n g t h a t t h e c r u s t is s t a t i o n a r y a t x = 0 a n d x = L, w e find

(11 .G.3 )

(11 .G.4)

(11 .G.5)

s o t h a t


T h i s r e s u l t m e a n s t h a t in o r d e r t o o v e r c o m e f o r c e s g e n e r a t e d w i t h i n t h e

c r u s t b y t h e ini t ia l d e f o r m a t i o n in E q . ( l l . G . l ) , w e w o u l d n e e d a f o r c e

w h i c h is o n l y a s m a l l f r a c t i o n of t h e a c t u a l b u o y a n t f o r c e g i v e n in E q .

(11 .G .2 ) . T h u s , v i r t u a l l y all of t h e f o r c e g e n e r a t e d b y t h e b u o y a n c y m u s t

g o i n t o o v e r c o m i n g t h e v i s c o u s d r a g of t h e m a n t l e ( t h e p r o c e s s w h i c h w e

c o n s i d e r e d in C h a p t e r 8) a n d a l m o s t n o n e i n t o o v e r c o m i n g t h e c r u s t a l

f o r c e s t h e m s e l v e s . T h i s is w h a t w e s e t o u t t o s h o w .

SUMMARY

W h e n a so l id is s u b j e c t e d t o e x t e r n a l f o r c e s o r t o r q u e s , it g e n e r a t e s

w i t h i n i tse l f f o r c e s a n d t o r q u e s w h i c h t e n d t o o p p o s e t h o s e b e i n g a p p l i e d

e x t e r n a l l y , a n d h e n c e t o b r i n g t h e e n t i r e s y s t e m i n t o a s t a t e of s t a t i c

e q u i l i b r i u m . F o r t h e c a s e of a n e l a s t i c so l id , a s i m p l e f o u r t h - o r d e r

d i f fe ren t i a l e q u a t i o n c a n b e w r i t t e n d o w n w h i c h r e l a t e s t h e a m o u n t of

d e f o r m a t i o n of t h e so l id t o t h e m a g n i t u d e of t h e e x t e r n a l f o r c e .

D e p e n d i n g o n t h e b o u n d a r y c o n d i t i o n s a n d t h e f o r c e s a c t i n g , t h i s

e q u a t i o n c a n b e u s e d t o d e s c r i b e t h e g e n e r a l f e a t u r e s of g e o l o g i c a l

f o r m a t i o n s l i ke l a c o l i t h s a n d m o u n t a i n c h a i n s , o r t h e b u c k l i n g of s t r u t s

w h e n l a r g e a x i a l l o a d s a r e a p p l i e d .

PROBLEMS

11.1. Consider a canti lever; i.e. a beam supported at one end only. Assume that the beam has Young ' s modulus E, moment of inertia I, weight per unit length q, and is of a length L.

(a) Wri te down the equat ion which describies the deformation of the beam as a function of length (see figure). This equat ion will have four undetermined coefficients.

(b) Wri te down the boundary condit ions at x = 0, hence determine two of the four cons tants .

(c) Wri te down the boundary condit ions at x = L, hence determine the remain-ing cons tants , showing that the deformation of the canti lever is given by

If w e n o w t a k e p a r a m e t e r s a p p r o p r i a t e t o F e n n o - S c a n d i a , n a m e l y p =

3.27 g / c c , E = 10 9 d y n e s c m 2 , L = 1400 k m , a n d e v a l u a t e I f o r a c r u s t

t h i c k n e s s of 35 k m , w e find ( s e t t i n g y = £ 0 )

(11 .G.7)

Problems 205

(d) Find the maximum deflection of the beam and the maximum internal to rque generated in the beam.

11 .3 . In Section l l . D , we considered a lacolith formed by upward flow through a straight crack. Show that if we consider upward flow through a point hole in the lower strata, the equat ion which descr ibes the shape of the surface layer is

where R is the radius of the lacolith and r the radial dis tance from the center . Wha t is the cons tant /3?

11.4. Consider the formation of a mountain train in which an initial deformat ion is present . Take as a model a crust of momen t I and Young ' s modulus E, embedded in a medium of spring cons tant k, and initially deformed to give a surface

x = L

y=P(R2-r2)\

= d0 sin —.

(see figure).

N o w suppose a load is applied to the ends of this deformation, as shown, so that the final configuration is

11 .2. Show that the equat ion equivalent to Eq . (11.C.6) for a solid beam is just


Assume all deformat ions are small, so that

y(x) = yo + y i ,

where yx is the extra bending due to the applied forces. Follow the steps leading to Eq . (1 I .E. 13) to find the final shape of the crust . Will the phenomenon of buckling occur here?

11.5. Consider a strut with unsuppor ted free ends , with an axial load P applied at x = 0 and x = L, and a force perpendicular to the strut at the ends be F 0 at x = 0, and - F 0 at x = L

(a) Show that the requirement that there be no torque at the ends leads to Eq. (11.F.4).

(b) Find a relation be tween P , F 0 and the undetermined constant B. (c) Show that in the limit F 0 = 0, the only allowed solution for the strut is y = 0

everywhere . Interpret this result .

11.6. A solid beam of Young ' s modulus E and cross-sect ional moment I is c lamped at one end, and allowed to extend vertically in a gravitational field as shown. If the beam has weight q(x) on it, and the internal forces act as shown:

(a) Show that the equat ion for the deformation is

(b) For the case q(x) = q = const , show that this reduces to

where

Problems 207

Where did this energy come from?

11.9. Calculate the shape of a beam which is c lamped at one end, and suppor ted (but not c lamped) at a level with the c lamped end at its other end. Le t the weight per unit length of the beam be q, and its length L.

11.10. Calculate the shape of a weightless beam clamped at both ends , but with a weight W applied at its center .

where

z = L - x. and

(c) Show that a solution to this equat ion is

and find a relation be tween the an. F r o m the fact that there is no to rque at z = 0, show that we can write

(d) At z = L, we must have P = 0. If aQ ^ 0, this means

Show by plotting the right-hand side of the above as a function of j8L 3 that there is a minimum value of /3L 3 which will allow a solution (jSLLn ~ 7.8). H e n c e find the max imum height to which a t ree can grow. Do any t rees in na ture come close to this limit?

11.7. Show that the work done in stretching a length of filament dl as in Fig. 11.5 is just

H e n c e , show that the energy stored in the bent b e a m is given by

11.8. H e n c e show that the energy stored in the canti lever in P rob lem 11.1 is


11.11. Consider a canti lever whose load per unit length is q, but which has a

charge per unit length a on it. Find the shape the canti lever will have in an

electrical field E directed vertically. Will it ever curve up instead of down?

11.12. Carry through the analysis in Problem 11.8 when a cons tant force per unit length B is exer ted in the horizontal direction on the beam. H e n c e discuss the effect of wind on vertical s t ructures .

REFERENCES

As in the case of hydrodynamics, there are many standard texts on the theory of elasticity. These include

L. D. Landau and E. M. Lifschitz, Theory of Elasticity, Pergamon Press, New York, 1959. The same comments apply to this as to the Landau and Lifschitz text on hydrodynamics cited in Chapter 1.

A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity, Dover Publications, New York, 1944.

This text, like Lamb's Hydrodynamics, is an exhaustive treatment of many interesting and complicated problems, but suffers from a somewhat dated notational scheme.

John Prescott, Applied Elasticity, Dover Publications, New York, 1961. A book which has many worked examples of complicated systems without the advanced mathematics used in many texts.

Gerard Nadeau, Introduction to Elasticity, Holt, Rinehart, and Winston, New York, 1964. Uses a somewhat cumbersome dyadic notation, but discusses many simple problems in an understandable way.

I. S. Sokolnikoff, Mathematical Theory of Elasticity, McGraw-Hill, New York, 1956. Math is easy to follow, but there is little relation to experiment.

A. M. Johnson, Physical Processes in Geology, Freeman, Cooper, San Francisco, 1970. An excellent and readable account of the geological processes by which various formations are created. This book is especially valuable for physicists because of the clear treatment of descriptive geology which accompanies each example.

Carl W. Condit, Scientific American, Vol. 230 # 2, p. 92, 1974. This is a very interesting discussion of wind bracing in tall buildings (see Problem 11.12).

See the references in Chapter 8 for readings on the Fenno-Scandian uplift.

12

General Properties of Solids— Dynamics

A. THE STRAIN TENSOR

W e h a v e s e e n t h a t t h e m a i n f e a t u r e w h i c h d i s t i n g u i s h e s so l i d s f r o m

l i q u i d s is t h e ab i l i t y t o g e n e r a t e i n t e r n a l f o r c e s t o o p p o s e e x t e r n a l l o a d s

p l a c e d u p o n t h e m . I n t h e c a s e of e l a s t i c s o l i d s , t h e s e i n t e r n a l f o r c e s a r e

r e l a t e d t o t h e d e f o r m a t i o n of t h e so l i d v i a H o o k e ' s l a w . U p t o t h i s p o i n t ,

h o w e v e r , w e h a v e n o t c o n s i d e r e d h o w a so l id m o v e s w h e n a f o r c e is

a p p l i e d , b u t o n l y t h e final s t a t i c d e f o r m a t i o n . I n o r d e r t o d i s c u s s t h e

d y n a m i c s of t h e r e s p o n s e of a so l id t o f o r c e s , it wil l first b e n e c e s s a r y t o

find a m o r e g e n e r a l w a y of d e s c r i b i n g b o t h t h e a p p l i e d f o r c e s a n d t h e

d e f o r m a t i o n of t h e so l id .

L e t u s c o n s i d e r t w o p o i n t s in a so l id s e p a r a t e d b y a d i s t a n c e dx ( s e e

F i g . 12.1). A f t e r a f o r c e is a p p l i e d , l e t t h e s e p a r a t i o n b e d x ' . W e c a n de f ine

t h e c h a n g e in r e l a t i v e p o s i t i o n b y a v e c t o r du in t h e e q u a t i o n

In England, we always let an institution strain until it breaks.

GEORGE BERNARD SHAW

Getting Married

dx' = dx + du, (12 .A.1)

w h i c h m e a n s t h a t t h e s q u a r e of t h e s e p a r a t i o n is j u s t

dx'2 = dx'i dx \ = (dXi + dUi)(dXi + dut)

209

210 General Properties of Solids—Dynamics

CD Fig. 12.1. The deformation of a solid.

w h e r e w e h a v e u s e d t h e r e l a t i o n s h i p

If w e shi f t s o m e d u m m y i n d i c e s , a n d r e a r r a n g e t h e s e c o n d t e r m , t h i s c a n

b e w r i t t e n a s

dx'2 = dx2 + 2uik dxt dxk, (12 .A.2)

w h e r e t h e t e n s o r uik i s j u s t g i v e n b y

(12 .A.3)

T h i s is a n e x t r e m e l y i m p o r t a n t q u a n t i t y , a s w e sha l l s e e , a n d is c a l l e d t h e

strain tensor.

F r o m t h e d e r i v a t i o n , it i s c l e a r t h a t t h e s t r a i n t e n s o r m u s t d e s c r i b e t h e

d e f o r m a t i o n of a so l id . S i n c e t h e d e r i v a t i v e s of ut d e s c r i b e t h e c h a n g e of

r e l a t i v e c o o r d i n a t e s in t h e so l i d [ s e e E q . ( 12 .A .2 ) ] , s o l o n g a s w e c o n f i n e

o u r a t t e n t i o n t o sma l l d e f o r m a t i o n s ( a s w e d i d in C h a p t e r 11), w e c a n d r o p

s e c o n d - o r d e r t e r m s in t h e d e r i v a t i v e s of uh s o t h a t

(12 .A.4)

T h i s is t h e f o r m w h i c h w e sha l l u s e t h r o u g h o u t t h e r e m a i n d e r of t h e

d i s c u s s i o n .

I n o r d e r t o u n d e r s t a n d w h a t t h e s t r a i n t e n s o r m e a n s , w e shal l l o o k a t

t w o e x a m p l e s of s t r a i n t e n s o r s a n d d e d u c e t h e a c t u a l d e f o r m a t i o n s t o

w h i c h t h e y c o r r e s p o n d . F o r t h e first e x a m p l e , c o n s i d e r a s t r a i n t e n s o r

g i v e n b y

/Mi 0 \

uik = I u2

\ 0 uj

I n s u c h a d i a g o n a l s t r a i n t e n s o r , t h e v a l u e s of dx' a r e g i v e n b y

dx i = (1 + Ui) dxh

The Strain Tensor 211

s o t h a t t h e v o l u m e of a n e l e m e n t o n t h e so l i d a f t e r t h e d e f o r m a t i o n w o u l d

j u s t b e

dV = dx\ dx2 dx3 = dxx dx2 dx3(\ + ux + u2 + u3) = dV(\ + uii),

w h e r e w e h a v e , a g a i n , d r o p p e d s e c o n d - o r d e r t e r m s in t h e d e f o r m a t i o n .

T h e q u a n t i t y Uu is t h e t r a c e of t h e s t r a i n t e n s o r , a n d w e s e e t h a t t h e

r e l a t i v e c h a n g e of v o l u m e of a n e l e m e n t is g i v e n b y

(12 .A.5)

T h u s , f r o m o u r first e x a m p l e , w e s e e t h a t t h e d i a g o n a l e l e m e n t s of t h e

s t r a i n t e n s o r s a r e r e l a t e d t o c h a n g e s in v o l u m e s in t h e so l id , a n d t h a t a

p u r e l y d i a g o n a l s t r a i n t e n s o r c o r r e s p o n d s t o e i t h e r a c o m p r e s s i o n o r

d i l a t i o n of t h e so l id .

F o r o u r s e c o n d e x a m p l e , c o n s i d e r a s t r a i n t e n s o r w h i c h h a s o n l y

o f f -d i agona l e l e m e n t s , s u c h a s

/ 0 ui2 0 \

uik = iu2l 0 0 \ 0 0 0 /

( s i n c e t h e t e n s o r is s y m m e t r i c , ux2 = w 2i). C o n s i d e r a so l id w h i c h b e f o r e

d e f o r m a t i o n c o n t a i n s t w o v e c t o r s ( s e e F i g . 12.2) A a n d B, in i t ia l ly a l o n g

t h e 1 a n d 2 a x e s , r e s p e c t i v e l y . A f t e r t h e d e f o r m a t i o n , t h e v e c t o r s wil l

h a v e s h i f t e d o v e r , a n d w e will h a v e ( b y de f in i t ion)

8A2 = u2iA,

8BX = u12B.

< > \

SB,

— X

Fig. 12.2. Rotation of position vectors in a solid.

A


s o t h a t t h e t o t a l c h a n g e in a n g l e b e t w e e n t h e t w o v e c t o r s is j u s t

a = a , + a2 = 2uX2. (12 .A.6)

T h u s , w e s e e t h a t t h e o f f -d iagona l e l e m e n t s of t h e s t r a i n t e n s o r a r e

r e l a t e d t o s h e a r d e f o r m a t i o n s in t h e so l id , a n d , in f a c t , c a n b e r e l a t e d t o

t h e s h e a r a n g l e a.

T h e s t r a i n t e n s o r , t h e n , g i v e s u s a w a y of d e s c r i b i n g t h e m o s t g e n e r a l

k i n d s of d e f o r m a t i o n s w h i c h c a n t a k e p l a c e in a so l id . T h e d i a g o n a l

e l e m e n t s c o r r e s p o n d t o t h e c o m p r e s s i o n o r d i l a t i o n of t h e so l id , w h i l e t h e

o f f -d i agona l e l e m e n t s c o r r e s p o n d t o s h e a r i n g .

B. THE STRESS TENSOR

N o w t h a t w e h a v e d e v e l o p e d a g e n e r a l i z e d w a y of d e s c r i b i n g t h e

d e f o r m a t i o n of a so l id w h e n f o r c e s a r e a p p l i e d , w e n e e d t o d e v e l o p a n

e q u a l l y g e n e r a l i z e d w a y of d e s c r i b i n g t h e f o r c e s t h e m s e l v e s . T h i s is

k n o w n a s t h e s t r e s s t e n s o r , a n d h a s a l r e a d y b e e n d i s c u s s e d ( a l t h o u g h

n o t u n d e r t h i s n a m e ) in C h a p t e r 8, w h e r e t h e t e n s o r <jik w a s i n t r o d u c e d t o

d e s c r i b e t h e v i s c o s i t y in a fluid [ s e e E q . (8 .A .3 ) ] .

L e t u s i n t r o d u c e t h e i d e a of a s t r e s s t e n s o r b y n o t i n g t h a t w h e n a b o d y

is d e f o r m e d , e a c h in f in i t e s ima l e l e m e n t in t h e b o d y f ee l s a f o r c e p e r u n i t

v o l u m e F e x e r t e d o n it b y i t s n e i g h b o r . F o r a n e l e m e n t in t h e i n t e r i o r of

t h e b o d y , t h e s e f o r c e s wil l c a n c e l o u t in t h e s t a t i c c a s e , b u t f o r a n e l e m e n t

in t h e s u r f a c e , t h e y wil l n o t ( s e e F i g . 12.3). T h i s , of c o u r s e , is t h e

m e c h a n i s m b y w h i c h a f o r c e is g e n e r a t e d a t t h e s u r f a c e of a b o d y t o

c a n c e l t h e a p p l i e d f o r c e s [ s e e E q s . (11 .C.6) a n d (11 .C .7 ) ] . T h e r e a d e r m a y

b e i n t e r e s t e d in c o m p a r i n g t h i s i d e a of f o r c e c a n c e l l a t i o n in t h e b o d y of a

so l id w i t h t h e i d e a of s u r f a c e t e n s i o n in a fluid ( C h a p t e r 5) o r in a n u c l e u s

( C h a p t e r 7 ) .

I n c o m p l e t e a n a l o g y t o t h e d e v e l o p m e n t in S e c t i o n 8 .A, w e c a n w r i t e

F o r sma l l a n g l e s , t h e a n g l e s t h r o u g h w h i c h e a c h v e c t o r h a s b e e n r o t a t e d

a r e g i v e n b y

(12 .B.1)

The Stress Tensor 213

Fig. 12.3. Forces on internal and surface volume elements in a solid.

fo r t h e f o r c e p e r u n i t v o l u m e in t h e i t h - d i r e c t i o n o n a v o l u m e e l e m e n t , s o

t h a t t h e t o t a l f o r c e in t h e i t h - d i r e c t i o n is j u s t

T h e t e n s o r cjik w h o s e d i v e r g e n c e is t h e b o d y f o r c e in a so l id is c a l l e d t h e

stress tensor, a n d is e x t r e m e l y i m p o r t a n t in t h e d i s c u s s i o n of s o l i d s . W e

s e e t h a t it c a n b e i n t e r p r e t e d [ E q . (12 .B .2 ) ] , a s t h e n e t f o r c e in t h e

i t h - d i r e c t i o n o n a s u r f a c e p e r p e n d i c u l a r t o t h e Jc th-d i rec t ion . T o m a k e t h i s

i d e a c l e a r , c o n s i d e r F i g . 12.4 in w h i c h a s u r f a c e in t h e y-z p l a n e is d r a w n .

T h i s s u r f a c e is p e r p e n d i c u l a r t o t h e x - d i r e c t i o n . I n g e n e r a l , t h e r e a r e t h r e e

t y p e s of f o r c e s t h a t c a n b e e x e r t e d o n it

(i) a f o r c e d i r e c t e d a l o n g t h e x - d i r e c t i o n , w h i c h w e w o u l d t e r m o~xx;

(ii) a f o r c e a l o n g t h e y - d i r e c t i o n , c a l l e d c r y x ;

(iii) a f o r c e a l o n g t h e z - d i r e c t i o n , c a l l e d azx.

(12 .B.2)

z

Fig. 12.4. An interpretation of the stress tensor.


T h e first of t h e s e is a c o m p r e s s i o n a l f o r c e , w h i l e t h e o t h e r t w o a r e s h e a r

f o r c e s .

T h e r e a r e t w o p o i n t s t o n o t e a b o u t t h e s t r e s s t e n s o r b e f o r e w e l o o k a t

s o m e e x a m p l e s . F i r s t , w e n o t e a g a i n t h a t in d e s c r i b i n g a f o r c e a c t i n g o n a

s u r f a c e , t w o t h i n g s m u s t b e spec i f i ed : T h e d i r e c t i o n of t h e f o r c e a n d t h e

d i r e c t i o n of t h e s u r f a c e . T h i s is w h y a s e c o n d r a n k t e n s o r p r o v i d e s t h e

m o s t n a t u r a l d e s c r i p t i o n of f o r c e s a c t i n g a t t h e s u r f a c e s of s o l i d s .

S e c o n d , in a l m o s t all of t h e p r o b l e m s w h i c h a r e e n c o u n t e r e d in d e a l i n g

w i t h s o l i d s , t h e s t r e s s t e n s o r is g i v e n . J u s t a s in d e a l i n g w i t h s t a t i c

d e f o r m a t i o n s w e w e r e g i v e n t h e l o a d a n d h a d t o d i s c o v e r t h e s h a p e of t h e

m a t e r i a l , in t h e m o r e g e n e r a l p r o b l e m s w h i c h w e sha l l d e s c r i b e , t h e f o r c e s

a c t i n g o n a so l id wi l l b e g i v e n , a n d w e sha l l w a n t t o find t h e r e s p o n s e of

t h e so l id ( d e s c r i b e d b y t h e s t r a i n t e n s o r ) . T h i s is e x a c t l y a n a l o g o u s t o t h e

u s u a l p r o b l e m in m e c h a n i c s , in w h i c h w e a r e g i v e n t h e f o r c e s a c t i n g o n a

b o d y , a n d t h e n r e q u i r e d t o find t h e s u b s e q u e n t m o t i o n . I t i s e a s y t o

s h o w [ see P r o b l e m ( 1 2 . 3 ) l t h a t t h e s t r e s s t e n s o r m u s t b e s y m m e t r i c — i . e .

t h a t <Jik = Cr k i .

F u r t h e r f a m i l i a r i z a t i o n w i t h trik i s p r o b a b l y b e s t d o n e t h r o u g h e x a m p l e s .

C o n s i d e r first a b o d y i m m e r s e d in a fluid, w h i c h e x e r t s a p r e s s u r e P o n

t h e s u r f a c e . S i n c e t h e p r e s s u r e b y de f in i t ion a c t s p e r p e n d i c u l a r t o t h e

s u r f a c e , t h e f o r c e e x e r t e d o n a s u r f a c e e l e m e n t is s i m p l y

Ft = - P dSt = - P8* dSk,

w h e r e t h e m i n u s s ign d e n o t e s a n i n w a r d f o r c e . F o r t h i s c a s e , t h e s t r e s s

t e n s o r is j u s t (Tik = — P8ik,

s o t h a t a p u r e c o m p r e s s i o n of d i l a t i o n c o r r e s p o n d s t o a d i a g o n a l s t r e s s

t e n s o r .

A n o t h e r i m p o r t a n t e x a m p l e of a s t r e s s t e n s o r c a n b e t a k e n f r o m t h e

field of e l e c t r i c i t y a n d m a g n e t i s m ( t h e r e a d e r u n f a m i l i a r w i t h t h i s field c a n

s k i p a h e a d t o S e c t i o n 12.C w i t h o u t l o s s of c o n t i n u i t y ) . T h e e l e c t r i c i t y a n d

m a g n e t i c fields, E a n d B , a r e g i v e n in t e r m s of c h a r g e a n d c u r r e n t

d e n s i t i e s , p a n d j , b y t h e M a x w e l l e q u a t i o n s

V - E = 4TTP, (12 .B .3)

V • B = 0 , (12 .B.4)

(12 .B .5)

The Stress Tensor 215

(12 .B.6)

S u p p o s e t h a t w e h a d a c o l l e c t i o n of c h a r g e s a n d c u r r e n t s e n c l o s e d in a

v o l u m e V. T h e n t h e t o t a l f o r c e a c t i n g o n t h e c h a r g e s a n d c u r r e n t w o u l d

i u s t b e

U s i n g E q . (12 .B .3) t o e l i m i n a t e p a n d E q . (12 .B .5) t o e l i m i n a t e j f r o m t h i s

e x p r e s s i o n , w e h a v e , a f t e r a d d i n g a n d s u b t r a c t i n g

t o t h e i n t e g r a n d

(12 .B.7)

T h e l e f t - h a n d s i d e of t h i s e q u a t i o n n o w is a f o r c e ( i .e . t h e t i m e

d e r i v a t i v e of t h e m o m e n t u m of t h e p a r t i c l e s ) a n d t h e t i m e d e r i v a t i v e of

( l / 4 7 r c ) ( E x B ) , w h i c h w e i d e n t i f y a s t h e m o m e n t u m of t h e field. T h u s ,

t h e l e f t - h a n d s i d e is j u s t t h e t i m e r a t e of c h a n g e of t h e t o t a l m o m e n t u m .

T h e r i g h t - h a n d s i d e , o n t h e o t h e r h a n d , c a n b e r e w r i t t e n u s i n g t h e r e s u l t of

t h e first v e c t o r i d e n t i t y in P r o b l e m 1.1 a n d E q . (12 .B.4) t o r e a d

w h i c h , in C a r t e s i a n t e n s o r n o t a t i o n , is j u s t

w h i c h is p r e c i s e l y t h e f o r m of E q . (12 .B .2 ) .

If w e w r i t e

(12 .B.8)

t h e n Tij is t h e Maxwell stress tensor, a q u a n t i t y f a m i l i a r f r o m e l e c -

t r o d y n a m i c s . I t i s , of c o u r s e , j u s t o n e e x a m p l e of a s t r e s s t e n s o r , a n d w e

h a v e c a l c u l a t e d it b y d e t e r m i n i n g d i r e c t l y t h e f o r c e s a c t i n g o n e a c h p o i n t

of o u r b o d y .


I t is i m p o r t a n t t o e m p h a s i z e t h a t in c a l c u l a t i n g t h e s t r e s s t e n s o r , w e a r e

a d d i n g n o t h i n g t o o u r k n o w l e d g e of t h e p h y s i c s of t h e s y s t e m . W e a r e

s i m p l y r e w r i t i n g t h e s t a t e m e n t s a b o u t t h e f o r c e s a c t i n g o n a b o d y in a w a y

w h i c h sha l l t u r n o u t t o b e v e r y c o n v e n i e n t f o r u s .

C. EQUATION OF MOTION FOR SOLIDS

H a v i n g n o w d e f i n e d t h e s t r e s s a n d s t r a i n t e n s o r s , w e h a v e a t o u r

d i s p o s a l c o m p l e t e l y g e n e r a l w a y s of d e s c r i b i n g b o t h t h e f o r c e s w h i c h a r e

a p p l i e d t o a so l id a n d t h e w a y in w h i c h t h e so l i d d e f o r m s in r e s p o n s e t o

t h e s e f o r c e s . T h e p r o b l e m n o w is t o r e l a t e t h e s e t w o d e s c r i p t i o n s — i . e . t o

f ind t h e d e f o r m a t i o n in a g i v e n so l id c o r r e s p o n d i n g t o a g i v e n f o r c e .

T h e r e is n o a priori r e l a t i o n b e t w e e n t h e s t r e s s a n d t h e s t r a i n . S u c h a

r e l a t i o n d e p e n d s e n t i r e l y o n t h e m a t e r i a l b e i n g s t r e s s e d . W e c a n e a s i l y

i m a g i n e m a n y d i f f e ren t k i n d s of r e s p o n s e t o a n a p p l i e d f o r c e . F o r

e x a m p l e , w e k n o w t h a t in s o m e c a s e s ( s e e S e c t i o n 2 .B) w e c a n t a l k a b o u t

e l a s t i c s o l i d s , w h e r e t h e d e f o r m a t i o n is d i r e c t l y p r o p o r t i o n a l t o t h e s t r e s s .

W e m i g h t p i c t u r e t h e m i c r o s c o p i c s t r u c t u r e of t h e m a t e r i a l a s in F i g . 12.5,

w h e r e t h e a t o m s a r e h e l d t o g e t h e r b y s p r i n g s . W h e n a f o r c e is a p p l i e d , t h e

m a t e r i a l wi l l d e f o r m u n t i l t h e s p r i n g s h a v e c o m p r e s s e d e n o u g h t o

c o u n t e r a c t t h e a p p l i e d f o r c e F .

O n t h e o t h e r h a n d , w e c o u l d i m a g i n e a m a t e r i a l in w h i c h t h e a t o m s w e r e

h e l d t o g e t h e r b y r ig id r o d s , s o t h a t if a f o r c e is a p p l i e d , t h e so l i d d o e s n o t

d e f o r m a t all u n t i l t h e a p p l i e d f o r c e r e a c h e s t h e p o i n t w h e r e it c a n b r e a k

t h e r o d s ( s e e F i g . 12.6). W e w o u l d t h e n s e e t h e m a t e r i a l f r a c t u r e

i n s t a n t a n e o u s l y .

A t h i r d p o s s i b l e k i n d of r e s p o n s e t o a f o r c e w o u l d b e o n e in w h i c h t h e

Fig. 12.5. An harmonic solid.

Equation of Motion for Solids 217

Fig. 12.6. A rigid solid.

f o r c e is r e l a t e d t o t h e rate of d e f o r m a t i o n ( i .e . t o t h e t i m e d e r i v a t i v e s of

t h e s t r a i n t e n s o r ) . W e s a w f o r c e s of t h i s t y p e in C h a p t e r 7 w h e n w e

d i s c u s s e d v i s c o s i t y , w h i c h g a v e r i s e t o a f o r c e w h i c h d e p e n d e d o n t h e

v e l o c i t y of t h e fluid. W e m i g h t p i c t u r e s u c h a so l id a s o n e in w h i c h t h e

b o n d s b e t w e e n t h e a t o m s a r e v e r y w e a k , s o t h a t a f o r c e w h i c h is a p p l i e d

c o n t i n u o u s l y r e s u l t s in a c o n t i n u o u s d e f o r m a t i o n . S u c h a s y s t e m w o u l d

b e c a l l e d a Newtonian solid, a n d is d i s c u s s e d in P r o b l e m 12.4.

T h e p o i n t of t h i s e x e r c i s e i s t o i l l u s t r a t e t h e r e m a r k m a d e a t t h e

b e g i n n i n g of t h i s s e c t i o n — t h e r e i s n o w a y w e c a n te l l f r o m first p r i n c i p l e s

h o w a so l id wi l l r e s p o n d t o a n a p p l i e d f o r c e . T h i s is a n e x a c t l og ica l

a n a l o g y t o t h e l e s s o n w e l e a r n e d in S e c t i o n l . D , w h e n w e f o u n d t h a t in

d e a l i n g w i t h a fluid s y s t e m , w e h a d t o h a v e a n e q u a t i o n of s t a t e , w h i c h

t o l d u s w h a t s o r t of fluid w e h a d in t h e s y s t e m . I n t h e c a s e of a so l id ,

s p e c i f y i n g t h e t y p e of m a t e r i a l in t h e s y s t e m c o r r e s p o n d s t o g i v i n g a

r e l a t i o n b e t w e e n t h e s t r e s s a n d t h e s t r a i n .

T h r o u g h o u t t h e r e s t of t h e t e x t , w e sha l l b e c o n c e r n e d p r i m a r i l y w i t h

e l a s t i c s o l i d s . I n s u c h s o l i d s , w e e x p e c t t h a t t h e s t r e s s wi l l b e p r o p o r t i o n a l

t o t h e s t r a i n . S i n c e crik is a s y m m e t r i c t e n s o r , a n d s i n c e it m u s t b e

p r o p o r t i o n a l t o t h e s t r a i n t e n s o r , t h e m o s t g e n e r a l f o r m of t h e s t r e s s -

s t r a i n r e l a t i o n s h i p m u s t b e

orik = kUn8ik +2fJLUik, (12 .C.1)

w h e r e A a n d /JL a r e c a l l e d t h e L a m e coe f f i c i en t s a n d differ f r o m o n e e l a s t i c

so l id t o a n o t h e r . E q u a t i o n (12 .C .1) is s i m p l y H o o k e ' s l a w in t e n s o r f o r m ,

a s wil l b e ve r i f i ed e x p l i c i t l y l a t e r .

T h u s , in o r d e r t o s u p p l y t h e " e q u a t i o n of s t a t e " f o r a so l id , w e m u s t

1. g i v e a s t r e s s - s t r a i n r e l a t i o n s h i p , w h i c h t e l l s u s w h a t g e n e r a l c l a s s of

s o l i d s w e h a v e [ E q . (12 .C.1) de f ines a g e n e r a l e l a s t i c s o l i d ] ,

2 . s p e c i f y t h e coe f f i c i en t s w h i c h s a y w h i c h p a r t i c u l a r so l i d in t h a t


(12.C.2) 0"n = T,

an = 0 o t h e r w i s e .

T h e n t h e t h r e e d i a g o n a l e q u a t i o n s f r o m E q . (12 .C .1) a r e

(TN = T = SnXUu + 2/XUN,

0 - 2 2 = 0 = 822Ûu + 2jLtw22, (12 .C.3)

cr 33 = 0 = 6 3 3 AMH + 2 p , M 3 3 .

A d d i n g t h e s e t h r e e e q u a t i o n s , a n d r e c a l l i n g t h a t

Uu = M 1 1 4- W 2 2 + W33,

w e h a v e

12.C.4)

w h i c h , if w e p l u g b a c k i n t o E q s . (12 .C.3) y i e l d s

a n d

12.C.5)

(12 .C.6)

f o r t h e t h r e e d i a g o n a l e l e m e n t s of t h e s t r a i n t e n s o r .

N o w M 1 1 is t h e d e f o r m a t i o n of t h e c y l i n d e r a l o n g t h e d i r e c t i o n of t h e

t e n s i o n . B y def in i t ion , t h i s is r e l a t e d t o t h e t e n s i o n b y

T = E M H ,

w h e r e E is Y o u n g ' s m o d u l u s . S i m i l a r l y , P o i s s o n ' s r a t i o i s

w h i c h , u p o n s u b s t i t u t i o n , y i e l d s

g e n e r a l c l a s s w e h a v e (e .g . , g i v i n g A a n d fx c o m p l e t e l y s p e c i f i e s t h e

e l a s t i c so l id ) .

I n o r d e r t o m a k e t h e t e n s o r f r o m H o o k e ' s l a w a l i t t le m o r e f a m i l i a r , l e t

u s l o o k a t s o m e e x a m p l e s . F i r s t , c o n s i d e r a c y l i n d e r u n d e r a t e n s i o n T, s o

t h a t t h e s t r e s s t e n s o r i s

Equation of Motion for Solids 219

a n d

(12 .C.8)

(12 .C .12)

w h i c h is u s u a l l y c a l l e d t h e bulk modulus of t h e m a t e r i a l .

I n w h a t f o l l o w s , t h e n , w e sha l l f e e l f r e e t o u s e a n y of t h e s e t h r e e s e t s of

e l a s t i c c o n s t a n t s t o de f ine o u r so l id , d e p e n d i n g o n w h i c h is m o s t c o n v e -

n i e n t in a p a r t i c u l a r p r o b l e m .

W e s e e , t h e n , t h a t t h e L a m e coef f i c ien t s a r e s i m p l y r e l a t e d t o E a n d cr,

t h e n u m b e r s w h i c h w e u s e d in C h a p t e r 11 t o de f ine a n e l a s t i c m a t e r i a l .

T h e r e a r e a n o t h e r s e t of c o n s t a n t s w h i c h a r e o f t e n u s e d a s a l t e r n a t i v e s

t o A a n d p, o r E a n d cr i n d e s c r i b i n g e l a s t i c s o l i d s . T o u n d e r s t a n d t h e s e ,

c o n s i d e r t w o e x a m p l e s : F i r s t , c o n s i d e r a p u r e s h e a r i n g f o r c e , s o t h a t

cr 12 = cr 2i = T,

an = 0 o t h e r w i s e .

T h e n w e h a v e

CTi2 = T = 2 j L L M i 2 ,

s o t h a t , r e c a l l i n g E q . (12 .A.6)

w h i c h c a n b e u s e d t o de f ine t h e shear modulus ( t h e p r o p o r t i o n a l i t y

c o n s t a n t b e t w e e n t h e a p p l i e d s h e a r a n d t h e a n g l e of d e f o r m a t i o n ) a s

(12 .C .10)

w h i c h i s , of c o u r s e , i d e n t i c a l w i t h t h e L a m e coef f ic ien t p,.

N e x t c o n s i d e r a so l i d u n d e r h y d r o s t a t i c c o m p r e s s i o n , s o t h a t

ani = - P 6 V (12 .C .11)

If w e t h e n f o l l o w t h e e x a c t s t e p s of E q . (12 .C.3) t o E q . (12 .C .4 ) , w e f ind

w h e r e w e h a v e u s e d E q . (12 .A .5 ) , a g e n e r a l p r o p e r t y of t h e s t r a i n t e n s o r .

W e c a n t h e n s e e t h a t t h e r a t i o of v o l u m e c h a n g e t o a p p l i e d p r e s s u r e is j u s t


D. BODY WAVES IN ELASTIC MEDIA

I n t h e c a s e of f lu ids , w e s a w t h a t a g r e a t d e a l of i n t e r e s t i n g i n f o r m a t i o n

c o u l d b e d e r i v e d b y l o o k i n g f o r w a v e - t y p e s o l u t i o n s of t h e e q u a t i o n s of

m o t i o n . I t is i n t e r e s t i n g t o a s k w h e t h e r t h e s a m e is t r u e f o r s o l i d s . W e

H a v i n g n o w w r i t t e n d o w n H o o k e ' s l a w a n d s e e n w h a t t h e L a m e

coef f i c i en t s r e p r e s e n t in t e r m s of d e f o r m a t i o n s of a so l id , w e c a n t u r n t o

t h e p r o b l e m of w r i t i n g d o w n t h e e q u a t i o n of m o t i o n f o r a n in f in i t e s ima l

e l e m e n t in t h e so l id . T h i s is a n a l o g o u s t o d e r i v i n g t h e E u l e r e q u a t i o n ,

s i n c e b o t h i n v o l v e N e w t o n ' s s e c o n d l a w . If ft i s t h e f o r c e in t h e

/ t h - d i r e c t i o n o n a n in f in i t e s ima l v o l u m e e l e m e n t , t h e n N e w t o n ' s s e c o n d

l a w f o r t h a t v o l u m e e l e m e n t is

(12 .C.13)

B u t w e k n o w t h a t [ s e e E q . (12 .B.1) ]

s o t h a t

(12 .C.14)

If w e u s e t h e de f in i t ion of t h e s t r a i n t e n s o r [ E q . (12 .A.4) ] a n d r e a r r a n g e

t e r m s , t h i s b e c o m e s

(12 .C.15)

(12 .C.16)

o r , in v e c t o r f o r m

T h i s is t h e b a s i c e q u a t i o n w h i c h d e s c r i b e s t h e t i m e - d e p e n d e n t r e s p o n s e of

a n e l e m e n t in a n e l a s t i c so l id t o a n a p p l i e d f o r c e . T h e r e m a i n d e r of t h i s

c h a p t e r wi l l b e d e v o t e d t o e x a m i n i n g t h e c o n s e q u e n c e s of t h e e q u a t i o n .

B e f o r e m o v i n g o n , h o w e v e r , w e sha l l , f o r t h e s a k e of c o m p l e t e n e s s , w r i t e

d o w n t w o f o r m s of t h e e q u a t i o n w h i c h sha l l b e u s e f u l l a t e r . If w e t a k e t h e

g r a d i e n t of E q . (12 .C .16 ) , w e find t h a t

(12 .C.17)

w h i l e if w e t a k e t h e c u r l of t h e e q u a t i o n , w e h a v e

(12 .C.18)

Body Waves in Elastic Media 221

sha l l s e e t h a t t h e r e a r e s e v e r a l d i f f e r en t t y p e s of w a v e s w h i c h c a n

p r o p a g a t e t h r o u g h a so l id , a n d w e sha l l s e e h o w t h i s i n f o r m a t i o n h a s

e n a b l e d u s t o d i s c o v e r t h e c o m p o s i t i o n of t h e i n t e r i o r of t h e e a r t h t h r o u g h

t h e d e v e l o p m e n t of t h e s c i e n c e of s e i s m o l o g y

F r o m E q s . (12 .C .17) a n d (12 .C .18 ) , it is c l e a r t h a t w a v e s wi l l e x i s t .

R a t h e r t h a n p r o c e e d f o r m a l l y f r o m t h e e q u a t i o n s of m o t i o n , h o w e v e r , l e t

u s l o o k a t e x a m p l e s of t w o d i f f e r en t t y p e s of w a v e s a n d s i m p l y v e r i f y t h a t

t h e y s a t i s fy t h e e q u a t i o n s of m o t i o n f o r a so l id .

W e sha l l first l o o k f o r s o l u t i o n s of t h e e q u a t i o n of t h e f o l l o w i n g t y p e : A

w a v e d i s t u r b a n c e of s o m e s o r t t r a v e l s in t h e x - d i r e c t i o n , a n d t h e

d i s p l a c e m e n t of t h e so l i d i s in t h e x - d i r e c t i o n a s w e l l . T h i s c o r r e s p o n d s t o

(12 .D.1 )

w h i c h r e d u c e s t o

(12 .D.2 )

W e h a v e u s e d t h e s u b s c r i p t / b e c a u s e t h i s is a l o n g i t u d i n a l w a v e , s i n c e

t h e d i s p l a c e m e n t is in t h e s a m e d i r e c t i o n a s t h e v e l o c i t y of t h e w a v e . I t

c a n b e i n t e r p r e t e d a s a c o m p r e s s i o n a l w a v e a s w e l l . T o s e e t h i s , l e t u s p l o t

ux a s a f u n c t i o n of x f o r fixed t. I n F i g . 12.7, ux p o s i t i v e c o r r e s p o n d s t o

Fig. 12.7. Velocities of particles for acoustic waves.

(12 .D.3)

T h i s , of c o u r s e , i s t h e e q u a t i o n of a w a v e t r a v e l i n g a l o n g t h e x - a x i s w i t h

v e l o c i t y ch w h e r e

a n d t h e e q u a t i o n of m o t i o n b e c o m e s

ux = ux(x- ct),

uy = uz = 0 ,

s o t h a t


t h e p a r t i c l e s m o v i n g t o t h e r i g h t ( w h i c h w e t a k e t o b e t h e d i r e c t i o n of t h e

w a v e ) , a n d ux n e g a t i v e c o r r e s p o n d s t o p a r t i c l e s m o v i n g t o t h e lef t . I n t h e

l o w e r p a r t of t h e figure w e s h o w t h e a c t u a l d i r e c t i o n of m o t i o n of

e l e m e n t s . W e s e e t h a t e l e m e n t s of t h e so l id t e n d t o m o v e t o w a r d e v e r y

o t h e r p o i n t w h e r e ux is z e r o , a n d a w a y f r o m t h e p o i n t s w h e r e ux is a n

e x t r e m u m . T h u s , t h e d e n s i t y in t h e f o r m e r r e g i o n s wi l l b e g r e a t e r t h a n t h e

d e n s i t y a r o u n d t h e l a t t e r . T h i s wi l l b e o b s e r v e d a s a p a t t e r n of d e n s i t y

v a r i a t i o n s w h i c h , a s t i m e p r o g r e s s e s , wi l l m o v e t o t h e r i gh t . T h i s is j u s t

w h a t a s o u n d w a v e i s , a n d h e n c e t h i s t y p e of w a v e is s o m e t i m e s c a l l e d a n

a c o u s t i c w a v e .

A m o r e u s u a l t y p e of w a v e is t h e t r a n s v e r s e w a v e , in w h i c h t h e

d i s p l a c e m e n t of t h e m a t e r i a l i s p e r p e n d i c u l a r t o t h e d i r e c t i o n of m o t i o n of

t h e w a v e . A w a v e o n a s t r i n g w o u l d b e a n e x a m p l e of s u c h a p h e n o m e n o n .

T h i s t y p e of w a v e , s h o u l d it e x i s t , w o u l d c o r r e s p o n d t o

uy = uy(x — ct), (12 .D.4)

uz = ux = 0 ,

w h i c h g i v e s

V • u = 0,

s o t h a t t h e e q u a t i o n of m o t i o n in t h e y d i r e c t i o n is j u s t

(12 .D.5)

w h i c h is a g a i n a w a v e e q u a t i o n f o r a t r a n s v e r s e w a v e ( t h e t e r m s

l o n g i t u d i n a l a n d t r a n s v e r s e r e f e r t o t h e d i r e c t i o n of t h e d i s p l a c e m e n t

r e l a t i v e t o t h e d i r e c t i o n of m o t i o n of t h e w a v e ) . T h e v e l o c i t y of t h e w a v e

is

(12 .D.6)

T h i s t y p e of t r a n s v e r s e w a v e in a so l id i s c a l l e d a shear wave, s i n c e t c a n

b e t h o u g h t of a s a s m a l l - s c a l e s h e a r i n g in t h e b o d y of t h e so l id .

I t is e a s y t o s e e t h a t e a c h of t h e s e w a v e s c o r r e s p o n d s t o a d i f f e r en t

f o r m of t h e e q u a t i o n of m o t i o n . T h e c o m p r e s s i o n a l w a v e c o r r e s p o n d s t o

v o l u m e c h a n g e s in t h e so l id , a n d h e n c e t o E q . (12 .C .17 ) , w h i l e t h e s h e a r

w a v e c o r r e s p o n d s t o E q . (12 .C .18) . O n e i m p o r t a n t c o n s e q u e n c e of t h e

f a c t t h a t t h e r e a r e t w o t y p e s of w a v e s w h i c h c a n b e e x c i t e d in a so l id ,

e a c h t r a v e l i n g w i t h a d i f f e ren t s p e e d , i s in s e i s m o l o g y . T o s e e t h i s , w e

n o t e t h a t f r o m E q s . (12 .D.3) a n d (12 .D.6) t h a t

Ci > ct,

Surface Waves in Solids 223

s o t h a t t h e c o m p r e s s i o n a l w a v e t r a v e l s f a s t e r t h a n t h e s h e a r w a v e . If w e

i m a g i n e a d i s t u r b a n c e s o m e w h e r e d e e p in t h e e a r t h , w i t h b o t h c o m p r e s -

s i o n a l a n d s h e a r w a v e s c o m i n g o u t , t h e c o m p r e s s i o n a l w a v e wi l l r e a c h t h e

s u r f a c e first . H e n c e , s e i s m o l o g i s t s r e f e r t o it a s t h e P w a v e , o r p r i n c i p l e

w a v e . T h e s h e a r w a v e a r r i v e s a t s o m e l a t e r t i m e , a n d h e n c e is c a l l e d t h e

S, o r s e c o n d a r y w a v e . T h e r e f o r e , t h e r e w o u l d b e t w o s h o c k s a r r i v i n g a t

t h e s u r f a c e a f t e r s u c h a d i s t u r b a n c e , a n d t h e t i m e d i f f e r e n c e b e t w e e n t h e i r

a r r i v a l s w o u l d d e p e n d o n t h e r e l a t i v e v a l u e s of ct a n d ct. T h e s e , in t u r n ,

d e p e n d o n t h e d e n s i t y a n d t h e k i n d of m a t e r i a l of w h i c h t h e e a r t h is

c o m p o s e d .

B y m e a s u r i n g t h e t i m e lag b e t w e e n t h e a r r i v a l of d i f f e r en t w a v e s f r o m a

d i s t u r b a n c e , o n e c a n o b t a i n i n f o r m a t i o n a b o u t t h e s t r u c t u r e of t h e e a r t h .

T h i s i s t h e a i m of t h e s c i e n c e of s e i s m o l o g y , w h i c h w e sha l l d i s c u s s l a t e r .

A s a n e x a m p l e of t h i s e f fec t , l e t u s c o n s i d e r a n e a r t h q u a k e a t T o k y o ,

a n d a s k w h a t t h e t i m e d i f f e r e n c e is b e t w e e n t h e P a n d S w a v e s a s

o b s e r v e d a t S a n F r a n c i s c o . L e t u s a s s u m e t h e e a r t h h a s a u n i f o r m

d e n s i t y , a n d t h a t t h e coe f f i c i en t s a r e e v e r y w h e r e c o n s t a n t a n d a r e e q u a l t o

t h o s e f o r f u s e d s i l i c a t e s . T h e s e w a v e s wi l l t r a v e l d i r e c t l y a c r o s s a c h o r d

of t h e e a r t h , w h i c h is 9.5 x 10 6 m l o n g . T h e r e f o r e , t h e t i m e d i f f e r e n c e wi l l

b e

A s w e sha l l s e e l a t e r , t h e f a c t t h a t t h e c o m p o s i t i o n of t h e e a r t h v a r i e s a s

a f u n c t i o n of d e p t h m a k e s t h e a c t u a l c a l c u l a t i o n of t h e p a t h s of s e i s m i c

w a v e s a n d of t h e p r o p e r t i e s of t h e e a r t h ' s i n t e r i o r q u i t e a b i t m o r e

difficult .

E. SURFACE WAVES IN SOLIDS

I n C h a p t e r 5 , w e s a w t h a t it is p o s s i b l e t o h a v e w a v e s in a fluid w h i c h

e x i s t o n l y in t h e s u r f a c e , a n d w h i c h d i e o u t r a p i d l y a s a f u n c t i o n of d e p t h .

I n t h i s s e c t i o n , w e sha l l s e e t h a t s u c h w a v e s c a n e x i s t in s o l i d s a s w e l l .

U n l i k e t h e P a n d S w a v e s w h i c h w e c o n s i d e r e d in t h e l a s t s e c t i o n , t h e

e x i s t e n c e of s u r f a c e w a v e s d e p e n d s o n a p p l y i n g b o t h t h e e q u a t i o n of

m o t i o n a n d t h e b o u n d a r y c o n d i t i o n s .

C o n s i d e r a s emi - in f in i t e so l id ( a s s h o w n in F i g . 12.8) in w h i c h a w a v e

p r o p a g a t e s w i t h v e l o c i t y

(12 .E .1 )


Fig. 12.8. Geometry for surface waves.

L e t u s a l s o a s s u m e t h a t t h e m a g n i t u d e of t h e d i s p l a c e m e n t of a so l id

e l e m e n t i n a n y d i r e c t i o n i s a f u n c t i o n of t h e d e p t h . T h i s m e a n s t h a t w e a r e

a s s u m i n g t h a t

^ = / , ( y ) e l ( k * - " ° ,

uy =fy(y)eKkx-t\ (12 .E .2 )

U z = / , ( y y ( t o - " ° .

If w e p u t t h e s e a s s u m e d f o r m s of t h e s o l u t i o n b a c k i n t o t h e e q u a t i o n of

m o t i o n , w e wil l h a v e

(12 .E .4 )

w h i c h h a s a s i t s s o l u t i o n f u n c t i o n s of t h e f o r m / = e ± y y . If w e t h r o w o u t

s o l u t i o n s w h i c h b e c o m e inf ini te a s y -> - «>, a n d n o t e t h a t w e will p r o v e

l a t e r t h a t y > 0, w e h a v e

Hi =Bieyiyeiikx~bit\ (12 .E .5 )

T h i s s o l u t i o n f o r ux, uy, a n d uz e x h i b i t s all t h e p r o p e r t i e s w e w i s h t o

a s s o c i a t e w i t h a s u r f a c e w a v e — e a c h c o m p o n e n t e x h i b i t s w a v e b e h a v i o r ,

b u t a s w e g o i n t o t h e i n t e r i o r of t h e m a t e r i a l , t h e d i s t u r b a n c e d i e s o u t

e x p o n e n t i a l l y ( b u t n o t e t h a t t h e t h r e e c o m p o n e n t s d o n o t d i e o u t a t t h e

s a m e r a t e ) . H o w e v e r , a s w a s i m p l i e d in t h e i n t r o d u c t i o n t o t h i s s e c t i o n , it

(12 .E .3 )

w h e r e c, = d if i = x, a n d ct if i = y, z.

I t m u s t b e e m p h a s i z e d t h a t a l t h o u g h ct [ g iven b y E q . (12 .D.3) ] a n d ct

[g iven b y E q . (12 .D.6) ] a r e v e l o c i t i e s of b o d y w a v e s , t h e y a r e n o t t h e

v e l o c i t y of a n y w a v e in t h e s u r f a c e . T h e y a r e s i m p l y d i f f e ren t c o m b i n a -

t i o n s of t h e p a r a m e t e r s p , A, a n d /x.

If w e s u b s t i t u t e t h e a s s u m e d f o r m s of ut i n t o t h e s e e q u a t i o n s , w e find

Surface Waves in Solids 225

is n e c e s s a r y t o s a t i s f y c o n d i t i o n s a t t h e b o u n d a r y a s w e l l a s t h e e q u a t i o n

of m o t i o n if w e w i s h t o s h o w t h a t s u c h w a v e s e x i s t .

T h e b o u n d a r y c o n d i t i o n , of c o u r s e , is s i m p l y t h e r e q u i r e m e n t t h a t t h e

p l a n e y = 0 b e a f r e e s u r f a c e , w h i c h m e a n s t h a t

cryx(y =0) = ayy(y =0) = cryz(y = 0) = 0 . (12 .E .6 )

L e t u s l o o k a t t h e c o n d i t i o n o n o- y z f irst . F r o m E q . (12 .C .1 ) , w e h a v e

B u t f r o m E q . ( 1 2 . E . 2 ) ,

(12 .E .7 )

b u t f r o m s y m m e t r y , duy/dz = 0 , s o w e h a v e

s o t h a t w e m u s t h a v e

Bz = 0 . (12 .E .8 )

I n o t h e r w o r d s , in s u r f a c e w a v e s of t h e t y p e w e a r e s t u d y i n g t h e r e c a n

b e n o d i s p l a c e m e n t in t h e z - d i r e c t i o n . I n t h e l a n g u a g e of s e i s m o l o g y ,

d i s p l a c e m e n t in t h e z - d i r e c t i o n is c a l l e d SH ( fo r s h e a r h o r i z o n t a l ) , s i n c e

t h e d i s p l a c e m e n t is h o r i z o n t a l t o t h e s u r f a c e in w h i c h t h e w a v e is

p r o p a g a t i n g . O n t h e o t h e r h a n d , SV ( s h e a r v e r t i c a l w a v e ) is o n e in w h i c h

uy is n o n z e r o .

F r o m t h e c o n d i t i o n o n axy, w e h a v e [ aga in u s i n g E q . (12 .C.1) ]

(Txy(y = 0) = 0 = ii[Bxyx + ikBy]eiikx-Mt\


(12 .E .9 )

If w e p u t t h e s e b a c k i n t o E q . ( 1 2 . E . 2 ) , w e f ind t h a t a t y = 0 , w e h a v e

(12 .E .10 )

w h i c h m e a n s t h a t t h e p a r t i c l e m o t i o n a s s o c i a t e d w i t h t h i s w a v e is in f a c t

r e t r o g r a d e e l l i p s e ( s e e F i g . 12.9).


wave direction

y = 0

Fig. 12.9. The motion of a particle in a surface wave.

T h i s t y p e of w a v e , w h i c h is c o n f i n e d t o t h e s u r f a c e a n d h a s n o SH

c o m p o n e n t , is c a l l e d t h e Rayleigh wave, a f t e r L o r d R a y l e i g h , w h o first

d i s c u s s e d it .

W h a t is t h e v e l o c i t y of a R a y l e i g h w a v e ? I n o r d e r t o a n s w e r t h i s , w e

m u s t a p p l y t h e final b o u n d a r y c o n d i t i o n . F r o m E q s . (12 .C .1 ) , ( 12 .E .2 ) , a n d

(12 .E .6 ) , w e h a v e

<r y y (y = 0 ) = (ifcRA + ( A +2fjL)yyBy)eiikx-fOt) = 0. ( 12 .E .11)

U s i n g E q . (12 .E .9 ) a n d r e a r r a n g i n g , w e h a v e

w h i c h c a n b e w r i t t e n , u s i n g E q s . ( 1 2 . E . 4 ) , ( 1 2 . E . 1 ) , ( 12 .D .3 ) , a n d (12 .D.6)

(12 .E .12)

T h i s e q u a t i o n d e t e r m i n e s c i n t e r m s of ct a n d ct (o r , c o n v e r s e l y , in

t e r m s of /x a n d A) .

N o w w e c o u l d , in p r i n c i p l e , g o a h e a d a n d s o l v e t h i s e q u a t i o n , i n s e r t i n g

f o r ct a n d cx s o m e q u a n t i t i e s a p p r o p r i a t e f o r t h e e a r t h ' s s u r f a c e .

H o w e v e r , o u r j o b is m a d e c o n s i d e r a b l y s i m p l e r if w e m a k e u s e of a n

e x p e r i m e n t a l o b s e r v a t i o n k n o w n a s Poisson's relation, w h i c h s t a t e s t h a t

f o r t h e e a r t h , it is a p p r o x i m a t e l y t r u e t h a t t h e L a m e coef f i c i en t s a r e a b o u t

e q u a l . T h i s , in t u r n , i m p l i e s t h a t

c 2 ^ 3 c 2 . (12 .E .13)

I n t h i s c a s e , t h e e q u a t i o n f o r c c a n b e s o l v e d s i m p l y t o g i v e

(12 .E .14)

W h i c h s ign s h o u l d w e p i c k ? T o a n s w e r t h i s q u e s t i o n , w e h a v e t o r e f e r

t o E q . ( 12 .E .5 ) , in w h i c h it w a s s h o w n t h a t a s u r f a c e w a v e c o u l d e x i s t in a

so l id . I n o r d e r f o r t h i s t o b e t r u e , it w a s n e c e s s a r y t h a t c2 < ct

2 < c2. T h u s ,

Waves in Surface Layers 227

o n l y t h e c h o i c e of t h e m i n u s s ign in E q . (12 .E .4 ) wi l l r e s u l t in a s u r f a c e

w a v e , a n d t h e o t h e r r o o t m u s t b e d i s c a r d e d a s e x t r a n e o u s . W e a r e lef t

w i t h t h e r e s u l t

c = 0 . 9 2 c „ (12 .E .15 )

s o t h a t t h e R a y l e i g h w a v e t r a v e l s a t a s l igh t ly s l o w e r v e l o c i t y t h a n t h e

s h e a r b o d y w a v e .

I n s e i s m o l o g y , t h e n , w e e x p e c t t h a t in a d d i t i o n t o t h e t w o b o d y w a v e s

d i s c u s s e d in S e c t i o n 12 .D , t h e r e wi l l b e a w a v e t r a v e l i n g a l o n g t h e s u r f a c e

of t h e e a r t h a s w e l l . T h i s m e a n s t h a t in a d d i t i o n t o t h e t w o s i g n a l s

d i s c u s s e d in t h e e x a m p l e of t h e T o k y o e a r t h q u a k e , a t h i r d s igna l wi l l b e

r e c e i v e d . T h i s s i g n a l wi l l a r r i v e a f t e r t h e S a n d P s i gna l s ( b e c a u s e it h a s a

l o w e r v e l o c i t y a n d f a r t h e r t o t r a v e l ) , a n d wi l l b e p u r e SV in n a t u r e .

S u c h w a v e s a r e , of c o u r s e , o b s e r v e d in n a t u r e . I n a d d i t i o n , it is a l s o

t r u e t h a t y e t a n o t h e r k i n d of s u r f a c e w a v e is o b s e r v e d , w h i c h is a p u r e SH

w a v e . A l t h o u g h s u c h a w a v e w o u l d n o t b e p o s s i b l e in a u n i f o r m

h o m o g e n o u s e a r t h , t h e y a r e p o s s i b l e in m o r e r e a l i s t i c m o d e l s of t h e e a r t h ,

a n d it i s t o t h i s p r o b l e m w e n o w t u r n o u r a t t e n t i o n .

F. WAVES IN SURFACE LAYERS

T h e r e a s o n t h a t w e f a i l ed t o p r e d i c t t h e e x i s t e n c e of SH s u r f a c e w a v e s

in t h e p r e v i o u s s e c t i o n w a s t h a t w e h a d t a k e n t o o s i m p l e a m o d e l f o r t h e

e a r t h . I n a c t u a l f a c t , t h e e a r t h is n o t a h o m o g e n e o u s m e d i u m , b u t h a s a

r a t h e r c o m p l e x s t r u c t u r e . T h i s wi l l b e d i s c u s s e d m o r e ful ly in t h e n e x t

c h a p t e r , b u t f o r o u r p u r p o s e s , w e n e e d o n l y o b s e r v e t h a t a b e t t e r m o d e l

f o r t h e s u r f a c e of t h e e a r t h w o u l d b e o n e in w h i c h t h e r e w a s a s u r f a c e

l a y e r of a d i f f e r en t m a t e r i a l f r o m t h e m a i n b o d y . T h e e x i s t e n c e of w a v e s

in s u c h a l a y e r w a s first n o t e d b y A . E . H . L o v e , a n d t h e y a r e u s u a l l y

c a l l e d Love waves.

S u p p o s e w e t a k e a s o u r m o d e l of t h e e a r t h ' s s u r f a c e t h e s i t u a t i o n

s h o w n in F i g . 12.10, w h e r e t h e r e i s a s emi - in f in i t e s o l i d e x t e n d i n g f r o m

y = 0 d o w n w a r d , w i t h d e n s i t y p ' a n d L a m e coef f ic ien t p , ' , a n d a l a y e r of

so l id of d e n s i t y p a n d coef f ic ien t p, f r o m y = 0 t o y = T.

If w e a g a i n c o n s i d e r a w a v e m o v i n g in t h e jc -d i rec t ion , t h e n a n SH w a v e

w o u l d c o r r e s p o n d t o a m o t i o n of t h e e l e m e n t s of t h e s o l i d s in t h e

z - d i r e c t i o n . T h u s , w e sha l l h a v e t o l o o k f o r s o l u t i o n s t o t h e e q u a t i o n s of

m o t i o n a n d t h e b o u n d a r y c o n d i t i o n s w h i c h a r e of t h e f o r m

uz = tty)eiikx-*\ (12 .F .1 )

Uy = UX = 0 ,


y = T

y = 0

Fig. 12.10. The geometry for Love waves.

in e a c h of t h e t w o m e d i a . F o l l o w i n g t h e s t e p s t h a t l ed t o E q . ( 1 2 . E . 4 ) , w e

find t h a t in e a c h m e d i u m , w e h a v e a n e q u a t i o n f o r / ( y ) of t h e f o r m

(12 .F .2 )

w h e r e ct is t h e s h e a r w a v e v e l o c i t y a p p r o p r i a t e t o e a c h m e d i u m . B y

a s s u m p t i o n , t h e r e is n o l o n g i t u d i n a l w a v e in t h i s s y s t e m .

A s b e f o r e , w e n e e d o n l y c o n s i d e r y' > 0 in t h e l o w e r m e d i u m , s i n c e t h e

m o t i o n m u s t s t a y finite a s y -> — oo. I n t h e s u r f a c e l a y e r , h o w e v e r , t h e r e is

n o r e s t r i c t i o n o n y, s o t h a t w e h a v e

/ ( y ) = A s in yy + B c o s yy

f o r t h e f u n c t i o n / ( y ) in t h e s u r f a c e l a y e r , a n d

f'(y) = Eey'y

(12 .F .3 )

(12 .F .4 )

f o r t h e f u n c t i o n f'(y) in t h e l o w e r m e d i u m , a n d F , A , a n d B a r e u n k n o w n

c o n s t a n t s . T h e q u a n t i t i e s y a n d y' differ in t h e v a l u e of ct w h i c h a p p e a r s

in E q . (12 .F .2 ) .

If t h i s d i s t u r b a n c e is t o r e p r e s e n t a p h y s i c a l l y r e a l i z a b l e s i t u a t i o n , t h r e e

b o u n d a r y c o n d i t i o n s m u s t b e sa t i s f ied :

(i) t h e m e d i u m m u s t b e c o n t i n u o u s a t y = 0, w h i c h m e a n s

i i , (y = 0 ) = i i i (y = 0 ) , (12 .F .5 )

(ii) t h e s t r e s s e s m u s t v a n i s h a t t h e f r e e s u r f a c e y = T, s o t h a t

o- y y (y = T) = ayx(y = T ) = ayz(y = T) = 0 , (12F .6 )

Summary 229

w h i c h is a n e q u a t i o n w h i c h r e l a t e s c = co/k t o ct, a n d h e n c e d e t e r m i n e s

t h e v e l o c i t y of t h e L o v e w a v e , j u s t a s E q . (12 .F .12 ) d e t e r m i n e d t h e

v e l o c i t y of t h e R a y l e i g h w a v e . I t s h o u l d b e n o t e d , h o w e v e r , t h a t u n l i k e

t h e R a y l e i g h w a v e , t h e L o v e w a v e will h a v e a v e l o c i t y d e p e n d e n t o n t h e

w a v e l e n g t h . T h i s t y p e of p h e n o m e n o n w a s o b s e r v e d in s u r f a c e w a v e s in

f luids in C h a p t e r 5.

SUMMARY

T h e s t r a i n a n d s t r e s s t e n s o r s p r o v i d e a g e n e r a l d e s c r i p t i o n of t h e

d e f o r m a t i o n of a so l i d a n d t h e a p p l i e d f o r c e s . T h e y a r e r e l a t e d , f o r a n

e l a s t i c so l id , b y H o o k e ' s l a w , a l t h o u g h o t h e r k i n d s of r e l a t i o n s a r e

p o s s i b l e . C o m b i n i n g H o o k e ' s l a w w i t h N e w t o n ' s s e c o n d l a w l ed t o a n

e q u a t i o n of m o t i o n f o r s o l i d s w h i c h , in t u r n , r e s u l t s in t h e e x i s t e n c e of

a c o u s t i c a n d s h e a r w a v e s in t h e b o d y of a so l id , a n d in R a y l e i g h w a v e s in

t h e s u r f a c e . If w e a d d t h e e x i s t e n c e of a s u r f a c e l a y e r , a s e c o n d k i n d of

s u r f a c e w a v e , t h e L o v e w a v e , is a l s o s e e n t o e x i s t .

(iii) t h e s t r e s s e s m u s t b e c o n t i n u o u s a t t h e i n t e r f a c e y = 0 .

F r o m c o n d i t i o n (i) , w e i m m e d i a t e l y find t h a t

E = B. (12 .F .7 )

I t is e a s y t o s e e t h a t c o n d i t i o n (ii) o n axy a n d cr y y is t r i v i a l ly sa t i s f ied ,

f h e c o n d i t i o n o n cr 2 y is j u s t

(12 .F .8 )

s o t h a t

C o n d i t i o n (iii) f o r cryz a t t h e i n t e r f a c e b e c o m e s

w h i c h b e c o m e s , u s i n g E q . ( 12 .F .1 ) ,

(12 .F .9 )

w h e r e t h e s e c o n d e q u a l i t y f o l l o w s f r o m E q . (12 .F .7 ) .

C o m b i n i n g t h i s r e s u l t w i t h E q . ( 1 2 . F . 8 ) , w e find

(12 .F .10)


PROBLEMS

12.1 . Wri te down H o o k e ' s law in tensor form in Cartesian, cylindrical, and spherical coordinates . (Hint: You may find it useful to go back to the definition of uik in te rms of a change in length.)

12.2. Show that the tensors Uik [defined in Eq . ( l . C . l l ) ] and aik [defined in Eq .

(8.A.9)] are s tress tensors in the sense of Section 12.B.

12.3. In addition to the internal forces canceling, leaving only a surface force, as discussed in Section 12.B, the internal momen t s in a solid must do the same.

(a) Show that the total momen t in a solid can be wri t ten

(b) Show that this can be conver ted to a surface integral, except for a te rm of

(c) H e n c e give an argument that the stress tensor must be symmetr ic .

12.4. In Section 12.C, we discussed the idea of a Newton ian solid. In such a solid,

H o o k e ' s law is replaced by an equat ion in which the stress is proport ional to the

t ime derivat ive, or ra te , of the strain, ra ther than to the strain itself. Using

arguments analogous to those leading to Eq . (12.C.1), write down the equat ion

relating stress and strain for such a solid. Show that in the case of an

incompressible solid, this becomes

where E is Young ' s modulus and n is the L a m e coefficient.

12.6. When will the Rayleigh wave arrive at San Franc isco in the example in Section 12.D?

12.7. In Chapter 8, we defined viscosity in te rms of a stress tensor and an argument based on Occam ' s razor . An al ternate way of defining viscosity is as follows: Consider a cylindrical tube with a fluid flowing in the z-direction. Take an element of the fluid and show that N e w t o n ' s second law is

the type

where 17 is a constant .

12.5. Show that for an incompressible elastic solid

(row),

where arz is the internal s tress generated by the fluid motion. If we define the

Problems 231

coefficient of viscosity by

show that we recover the Nav ie r -S tokes equat ion.

12.8. Consider a circular cylinder which is being twisted by a force F applied tangentially. Le t the result of the force be that an e lement at the edge is moved through an angle <£, as shown. Show that the to rque on the cylinder is related to the angle by

What is the energy stored in the twisted cylinder?

12.9. It should be obvious that Eq . (12.C.16) can be applied in the case where no motion is present in the solid (i.e. for the static cases t reated in the previous chapter) . T o make this point , consider a solid sphere of inner and outer radii a and b, respect ively, with pressure P i inside and P2 outside.

(a) F rom the symmet ry of the problem, show that the equat ion for the displacement is just

T h e quanti ty Tl/cp is called the torsional rigidity.

' ! / I /

/ /

/ /

V(V • u) = 0.

(b) Hence show that the strain tensor is given by

and

where A and B are as yet undetermined cons tan ts .


(c) F rom part (b), show that the radial componen t of the stress tensor is

where

and

12.10. F rom the results of Problem 12.9, der ive the s t resses for the following two limiting cases :

(a) A thin spherical shell of th ickness h and radius R sur rounded by a vacuum and maintaining a pressure P inside.

(b) A spherical cavity in an infinite medium, with a pressure P inside of it. Can you think of any applications where these limits might be useful?

12.11. Consider the case of a solid which is undergoing a plane deformation: i.e. a deformation in which uz = 0 everywhere in the solid.

(a) For the static case , show that the equat ions of motion can be reduced to two equat ions in the stress tensor .

(b) Define a function x by the relations

and show that these forms of the tensor automatically satisfy the equat ions in part

(a).

(c) Hence show that the function x> called the stress function, must satisfy the

equat ion

V 2 (V 2 *) = 0,

which is called the biharmonic equation.

12.12. In Section 12.C, we derived the equat ions of motion in te rms of the stress tensor . This is, of course , the most usual and useful form of this equat ion. There is another form, however , which can be wri t ten solely in te rms of the strain tensor . U s e H o o k e ' s law to write it down.

12.13. Consider a cylinder of radius a rotating with f requency co about its axis of

symmetry .

(a) Wri te down the equat ion of motion for such a system. (b) Show that the solution to the equat ion is

12.14. In the text , we consider elastic waves in infinite or semi-infinite media only. Le t us ask what happens when we consider waves in thin rods or sheets of the type considered in Chapter 11.

References 233

(a) Consider a longitudinal wave in a thin rod. Le t the wave travel in the z-direction. Show that the velocity of the wave in this case is

H o w does this compare to waves in an infinite medium? (b) Consider a longitudinal wave traveling in a thin plate in the z-direction.

Show that in this case we find

for the velocity of the d is turbance associated with uz.

12.15. Consider a beam of the type discussed in Section l l . C which can be bent , but need no longer be in static equilibrium.

(a) Show that the equat ion of motion for such a system is

(b) Determine the frequencies at which the rod may vibrate , if it is c lamped at one end and free at the other .

(c) H e n c e show that the smallest f requency is

where q is the mass per unit length of the rod. (Hint: Assume that the solutions of the equat ion are separable , and that the integration of the X(x) equat ion is a sum of t r igonometr ic and hyper t r igonometr ic functions of x.) This is the theory of the tuning fork.

K. E. Bullen, An Introduction to the Theory of Seismology, Cambridge U.P., 1965. An excellent modern discussion of the theory of solids as applied to seismology. The mathematics level is fairly high.

Sir Harold Jeffreys, The Earth, Cambridge U.P., 1970. One of the classic texts in geophysics. Very complete and readable, with many examples and several sections on the origin of the earth.

For related reading, see

J. D. Jackson, Classical Electrodynamics, John Wiley and Sons, New York, 1972. This text contains a discussion of the Maxwell stress tensor in Chapter 6.

REFERENCES

13

Applications of Seismology: Structure of the Earth and Under-ground Nuclear Explosions

A vast, limitless expanse of water . . . spread before us. . . . 'The Central Sea' said my uncle

JULES VERNE

A Journey to the Center of the Earth

A. SEISMIC RAYS

I n t h e p r e v i o u s c h a p t e r , t h e e x i s t e n c e of w a v e s in e l a s t i c s o l i d s w a s

d i s c u s s e d a n d t h e w a y in w h i c h t h e s e w a v e s c o u l d b e u s e d t o g a i n

i n f o r m a t i o n a b o u t t h e s t r u c t u r e of t h e e a r t h w a s h i n t e d a t . Al l of t h i s

d i s c u s s i o n ( e x c e p t f o r t h e d e v e l o p m e n t of L o v e w a v e s ) w a s d o n e in t h e

c o n t e x t of a u n i f o r m e a r t h of c o n s t a n t d e n s i t y a n d L a m e coe f f i c i en t s . W e

k n o w , of c o u r s e , t h a t t h e e a r t h is r e a l l y n o t s o s i m p l e a s t h a t , a n d o n e of

t h e m a i n g o a l s of s e i s m o l o g y is t o t r y t o d i s c o v e r t h e d e t a i l s of t h e

s t r u c t u r e of t h e e a r t h . T h e p r o b l e m c a n b e p u t in t h e f o l l o w i n g w a y :

G i v e n t h a t w e c a n o n l y m a k e m e a s u r e m e n t s a t t h e s u r f a c e of t h e e a r t h ,

w h a t c a n w e d o t o d i s c o v e r t h e s t r u c t u r e of t h e i n t e r i o r ?

O n e o b v i o u s w a y t o a n s w e r t h i s q u e s t i o n is t o t r y t o m e a s u r e s o m e t h i n g

t h a t p a s s e s t h r o u g h t h e i n t e r i o r , a n d is a f f ec t ed b y it . T h i s is s o m e w h a t

a n a l o g o u s t o a p h y s i c i a n t a k i n g X - r a y s of t h e h u m a n b o d y , a n d l e a r n i n g

f r o m t h e a b s o r p t i o n of t h e r a d i a t i o n t h e c o n d i t i o n in t h e i n t e r i o r .

T h e o n l y " r a d i a t i o n " of t h i s t y p e t h a t w e h a v e a t o u r d i s p o s a l a r e t h e

w a v e s , d i s c u s s e d in t h e l a s t c h a p t e r , w h i c h c a n p r o p a g a t e in s o l i d s .

O b v i o u s l y , t h e s u r f a c e w a v e s wi l l b e of l i m i t e d v a l u e in e x p l o r i n g t h e

d e e p i n t e r i o r of t h e e a r t h , a n d w e wil l c o n f i n e o u r a t t e n t i o n t o b o d y w a v e s

f o r t h e m o m e n t . L e t u s r e c o n s i d e r t h e e x a m p l e of S e c t i o n 12 .D in w h i c h

234

Seismic Rays 235

Fig. 13.1 Seismic rays for a Tokyo earthquake.

a n e a r t h q u a k e o c c u r r e d in T o k y o . If t h e e a r t h w e r e u n i f o r m , t h e s e i s m i c

w a v e s w o u l d p r o p a g a t e o u t f r o m T o k y o , a n d c o u l d b e m e a s u r e d a t m a n y

p l a c e s o n t h e e a r t h ( s e e F i g . 13.1). A t e a c h o b s e r v i n g s t a t i o n , t h e t i m e of

t h e a r r i v a l s of t h e P a n d S w a v e s c o u l d b e m e a s u r e d . I n t h i s p r o b l e m ,

t h e r e a r e t h r e e n u m b e r s w h i c h w e d o n o t k n o w , b u t w o u l d l i k e t o l e a r n .

T h e s e a r e t h e d e n s i t y of t h e e a r t h a n d t h e t w o L a m e coe f f i c i en t s . T h u s , if

w e c o u l d m e a s u r e t h r e e d i f f e r en t t i m e i n t e r v a l s ( c o r r e s p o n d i n g t o t h r e e

d i f f e r en t o b s e r v a t i o n s t a t i o n s ) , w e c o u l d c o m p l e t e l y d e t e r m i n e p , A, a n d

ix. W h i l e d a t a f r o m a s ing le o b s e r v a t i o n p o i n t c a n n o t te l l u s m u c h a b o u t

t h e i n t e r i o r of t h e e a r t h , d a t a f r o m m a n y s t a t i o n s , t a k e n t o g e t h e r , c a n d o

s o q u i t e w e l l . T h i s , i n c i d e n t a l l y , i s t h e r e a s o n t h a t i n t e r n a t i o n a l c o o p e r a -

t i o n h a s b e e n s o i m p o r t a n t in t h e d e v e l o p m e n t of s e i s m o l o g y .

T h e e a r t h , of c o u r s e , is n o t u n i f o r m . T h e r e a l p r o b l e m of s e i s m o l o g y is

t o d i s c o v e r t h e d e n s i t y a n d e l a s t i c p r o p e r t i e s of t h e m a t e r i a l s i n s i d e of t h e

e a r t h a s a f u n c t i o n of d e p t h a n d p o s i t i o n . T h i s m e a n s t h a t i n s t e a d of

t r y i n g t o fix t h r e e u n k n o w n c o n s t a n t s , a s in t h e a b o v e e x a m p l e , t h e

s e i s m o l o g i s t is a c t u a l l y t r y i n g t o fix d e n s i t i e s a n d e l a s t i c c o n s t a n t s ( a n d

d e v i a t i o n s f r o m e l a s t i c i t y ) a s a f u n c t i o n of d e p t h in t h e e a r t h . A s a s t a r t

t o w a r d s o l v i n g t h i s p r o b l e m , le t u s a s k h o w a s e i s m i c r a y p r o p a g a t e s

t h r o u g h a m e d i u m w h i c h is n o t u n i f o r m .

T o d o t h i s , l e t u s c o n s i d e r w h a t h a p p e n s w h e n a p l a n e w a v e ( e i t h e r S o r

P ) a r r i v e s a t a b o u n d a r y b e t w e e n t w o l a y e r s , e a c h c h a r a c t e r i z e d b y a

d i f f e ren t v e l o c i t y ( w h i c h , in t u r n , i s r e l a t e d t o d i f f e r en t e l a s t i c c o n s t a n t s ) .

W e c a n s e e w h a t wi l l h a p p e n t o t h e w a v e b y i n v o k i n g Huygens principle,

f a m i l i a r f r o m o p t i c s , w h i c h t e l l s u s t h a t e a c h p o i n t of a w a v e f r o n t c a n b e

t h o u g h t of a s e m i t t i n g a s p h e r i c a l w a v e l e t w i t h t h e w a v e a t a n y o t h e r

236 Applications of Seismology

Fig. 13.2(a). The propagation of a wave by Huygens wavelets.

p o i n t b e i n g g i v e n b y t h e s u m of t h e w a v e l e t s . F o r e x a m p l e , a p l a n e w a v e

p r o p a g a t i n g in a u n i f o r m m e d i u m c o u l d b e t h o u g h t of a s s h o w n in F i g .

13.2(a) , w i t h e a c h p o i n t of t h e f r o n t e m i t t i n g a w a v e l e t , a n d t h e s e w a v e l e t s

a d d i n g u p t o g i v e t h e w a v e f r o n t f a r t h e r d o w n s t r e a m .

W h e n s u c h a w a v e e n c o u n t e r s a b o u n d a r y , h o w e v e r , t h e s i t u a t i o n wil l

b e a s p i c t u r e d in F i g . 13.2(b) . T h e w a v e l e t s e m i t t e d f r o m t h e p o i n t P wi l l

t r a v e l a t a v e l o c i t y v\ c h a r a c t e r i s t i c of t h e s e c o n d m e d i u m , w h i l e t h o s e

e m i t t e d a t Q wil l c o n t i n u e t o t r a v e l w i t h v e l o c i t y v. If it t a k e s t i m e t f o r

t h e w a v e l e t f r o m Q t o t r a v e l t o t h e i n t e r f a c e , t h e n t h e w a v e l e t f r o m P wi l l

h a v e t r a v e l e d a d i s t a n c e v' t. H e n c e t h e n e w w a v e f r o n t wi l l b e t h e l i ne

AB, a n d w e s e e t h a t t h e n e t e f fec t of t h e i n t e r f a c e is t o c h a n g e t h e

d i r e c t i o n of t h e w a v e . T h i s p h e n o m e n o n , k n o w n a s r e f r a c t i o n , is f a m i l i a r

in o p t i c s . F r o m t h e g e o m e t r y in F i g . 13 .2(b) , it is e a s y t o s e e t h a t

s in 6 s in 6' n

— = — , (13 .A.1)

w h i c h is t h e f a m i l i a r Sneirs law f o r r e f r a c t i o n .

Fig. 13.2(b). Geometry forthe derivation of Snell's law.

Seismic Rays 237

Fig. 13.3. Propagation of a seismic wave in a layered medium.

L e t u s n o w c o n s i d e r t h e i n t e r i o r of t h e e a r t h a s a s e r i e s of l a y e r s , a s

s h o w n in F i g . 1 3 . 3 . T h e n a w a v e w h i c h s t a r t s off a t a n a n g l e 0i wi l l b e

s u c c e s s i v e l y r e f r a c t e d a t e a c h i n t e r f a c e , w i t h t h e r e l a t i o n s h i p

Vi V2 V2 t>3

f o l l o w i n g f r o m S n e l l ' s l a w . B y g e o m e t r y , h o w e v e r , w e h a v e

L = ri s in 6[ = r2 s in 62.

F r o m t h e a b o v e t w o e q u a t i o n s , w e s e e i m m e d i a t e l y t h a t

ri s in 0i r2 s in 02 rx s in 0i r2 s in 62

v3 v2

T h e e x t e n s i o n of t h i s t y p e of r e l a t i o n s h i p t o a n inf in i te n u m b e r of l a y e r s

( w h i c h w o u l d r e p r e s e n t a c o n t i n u o u s l y c h a n g i n g i n t e r i o r ) y i e l d s t h e

g e n e r a l l a w

r s in 0 , - A ^ P =—-—, ( 1 3 . A . 2 )

s in 0i _ s in 0'i s in 0 2 _ s in S2

Vi


w h e r e p i s a c o n s t a n t a l o n g t h e e n t i r e r a y (it i s c a l l e d t h e r a y p a r a m e t e r ) .

I t f o l l o w s t h a t w a v e s t r a v e l i n g t h r o u g h t h e e a r t h d o n o t , in f a c t f o l l o w

s t r a i g h t l i ne s a s in F i g . 13 .1 , b u t c u r v e s , a s in F i g . 13.4.

T h e r e a r e s e v e r a l p o i n t s w h i c h s h o u l d b e m a d e b e f o r e p r o c e e d i n g .

F i r s t , i t s h o u l d b e o b v i o u s t h a t , in g e n e r a l , S a n d P w a v e s s t a r t i n g f r o m

t h e s a m e p o i n t wil l h a v e d i f f e r en t p a t h s in t h e i n t e r i o r , s i n c e , in g e n e r a l ,

t h e d e p e n d e n c e s of ct a n d a o n r wi l l n o t b e t h e s a m e . S e c o n d , in

a d d i t i o n t o t h e p h e n o m e n o n of r e f r a c t i o n in t h e e a r t h , s e i s m i c w a v e s ( l ike

a n y o t h e r w a v e s ) c a n b e r e f l e c t e d a t i n t e r f a c e s a s w e l l . T h i s wi l l b e s h o w n

in P r o b l e m s 13 .3 , 13.4, a n d 13.5 .

T h e g e n e r a l p r o b l e m f a c e d b y t h e s e i s m o l o g i s t , t h e n , i s t o u n d e r s t a n d

t h e r e l a t i o n b e t w e e n t h e t i m e a n d p l a c e of a r r i v a l of a s e i s m i c w a v e , a n d

t h e t r a j e c t o r y w h i c h it h a s f o l l o w e d t h r o u g h t h e e a r t h . L e t u s e x a m i n e t h i s

p r o b l e m in m o r e d e t a i l . W e k n o w t h a t e a c h s e i s m i c w a v e is c h a r a c t e r i z e d

b y a p a r a m e t e r p , a n d t r a v e l s t h r o u g h t h e e a r t h s u b t e n d i n g a n a n g l e A at

t h e c e n t e r , t a k i n g a t i m e T t o g e t f r o m P 0 , t h e p o i n t of e m i s s i o n , t o Q 0 , t h e

p o i n t of d e t e c t i o n ( s e e F i g . 13.4).

If w e d e n o t e b y s t h e d i s t a n c e a l o n g t h e c u r v e P 0 Q o , t h e n b y s i m p l e g e o m e t r y E q . (13 .A .2 ) ,

(13 .A.4)

Fig. 13.4. The path traversed by a seismic ray.

b u t in g e n e r a l ,

ds2=dr2+r2da\

(13 .A.3)

s o t h a t , if w e l e t TJ = r/v,

Seismic Rays 239

If w e i n t e g r a t e f r o m t h e p o i n t of e m i s s i o n , Po , t o t h e h a l f w a y p o i n t a l o n g

t h e t r a j e c t o r y (a t r = r O , w e h a v e

(13 .A.5)

f o r t h e a n g l e s u b t e n d e d b y t h e t r a j e c t o r y . I n a c o m p l e t e l y a n a l o g o u s w a y ,

w e c a n d e r i v e a n e x p r e s s i o n f o r t h e t r a n s i t t i m e

(13 .A.6)

f r o m t h e f a c t t h a t ds = vdt.

T h e s e e q u a t i o n s r e l a t e t h e a n g l e A a n d t i m e T, b o t h of w h i c h a r e

m e a s u r a b l e q u a n t i t i e s , t o i n t e g r a l s i n v o l v i n g v(r) a n d t h e r a y p a r a m e t e r p .

B u t s i n c e

(13 .A.7)

p c a n a l s o b e d e t e r m i n e d b y s u r f a c e m e a s u r e m e n t s . T h u s , b y m e a s u r i n g

a r r i v a l t i m e s of w a v e s a t d i f f e ren t p o i n t s a b o u t t h e e a r t h , w e c a n

d e t e r m i n e v(r) in t h e i n t e r i o r . I n a c t u a l p r a c t i c e , t h e r e i s m o r e d a t a t h a n

j u s t s e i s m i c a r r i v a l t i m e s . W e h a v e a l s o t h e f r e e v i b r a t i o n s o f t h e e a r t h

( s e e C h a p t e r 7) a n d s o m e g o o d t h e o r e t i c a l c o n j e c t u r e s a b o u t t h e c h e m i c a l

c o m p o s i t i o n of t h e i n t e r i o r w h i c h m u s t b e fit i n t o t h e r e s u l t s a s w e l l .

T h e g e n e r a l p i c t u r e o f t h e e a r t h ' s i n t e r i o r w h i c h h a s a r i s e n f r o m s u c h

s t u d i e s i s i l l u s t r a t e d in F i g . 13 .5 . T h e o u t e r l a y e r o f t h e e a r t h , t h e c r u s t , i s

o n l y a b o u t 15 k m t h i c k . U n d e r t h e c r u s t is a so l id m a n t l e , w h i c h i s i t se l f

crust

outer core

mantle

Fig. 13.5. The general structure of the earth.


u s u a l l y d i v i d e d i n t o u p p e r , m i d d l e , a n d l o w e r r e g i o n s . T h e m a n t l e e x t e n d s

t o a d e p t h of a b o u t 2800 k m . B e t w e e n t h e m a n t l e a n d t h e c r u s t is a s h a r p

t r a n s i t i o n a l r e g i o n k n o w n a s t h e Mohorovicic discontinuity. W e b e l i e v e

t h a t t h e c o n t i n e n t s , w h i c h a r e p a r t of t h e c r u s t , a c t u a l l y f loat o n t h e m a n t l e ,

a n d h a v e m o v e d a r o u n d d u r i n g g e o l o g i c a l t i m e s . T h e s u b j e c t of continental

drift is a f a s c i n a t i n g o n e , a n d o n e of t h e m o r e i m p o r t a n t i d e a s of m o d e r n

g e o p h y s i c s .

B e l o w t h e m a n t l e , t h e r e is t h e c o r e . T h e o u t e r c o r e , e x t e n d i n g d o w n t o

a b o u t 5000 k m , is l i qu id m e t a l , a n d it is t h o u g h t t h a t t h e m o t i o n s of t h i s

l i qu id c o r e a r e i m p o r t a n t in g e n e r a t i n g t h e m a g n e t i c field of t h e e a r t h . A t

t h e v e r y c e n t e r of t h e e a r t h is t h e i n n e r c o r e , c o m p o s e d of so l id m e t a l s .

T h u s , w e s e e t h a t o n e a p p l i c a t i o n of t h e t h e o r y of e l a s t i c i t y is t o g i v e u s

a n i n c r e a s i n g l y d e t a i l e d p i c t u r e of t h e e a r t h o n w h i c h w e l ive . A s i d e f r o m

t h e o b v i o u s p r a c t i c a l a d v a n t a g e s of s u c h k n o w l e d g e , t h i s a l s o g i v e s u s

i m p o r t a n t i n f o r m a t i o n a b o u t t h e p r o c e s s b y w h i c h t h e e a r t h , a n d h e n c e

t h e s o l a r s y s t e m , w e r e f o r m e d .

B. UNDERGROUND NUCLEAR EXPLOSIONS

A n o t h e r a p p l i c a t i o n of t h e k n o w l e d g e of w a v e s in so l i d s is in t h e field

of a r m s c o n t r o l . T h e ab i l i t y t o l imi t t h e d e v e l o p m e n t of n u c l e a r w e a p o n s

d e p e n d s d i r e c t l y o n t h e ab i l i t y t o d e t e c t n u c l e a r t e s t s . W h e n s u c h t e s t s a r e

c a r r i e d o u t in t h e a t m o s p h e r e , t h e d e t e c t i o n is r e l a t i v e l y s i m p l e , s i n c e

p r e v a i l i n g w i n d s wil l c a r r y r a d i o a c t i v e d e b r i s a c r o s s n a t i o n a l b o u n d a r i e s

t o d e t e c t i n g s t a t i o n s . U n d e r g r o u n d t e s t s , h o w e v e r , a r e n o t s o e a s y t o

d e t e c t , s i n c e t h e d e b r i s is c o n f i n e d ( b a r r i n g a n a c c i d e n t a l r e l e a s e of

r a d i o a c t i v e m a t e r i a l s i n t o t h e a t m o s p h e r e ) . I n f a c t , t h e o n l y i n d i c a t i o n

t h a t s u c h a t e s t h a s o c c u r r e d w h i c h w o u l d b e d e t e c t a b l e a t l a r g e d i s t a n c e s

f r o m t h e s i t e of t h e t e s t w o u l d b e t h e s e i s m i c s igna l g e n e r a t e d b y t h e

e x p l o s i o n . T h i s , in t u r n , l e a d s u s t o t h e q u e s t i o n of h o w s e i s m i c w a v e s a r e

g e n e r a t e d .

B e f o r e t u r n i n g t o t h i s q u e s t i o n , h o w e v e r , le t u s r e v i e w br ie f ly t h e

s e q u e n c e of e v e n t s w h i c h f o l l o w s a n u c l e a r e x p l o s i o n . I m m e d i a t e l y

f o l l o w i n g t h e b l a s t , t r e m e n d o u s p r e s s u r e ( o n t h e o r d e r of 10 6 a t m o s -

p h e r e s ) a r e p r e s e n t . T h e s u d d e n r e l e a s e of e n e r g y c o m p l e t e l y s t r i p s t h e

a t o m s in t h e n e i g h b o r h o o d of t h e b l a s t , a n d t w o t h i n g s o c c u r : (1) a b u r s t

of e l e c t r o m a g n e t i c r a d i a t i o n m o v e s a w a y f r o m t h e b l a s t s i t e , a n d (2) t h e

d e b r i s of t h e b l a s t m o v e s a w a y a l s o , f o r m i n g a s h o c k f r o n t . A t t h e

b e g i n n i n g , t h e r a d i a t i o n f r o n t m o v e s q u i c k l y , h e a t i n g u p t h e s u r r o u n d i n g

m a t e r i a l a n d f o r m i n g a n e x p a n d i n g " f i r e b a l l " of h i g h t e m p e r a t u r e g a s e s .

A s t h e fireball e x p a n d s , i t s t e m p e r a t u r e d r o p s ( w h y ? ) a n d t h e e x p a n s i o n

Underground Nuclear Explosions 241

s l o w s d o w n . A t s o m e p o i n t , c a l l e d breakaway, t h e s h o c k w a v e o v e r t a k e s

t h e r a d i a t i o n f r o n t a n d m o v e s a h e a d of i t .

I n a t m o s p h e r i c e x p l o s i o n s , t h i s is a c o m p l e t e d e s c r i p t i o n of t h e b l a s t

p h e n o m e n o n . I n u n d e r g r o u n d e x p l o s i o n s , h o w e v e r , t h e r e is a n o t h e r

q u a n t i t y w h i c h e n t e r s a n d t h a t is t h e s i z e of t h e c a v i t y in w h i c h t h e

e x p l o s i o n o c c u r s . F o r t h e s a k e of s i m p l i c i t y , w e wil l a s s u m e t h r o u g h o u t

t h e r e s t of t h i s s e c t i o n t h a t w e a r e d e a l i n g w i t h a s p h e r i c a l l y s y m m e t r i c

g e o m e t r y . If t h e r a d i u s of t h e c a v i t y is l e s s t h a n t h e r a d i u s a t w h i c h

b r e a k a w a y o c c u r s , t h e n t h e f i rebal l wil l a c t u a l l y s t r i k e t h e c a v i t y w a l l s ,

v a p o r i z i n g t h e m . S i n c e m o r e e n e r g y is r e q u i r e d t o v a p o r i z e r o c k t h a n t o

h e a t u p a i r , t h e f i rebal l wil l b e s l o w e d d o w n . W h e n t h e s h o c k f r o n t

c a t c h e s u p w i t h t h e f i rebal l a n d m o v e s a h e a d , o n e of t w o t h i n g s m a y

h a p p e n : (i) t h e s h o c k f r o n t wil l h a v e suff ic ient e n e r g y t o c o n t i n u e

v a p o r i z i n g t h e r o c k , (ii) t h e s h o c k f r o n t wi l l h a v e o n l y e n o u g h e n e r g y t o

m e l t t h e s u r r o u n d i n g r o c k . I n e i t h e r c a s e , a s t h e s h o c k w a v e p r o c e e d s o u t

f r o m t h e b l a s t s i t e t h e d a m a g e w h i c h it d o e s d e c r e a s e s . A t l a r g e d i s t a n c e s ,

t h e r o c k wil l b e f r a c t u r e d , b u t it is c l e a r t h a t a t s o m e d i s t a n c e , w h i c h w e

shal l d e n o t e b y ( t h e " s e i s m i c " r a d i u s ) , t h e d e f o r m a t i o n of t h e r o c k

c a u s e d b y t h e s h o c k f r o n t will n o t e x c e e d t h e e l a s t i c l i m i t s , a n d t h e r o c k

wi l l s i m p l y b e d e f o r m e d e l a s t i c a l l y , w h i c h m e a n s t h a t it wi l l e x e r t i n t e r n a l

f o r c e s w h i c h wil l b r i n g it b a c k t o i t s o r ig ina l p o s i t i o n . W e s p e a k of t h e

s h o c k w a v e " d e c a y i n g " i n t o a n e l a s t i c w a v e a t t h i s p o i n t . T h e q u e s t i o n

w h i c h w e m u s t a s k h a s t o d o w i t h r e l a t i n g t h e d e f o r m a t i o n a t R{r t o t h e

s e i s m i c w a v e w h i c h w o u l d b e d e t e c t e d a t l a r g e d i s t a n c e s .

I t s h o u l d b e o b v i o u s f r o m t h e f o r e g o i n g d i s c u s s i o n t h a t it is p o s s i b l e t o

h e i g h t e n o r r e d u c e t h e e f f ec t s of t h e b l a s t a t R^ b y c h o o s i n g t h e c a v i t y

r a d i u s t o b e g r e a t e r o r l e s s t h a n t h e fireball r a d i u s , a n d b y c h o o s i n g t h e

m a t e r i a l s u r r o u n d i n g t h e b l a s t s i t e . T h u s , a s m a l l c a v i t y in so l id r o c k (a

" t a m p e d " e x p l o s i o n ) w o u l d p r o d u c e m u c h g r e a t e r s e i s m i c s igna l s t h a n a

l a r g e c a v i t y in a v e r y p o r o u s m a t e r i a l . T h i s p r o b l e m , w h i c h i n v o l v e s t h e

coupling of t h e e x p l o s i o n t o s e i s m i c w a v e s , is o b v i o u s l y of g r e a t i n t e r e s t

t o t h o s e c o n c e r n e d w i t h a r m s c o n t r o l . A m u c h m o r e d e t a i l e d d i s c u s s i o n is

g i v e n in t h e t e x t b y R o d e a n (1971) c i t e d a t t h e e n d of t h e c h a p t e r .

T h e p r o b l e m of d e t e c t i n g a n u n d e r g r o u n d t e s t , t h e n , b e c o m e s o n e of

u n d e r s t a n d i n g w h a t s o r t of s e i s m i c s i gna l s s u c h a t e s t w o u l d g e n e r a t e . L e t

u s c o n s i d e r a s p h e r i c a l l y s y m m e t r i c s i t u a t i o n s u c h a s t h a t in F i g . 13.6, in

w h i c h s o m e k n o w n d i s p l a c e m e n t of t h e m a t e r i a l t a k e s p l a c e a t r = R„,

a n d w a v e s p r o p a g a t e o u t . W e k n o w t h a t t h e e q u a t i o n s w h i c h g o v e r n t h e

d i s p l a c e m e n t s of t h e so l id a t l a r g e r a d i i a r e

(13 .B.1)


Fig. 13.6. Coordinates for the underground nuclear explosion.

f o r d i s p l a c e m e n t s in t h e r-direction, a n d

(13 .B.2)

f o r d i s p l a c e m e n t s in t h e 0 - d i r e c t i o n . A s imi l a r e q u a t i o n c a n b e w r i t t e n f o r

t h e </>-direction, of c o u r s e .

If w e m a k e t h e s u b s t i t u t i o n

t h e n E q . (13 .B.1) b e c o m e s

(13 .B.3)

(13 .B.4)

w h i c h is j u s t t h e w a v e e q u a t i o n . W i t h o u t l o s s of g e n e r a l i t y , w e wi l l

c o n s i d e r o n l y p l a n e w a v e s o l u t i o n s , s o t h a t

(13 .B.5)

(13 .B.6)

I n o r d e r t o d e t e r m i n e t h e c o n s t a n t s A a n d B, it i s n e c e s s a r y t o r e f e r t o

t h e b o u n d a r y c o n d i t i o n s a t t h e p o i n t r = R^. W e k n o w t h a t a t t h i s p o i n t

t h e r e is n o e x t e r n a l f o r c e o n t h e r o c k u n t i l t h e t i m e of t h e e x p l o s i o n , t h e n

f o r c e s a r e a p p l i e d t o t h e m a t e r i a l , a n d t h e s e f o r c e s wi l l d i e o u t g r a d u a l l y a

l o n g t i m e a f t e r t h e e x p l o s i o n . I n g e n e r a l , t h e a p p l i e d f o r c e a t t h e s e i s m i c

r a d i u s w o u l d l o o k l ike t h e o n e s h o w n in F i g . 13.7. W h a t e v e r t h e a c t u a l

f u n c t i o n a l d e p e n d e n c e of t h e f o r c e , h o w e v e r , it is c l e a r t h a t w e c a n a l w a y s

w r i t e

B y e x a c t l y s imi l a r s t e p s , w e w o u l d h a v e

Underground Nuclear Explosions 243

w h i c h l e a d s i m m e d i a t e l y t o t h e r e s u l t

(13 .B.10)

Fig. 13.7. A typical applied force at the seismic radius.

s o t h a t w e c a n , f o r t h e s a k e of o u r p r o b l e m , c o n s i d e r o n l y t h e F o u r i e r

c o m p o n e n t

F(t) = F0eiM»t.

W e n o w a s k t h e c r i t i c a l q u e s t i o n . I n w h i c h d i r e c t i o n is t h i s f o r c e

p o i n t e d ? F o r a n u n d e r g r o u n d e x p l o s i o n , w e w o u l d e x p e c t t h e f o r c e t o b e

m a i n l y r a d i a l , s o t h a t t h e s e i s m i c r a d i u s ,

trrr(t) = F(t), (13 .B.7)

(Jre = (Jr = 0 .

F o r a n e a r t h q u a k e , o r o t h e r n a t u r a l s o u r c e of t h e s e i s m i c s igna l , o n t h e

o t h e r h a n d , w e w o u l d e x p e c t t h a t ov0 a n d orr(f> w o u l d n o t v a n i s h a t r = R^.

T h i s , t h e n , is t h e m a i n d i f f e r e n c e b e t w e e n u n d e r g r o u n d e x p l o s i o n s a n d

n a t u r a l l y o c c u r r i n g e v e n t s . W e m u s t n o w s e e w h a t ef fec t t h i s d i f f e r e n c e in

b o u n d a r y c o n d i t i o n s wi l l h a v e o n s e i s m i c s igna l s f a r f r o m t h e e v e n t .

F r o m H o o k e ' s l a w f o r t h e c a s e of s p h e r i c a l s y m m e t r y , w e h a v e

(13 .B.8)

s o t h a t a t t h e s e i s m i c r a d i u s , c o m b i n i n g E q s . (13 .B .8 ) , (13 .B .7 ) , a n d

(13 .B .5 ) , w e h a v e

(13 .B.9)

a n d


w h i c h l e a d s i m m e d i a t e l y t o t h e r e s u l t

B =0. (13 .B.12)

T h u s , in o u r s impl i f ied m o d e l , t h e s igna l c h a r a c t e r i s t i c of a n u n d e r g r o u n d

e x p l o s i o n w o u l d b e a m i s s i n g S w a v e . Of c o u r s e , in a r e a l s i t u a t i o n , t h e

a p p l i e d f o r c e w o u l d n e v e r b e e x a c t l y r a d i a l , a n d s o m e S w a v e w o u l d b e

g e n e r a t e d . N e v e r t h e l e s s , a s h a r p d i m i n u t i o n of S w a v e is o n e c o m m o n l y

a c c e p t e d c r i t e r i o n f o r d i s c r i m i n a t i n g b e t w e e n s m a l l e a r t h q u a k e s a n d

u n d e r g r o u n d t e s t s .

A m o r e i m p o r t a n t t o o l , w h i c h w e sha l l n o t d i s c u s s in d e t a i l , a r i s e s f r o m

t h e f a c t t h a t t h e o u t g o i n g s e i s m i c w a v e s f r o m a n u n d e r g r o u n d e v e n t wil l

s t r i k e t h e s u r f a c e n e a r t h e e v e n t a n d g e n e r a t e R a y l e i g h s u r f a c e w a v e s .

A l t h o u g h t h e t h e o r y of h o w R a y l e i g h w a v e s a r e g e n e r a t e d in t h i s m a n n e r

is n o t r e a l l y w e l l w o r k e d o u t , it d o e s t u r n o u t t h a t s o u r c e s w h i c h g e n e r a t e

b o t h S a n d P w a v e s a r e m u c h m o r e eff icient in c r e a t i n g R a y l e i g h w a v e s

a t a f r e e s u r f a c e t h a n a r e s o u r c e s g e n e r a t i n g o n l y P w a v e s . T h i s m e a n s

t h a t a s e c o n d c o n s e q u e n c e of E q . (13 .B .12) is t h a t in a d d i t i o n t o t h e

a b s e n c e of t h e S b o d y w a v e s ( a n a b s e n c e w h i c h is s o m e w h a t difficult t o

d e t e c t f o r s m a l l e x p l o s i o n s w i t h p r e s e n t t e c h n i q u e s ) , t h e r e s h o u l d b e a

g r e a t r e d u c t i o n in s u r f a c e w a v e s a s w e l l . T h i s h a s , in f a c t , b e e n o b s e r v e d ,

a n d is d i s c u s s e d in s o m e of t h e r e f e r e n c e s a t t h e e n d of t h e c h a p t e r .

W e s e e t h e n , t h a t a r e l a t i v e l y s i m p l e m o d e l of t h e s e i s m i c r e s p o n s e t o

a n u n d e r g r o u n d e x p l o s i o n c a n e x p l a i n s o m e of t h e i d e a s w h i c h a r e n o w

b e i n g e x a m i n e d in r e s e a r c h o n n u c l e a r a r m s c o n t r o l .

SUMMARY

W e h a v e s e e n h o w t h e k n o w l e d g e a b o u t w a v e s in so l i d s c o u l d b e

a p p l i e d t o t w o s e p a r a t e p r o b l e m s . F i r s t , w e s a w t h a t b o d y w a v e s t r a v e l i n g

t h r o u g h t h e e a r t h w o u l d f o l l o w t r a j e c t o r i e s w h i c h d e p e n d e d o n t h e e l a s t i c

c o n s t a n t s in t h e i n t e r i o r . T h i s b e c o m e s t h e n a m e t h o d of finding o u t a b o u t

t h e s t r u c t u r e of t h e i n t e r i o r of t h e e a r t h .

S e c o n d , w e s a w t h a t u n d e r g r o u n d n u c l e a r e x p l o s i o n s a n d e a r t h q u a k e s

a r e q u i t e d i f f e ren t a s f a r a s t h e t y p e of s e i s m i c w a v e s w h i c h t h e y g e n e r a t e

T h u s , t h e a m p l i t u d e of t h e P w a v e a t a l a r g e d i s t a n c e f r o m t h e s o u r c e is

d i r e c t l y p r o p o r t i o n a l t o t h e m a g n i t u d e of t h e a p p l i e d f o r c e s . T h e S w a v e ,

o n t h e o t h e r h a n d , m u s t b e d e t e r m i n e d f r o m t h e r e q u i r e m e n t t h a t a t

r = R„,

(13 .B.11)

Problems 245

a r e c o n c e r n e d . A n e x p l o s i o n w o u l d b e e x p e c t e d t o h a v e m u c h s m a l l e r S

w a v e s a n d s u r f a c e w a v e s t h a n a n e a r t h q u a k e .

PROBLEMS

13.1 . For the example of the Tokyo ear thquake of Section 12.D, cons t ruc t a table of t ime intervals be tween the event and the arrivals of the S and P waves at 10 different points a round the world (you may choose your own points) , assuming a uniform ear th .

13.2. Consider a ray starting at P 0 and ending at Q 0 , as in Fig. 13.4, and let T and A be the travel t ime and subtended angle for this ray, and p be the ray parameter . If a ray starts from a neighboring point, and has T + dT, A + dA, and p + dp for the corresponding values , show that

13.4. (a) Show that if an SV wave were incident on the surface in Prob lem 13.3, of magnitude B , and the reflected P and S waves had ampli tude Ax and Bu

13.3. Consider a free surface at z = 0 with a P wave incident with angle 6. Take the incident wave to be of the form

(a) Wri te down the boundary condit ions at the surface z = 0. (b) Assume that there will be bo th a reflected P wave and a reflected S wave ,

and take their form to be

il;p = Aleilk(x+ztane)-"t]

and _ D „ i[k(x+z tan <f>)-o)t]

yjs — ri\e

Show that it is not possible to satisfy the boundary condit ions if Bx = 0 so that there must be a reflected S wave .

(c) F r o m the equat ions of motion, show that

(d) Show that the coefficients of the reflected wave are given by

and that the result of par t (c) still follows. (b) H e n c e show that if cos <f> > ctld, the reflected P wave will die out rapidly as

we leave the surface, and the ampli tudes of the incident and reflected S wave will be equal .

(c) Show that for an incident SH wave , the reflected wave is a lways equal in ampli tude to the incident wave , and no P wave is generated at the surface.

13.5. Consider now a wave incident from below on an interface at z = 0, with the material in the lower half plane character ized by L a m e coefficient fx and A, and the material in the upper half plane character ized by /JL' and A'. Assume the ampli tudes of the waves are as follows:

A incident P wave , B incident SV wave , A i reflected P wave , Bi reflected S V wave , A' t ransmit ted P wave , B' t ransmit ted S V wave , C incident SH wave , C i reflected SH wave , C t ransmit ted SH wave ,

and assume that the angles associated with the direct ions of the t ransmit ted P and S waves are 0' and </>', respect ively.

(a) Wri te down the equat ions of motion in each medium and the condit ions which must hold at z = 0.

(b) Show that the equat ions for the SH wave are independent of the equat ions for the P and S V waves (as was seen in Prob lems 13.3 and 13.4 above) , and that

C + G = C \

fx tan <f>(C - Ci) = n' tan cb'C.

(c) Derive Snell 's law for refraction from the boundary condit ions in part (a). (d) Wri te down the four (rather complicated) equat ions which determine AUBU

A' and B'. 13.6. A liquid can be character ized by the s ta tement that ix = 0. Given the results of Problem 13.3, can you explain why no S waves are observed directly opposi te an ear thquake , al though P waves are?

13.7. A rough parameter izat ion of the velocity of seismic waves as a function of


respect ively, that retracing the steps in Problem 13.3 would give

References 247

depth , which is useful in calculat ions, is

v = arb,

where a and b are cons tan ts . Fo r the special case b = 1, consider a signal originating at a lati tude 0 . T h e signal is observed at a point Q 0 , at lat i tude <£. Find the deepes t penetra t ion of the ray as a function of .

13.8. Der ive the equat ion analogous to E q s . (13.B.10) and (13.B.12) for an ea r thquake in which the boundary condit ions are

(Trr = CTre = CTVrf, = F(t)

at some radius R<r. Is this a reasonable model of an ea r thquake?

REFERENCES

All of the geophysics texts cited in Chapter 12 contain discussions of seismic waves.

Bruno Rossi, Optics, Addison-Wesley, Reading, Mass., 1957. Chapter 1 contains an excellent description of Huygens principle applied to light.

F. D. Stacey, Physics of the Earth, John Wiley and Sons, New York, 1969. A descriptive, mainly nonmathematical discussion of seismology, the earth's magnetic and gravitational fields, and the internal structure of the earth.

R. H. Tucker, A. H. Cook, H. M. Iyer, and F. D. Stacey, Global Geophysics, American Elsevier, New York, 1970.

A descriptive book covering seismology and the earth's structure. F. G. Blake, Jr., "Spherical Wave Propagation in Solid Media," J. Acoustical Soc. America 24, 211 (1952).

A concise description of wave propagation. B. Gutenberg, Physics of the Earth's Interior, Academic Press, New York, 1959.

Detailed discussion of the layers and regions of the interior. S. K. Runcorn (ed.), Continental Drift, Academic Press, New York, 1962.

Collection of papers on all phases of the problem of continental drift. Physics Today, March 1974.

An entire issue devoted to discussions of modern ideas in geophysics. H. C. Rodean, "Nuclear-Explosion Seismology," U.S.A.E.C. Technical Information Bulletin (TID-25572), 1971.

Detailed description of effects of underground explosions, the problem of coupling, and the generation of seismic waves. Extensive bibliography.

H. R. Myers, "Extending the Nuclear Test Ban," Scientific American 226, 13 (1972). A good review article on the present status of our abilities to detect underground nuclear explosions. No mathematics.

14

Applications to Medicine: Flow of the Blood and the Urinary Drop Spectrometer

Physics is love, engineering is marriage.

NORMAN MAILER

Of a Fire on the Moon

A. INTRODUCTION

T h r o u g h o u t t h e t e x t u p t o t h i s p o i n t , w e h a v e c o n c e r n e d o u r s e l v e s w i t h

m o r e o r l e s s " c o n v e n t i o n a l " a p p l i c a t i o n s of t h e p h y s i c s of f lu ids a n d

s o l i d s t o a r e a s of b a s i c r e s e a r c h in s u c h f ie lds a s a s t r o n o m y a n d

g e o p h y s i c s , t o u c h i n g o n l y br ie f ly a n d o c c a s i o n a l l y o n t o p i c s w h i c h m i g h t

b e c o n s i d e r e d " a p p l i e d p h y s i c s . " Y e t it i s c l e a r t h a t w i t h s o m u c h of t h e

w o r l d a r o u n d u s c o m p o s e d of m a t e r i a l s w h i c h a r e a p p r o x i m a t e l y i d e a l

f luids o r s o l i d s , o n e of t h e p r i m e r e a s o n s f o r s t u d y i n g t h e s u b j e c t s in t h i s

t e x t i s in o r d e r t o b e a b l e t o a p p l y t h e s i m p l e p h y s i c a l p r i n c i p l e s w h i c h w e

h a v e l e a r n e d t o r e a l s i t u a t i o n s .

P e r h a p s in n o a r e a is t h i s m o r e t r u e t h a n in t h e a r e a of t h e a p p l i c a t i o n s

of p h y s i c s t o m e d i c i n e a n d t o a n u n d e r s t a n d i n g of t h e h u m a n b o d y . T h e

b o d y i s , a f t e r a l l , a s y s t e m w h i c h o p e r a t e s a c c o r d i n g t o t h e s a m e p h y s i c a l

l a w s a s d o o t h e r n a t u r a l s y s t e m s . T h e r e a r e m a n y p a r t s of t h e b o d y w h e r e

it s e e m s o b v i o u s t h a t a s i m p l e p h y s i c a l m o d e l w o u l d e x p l a i n a g r e a t d e a l

of t h e o b s e r v e d b e h a v i o r . T h e s k e l e t o n , f o r e x a m p l e , c a n b e t h o u g h t of a s

a s t r u c t u r a l s y s t e m in w h i c h e x t e r n a l l o a d s a r e c o u n t e r a c t e d b y i n t e r n a l l y

g e n e r a t e d f o r c e s , j u s t a s w a s t h e c a s e f o r m o u n t a i n c h a i n s in C h a p t e r 1 1 .

248

Introduction 249

T h e r e a r e m a n y fluid s y s t e m s in t h e b o d y , t h e m o s t o b v i o u s of w h i c h is

t h e c i r c u l a t o r y s y s t e m . B u t e v e n a t t h e c e l l u l a r l e v e l c l a s s i c a l p r o c e s s e s of

o s m o s i s a n d d i f fus ion t h r o u g h m e m b r a n e s a r e e x t r e m e l y i m p o r t a n t .

I n t h i s c h a p t e r , w e sha l l d i s c u s s s o m e s i m p l e m o d e l s f o r t w o p h y s i c a l

s y s t e m s : T h e flow of b l o o d t h r o u g h a n a r t e r y , a n d t h e b e h a v i o r of t h e

e x t e r n a l u r i n e s t r e a m . T h e first of t h e s e is a n o ld p r o b l e m w h i c h h a s

r e c e i v e d a g r e a t d e a l of a t t e n t i o n in t h e p a s t , w h i l e t h e s e c o n d r e p r e s e n t s

a r e l a t i v e l y n e w a p p l i c a t i o n of p h y s i c a l r e a s o n i n g t o d i a g n o s t i c m e d i c i n e .

T h e c i r c u l a t o r y s y s t e m c a n , w i t h a g r e a t d e a l of o v e r s i m p l i f i c a t i o n , b e

c o n s i d e r e d a s s h o w n in F i g . 14 .1 .

capillaries

Fig. 14.1. A schematic view of the circulatory system.

T h e b l o o d is p u m p e d f r o m t h e h e a r t a n d l u n g s t h r o u g h a s y s t e m of

b r a n c h i n g a r t e r i e s , w h o s e s i z e d i m i n i s h e s w i t h d i s t a n c e f r o m t h e h e a r t .

E v e n t u a l l y , it f l ows t h r o u g h t h e n e t w o r k of c a p i l l a r i e s a n d b a c k i n t o t h e

v e n o u s s y s t e m , w h i c h r e t u r n s it t o t h e h e a r t a n d l u n g s .

T h e b a s i c p r o b l e m of b l o o d f low c a n b e s t a t e d a s f o l l o w s : G i v e n t h e

t i m e d e p e n d e n c e of t h e p r e s s u r e a n d t h e f low a t t h e e x i t of t h e h e a r t , a n d

g i v e n t h e c o m p o s i t i o n a n d l a y o u t of t h e a r t e r i a l a n d v e n o u s s y s t e m s , w h a t

wil l t h e flow a n d p r e s s u r e b e a t a n y p o i n t in t h e b o d y ? T h i s is a n

e x t r e m e l y c o m p l i c a t e d p r o b l e m , a n d w e a r e a l o n g w a y f r o m b e i n g a b l e t o

d e s c r i b e t h e c i r c u l a t o r y s y s t e m m a t h e m a t i c a l l y . P e r h a p s a f e w r e m a r k s

a b o u t t h e c o m p l e x i t y of t h e s y s t e m wil l h e l p t h e r e a d e r t o u n d e r s t a n d

w h y .

T h e first p r o b l e m is t h e n a t u r e of b l o o d i tself . S t r i c t l y s p e a k i n g , it is n o t

a fluid in t h e c l a s s i c a l s e n s e in w h i c h w e h a v e u s e d t h e t e r m u p t o t h i s

250 Applications to Medicine

p o i n t , b u t is a s u s p e n s i o n of s m a l l p a r t i c l e s in a fluid ( k n o w n a s t h e

p l a s m a ) . T h e m o s t i m p o r t a n t of t h e s e p a r t i c l e s f r o m t h e p o i n t of v i e w of

t h e c i r c u l a t i o n a r e t h e r e d b l o o d c e l l s , w h i c h a r e r o u g h l y t h e s h a p e of a

d o u g h n u t w i t h t h e c e n t e r p a r t i a l l y filled in , a n d a r e t y p i c a l l y a b o u t 7

m i c r o n s a c r o s s (1 m i c r o n = 1 0 ~ 4 c m ) . W h e n w e a r e d e a l i n g w i t h a r t e r i e s ,

w h o s e d i m e n s i o n s a r e t y p i c a l l y in t h e m i l l i m e t e r r a n g e , t h i s is n o t t o o

i m p o r t a n t a n e f fec t , b u t 5 - 1 0 fx is t h e s i z e of a t y p i c a l c a p i l l a r y . T h i s

m e a n s t h a t f low in t h e a r t e r y will b e q u i t e d i f f e r en t in c h a r a c t e r f r o m t h a t

in a c a p i l l a r y . I n t h e f o r m e r , t h e s i z e of t h e v e s s e l is v e r y l a r g e c o m p a r e d

t o t h e s i z e of t h e c e l l s , s o t h a t it is r e a s o n a b l e t o t r e a t t h e b l o o d a s a

c l a s s i c a l fluid. I n c a p i l l a r i e s , h o w e v e r , t h e ce l l s m u s t g o t h r o u g h o n e a t a

t i m e ( t h e p r o c e s s is s i m i l a r t o p u s h i n g a c o r k t h r o u g h a b o t t l e n e c k ) . I n

v e s s e l s of i n t e r m e d i a t e s i z e , l i ke t h e a r t e r i o l e s , t h e p r o b l e m is e v e n m o r e

c o m p l e x .

E v e n if w e r e s t r i c t o u r a t t e n t i o n t o t h e a r t e r i e s , w e i m m e d i a t e l y

e n c o u n t e r diff icul t ies w h i c h w e h a v e n o t r u n i n t o b e f o r e . W e h a v e a l w a y s

a r g u e d t h a t it is a g o o d a p p r o x i m a t i o n t o t r e a t l i q u i d s a s i n c o m p r e s s i b l e ,

s o t h a t t h e e q u a t i o n of c o n t i n u i t y t a k e s o n a p a r t i c u l a r l y s i m p l e f o r m . I n

a d d i t i o n , w e h a v e a l w a y s b e e n a b l e t o a s s u m e t h a t t h e coef f ic ien t of

v i s c o s i t y of a fluid, a s d e f i n e d in E q . (8 .A.9) w a s a c o n s t a n t , i n d e p e n d e n t

of t h e m o t i o n of t h e fluid. B e c a u s e of i t s p e c u l i a r c o m p o s i t i o n , n e i t h e r of

t h e s e a s s u m p t i o n s is t r u e f o r b l o o d . I t i s , in f a c t , a r e l a t i v e l y c o m p r e s s i b l e

fluid, a n d i t s coef f ic ien t of v i s c o s i t y d e p e n d s m a r k e d l y o n t h e v e l o c i t y .

T h i s m e a n s t h a t t h e N a v i e r - S t o k e s e q u a t i o n b e c o m e s e x t r e m e l y c o m p l i -

c a t e d e v e n if w e c a n t r e a t b l o o d a s a c l a s s i c a l f luid, a n d e x p l a i n s t h e

r e l a t i v e l y p r i m i t i v e s t a t e of t h e t h e o r y of b l o o d f low.

A s e c o n d i m p o r t a n t c o m p l i c a t i o n in t h e p r o b l e m of b l o o d f low is t h e

f a c t t h a t t h e b o u n d a r y c o n d i t i o n s a r e n o l o n g e r of t h e s i m p l e f o r m w e

h a v e g r o w n a c c u s t o m e d t o . T h e w a l l s of t h e a r t e r i e s a r e n o t r ig id , b u t a r e

in f a c t d e f o r m a b l e s o l i d s . T h u s , w h e n a p u l s e c o m e s d o w n t h e a r t e r y , t h e

w a l l s e x p a n d . N o r is t h e a r t e r i a l w a l l n e c e s s a r i l y of t h e s i m p l e t y p e w h i c h

c a n b e d e s c r i b e d b y H o o k e ' s l a w f o r a n e l a s t i c m a t e r i a l . I n f a c t , t h e

a r t e r i a l w a l l is c o m p o s e d of a r a t h e r c o m p l i c a t e d m a t e r i a l w h o s e

p r o p e r t i e s u n d e r s t r e s s fall i n t o t h e g e n e r a l c l a s s of m a t e r i a l s c a l l e d

viscoelastic. T h i s m e a n s t h a t t h e r e s p o n s e t o a n a p p l i e d f o r c e d e p e n d s o n

t h e r a t e a t w h i c h t h a t f o r c e is a p p l i e d , a s w o u l d b e a p p r o p r i a t e fo r a

N e w t o n i a n so l id ( s e e S e c t i o n 12 .C) , a s w e l l a s t h e u s u a l r e s t o r i n g f o r c e

w h i c h is p r o p o r t i o n a l t o t h e m a g n i t u d e of t h e a p p l i e d f o r c e . I n a d d i t i o n , a t

l a r g e d e f o r m a t i o n s , t h e s t r u c t u r e of t h e a r t e r i a l w a l l i t se l f c o m e s i n t o

p l a y . I t is c o m p o s e d of t w o s u b s t a n c e s , e l a s t i n a n d c o l l a g e n (a t h i r d

Introduction 251

elastin

' \ col lagen fibers

Fig. 14.2(a). The arterial wall at rest.

s t r u c t u r a l c o m p o n e n t , s m o o t h m u s c l e , i s n o t t h o u g h t t o h a v e m u c h ef fec t

o n t h e e l a s t i c p r o p e r t i e s of t h e w a l l ) . T h e e l a s t i n is a r u b b e r y , e x t e n s i b l e

m a t e r i a l , w h i l e t h e c o l l a g e n is m o r e l i ke a fiber w h i c h h a s a h i g h r e s i s t a n c e

t o d e f o r m a t i o n . T h e c o l l a g e n is s t r u n g v e r y l o o s e l y in t h e w a l l , w i t h a lo t

of s l a c k ( s e e F i g . 14.2(a)) , s o t h a t f o r s m a l l d e f o r m a t i o n s , it h a s n o e f fec t

o n t h e w a l l s . W h e n t h e w a l l is s t r e s s e d s o t h a t t h e s l a c k is t a k e n u p ,

h o w e v e r , w e h a v e t h e s i t u a t i o n in F i g . 14 .2(b) , in w h i c h t h e c o l l a g e n n o w

t a k e s o v e r a n d p r e v e n t s f u r t h e r d e f o r m a t i o n of t h e a r t e r y . T h e b i o l o g i c a l

u s e f u l n e s s of s u c h a s y s t e m is o b v i o u s , b u t e q u a l l y o b v i o u s is t h e f a c t t h a t

s u c h a s t r u c t u r e is e x t r e m e l y difficult t o d e s c r i b e m a t h e m a t i c a l l y .

Fig. 14.2(b). The arterial wall under tension.

N e v e r t h e l e s s , it is t h e j o b of t h e s c i e n t i s t t o d e a l w i t h c o m p l i c a t e d

s y s t e m s w h e n t h e y o c c u r in n a t u r e . T h e g e n e r a l l i ne of a t t a c k w h i c h is

u s u a l l y f o l l o w e d is t o m a k e a s e r i e s of a p p r o x i m a t i o n s w h i c h s imp l i fy t h e

p r o b l e m t o t h e p o i n t w h e r e it is m a t h e m a t i c a l l y t r a c t a b l e , a n d t h e n h o p e

t h a t t h e s o l u t i o n w h i c h is o b t a i n e d h a s t h e m a i n f e a t u r e s of t h e s y s t e m

w h i c h w e a r e t r y i n g t o d e s c r i b e .

W e sha l l f o l l o w t h i s l i ne in d e a l i n g w i t h t h e p r o b l e m of b l o o d f low in t h e

a r t e r i e s . W e sha l l b e g i n b y d i s c u s s i n g t h e r e s p o n s e of a n e l a s t i c a r t e r i a l


w a l l t o p r e s s u r e , b o t h s t a t i c a n d t i m e d e p e n d e n t , a n d t h e n t u r n t o t h e full

p r o b l e m .

B. RESPONSE OF ELASTIC ARTERIAL WALLS TO PRESSURE

W e sha l l b e g i n b y e x a m i n i n g a p r o b l e m w h o s e a p p l i c a b i l i t y t o t h e f low

of b l o o d in a n a r t e r y is o b v i o u s . C o n s i d e r a c y l i n d r i c a l e l a s t i c t u b e w h i c h

c o n t a i n s s o m e fluid w h o s e p r e s s u r e ( n o t n e c e s s a r i l y c o n s t a n t ) is k n o w n .

F o r t h e m o m e n t , w e a s s u m e t h e p r e s s u r e d o e s n o t v a r y a l o n g t h e l e n g t h

of t h e t u b e . H o w d o e s t h e t u b e r e s p o n d ?

L e t u s a s s u m e t h a t t h e r e i s a z i m u t h a l s y m m e t r y , a n d c o n s i d e r o n e

in f in i t e s ima l v o l u m e e l e m e n t in t h e m a t e r i a l . T h e s t r e s s e s w h i c h a c t o n

t h i s e l e m e n t a r e s h o w n in F i g . 14 .3 . If w e c o m p u t e t h e f o r c e s in t h e

r - d i r e c t i o n , w e h a v e

w h e r e dz i s t h e h e i g h t of t h e v o l u m e e l e m e n t . T h e e q u a t i o n f o l l o w s f r o m

t h e de f in i t ion of t h e s t r e s s t e n s o r a s a f o r c e p e r u n i t a r e a . F r o m N e w t o n ' s

s e c o n d l a w , t h i s m u s t b e

Fig. 14.3. A section of the arterial wall.

Fr = - Wrrir dO dz) + (cTrr + darr)((r + dr) dd dz)

(14 .B.1)

Fr = pr dr dd dz (14 .B.2)

w h e r e p is t h e d e n s i t y of t h e m a t e r i a l . T h e e q u a t i o n of m o t i o n in t h e

Response of Elastic Arterial Walls to Pressure 253

r - d i r e c t i o n is t h e n

( 1 4 . B . 3 )

w h i c h is j u s t N e w t o n ' s s e c o n d l a w in c y l i n d r i c a l c o o r d i n a t e s .

I n o r d e r t o p r o c e e d f r o m t h i s p o i n t , w e m u s t k n o w s o m e t h i n g a b o u t t h e

n a t u r e of t h e m a t e r i a l in t h e a r t e r i a l w a l l . T h i s w a s d i s c u s s e d in S e c t i o n

1 4 . A . F o r t h e m a t h e m a t i c a l p r o b l e m s w h i c h f o l l o w , w e sha l l m a k e t w o

a s s u m p t i o n s a b o u t a r t e r i a l w a l l s . F i r s t , w e sha l l a s s u m e t h a t t h e y a r e

e l a s t i c , a n d h e n c e o b e y H o o k e ' s l a w . S e c o n d , w e sha l l a s s u m e t h a t t h e y

a r e c o m p o s e d of a n i n c o m p r e s s i b l e m a t e r i a l , s o t h a t

V • u = 0 .

W e sha l l d i s c u s s t h e s e a n d o t h e r a s s u m p t i o n s in m o r e d e t a i l l a t e r . F o r t h e

m o m e n t , w e n o t e s i m p l y t h a t if t h e y a r e t r u e , H o o k e ' s l a w t a k e s t h e f o r m

( s e e P r o b l e m 1 2 . 5 )

( 1 4 . B . 4 )

s o t h a t t h e e q u a t i o n of m o t i o n b e c o m e s

( 1 4 . B . 5 )

w h i c h c a n b e s o l v e d f o r ur.

L e t u s w o r k o u t a c o u p l e of e x a m p l e s t o s e e h o w t h e t u b e r e s p o n d s . F o r

t h e f irs t , l e t u s t a k e t h e s i m p l e s t p o s s i b l e c a s e — t h e c a s e w h e r e t h e

i n t e r n a l p r e s s u r e is a c o n s t a n t . T h e n t h e l e f t - h a n d s i d e of E q . ( 1 4 . B . 5 ) wi l l

v a n i s h , a n d w e wi l l h a v e a s e c o n d - o r d e r e q u a t i o n f o r ur. If Ri i s t h e i n n e r

r a d i u s of t h e t u b e , a n d JR d t h e o u t e r , t h e n t h e t w o b o u n d a r y c o n d i t i o n s

w h i c h m u s t b e sa t i s f ied a r e

crrr = - P ( 1 4 . B . 6 )

a t r = Rh a n d

crrr = 0 ( 1 4 . B . 7 )

a t r = R0. If w e g u e s s a s o l u t i o n f o r ur of t h e f o r m

ur oc r

l


a n d s u b s t i t u t e b a c k i n t o E q . (14 .B .5 ) , w e find / = ± 1 , s o t h a t t h e m o s t

g e n e r a l s o l u t i o n f o r ur is j u s t

w h i l e E q . (14 .B.6) y i e l d s

s o t h a t t h e s o l u t i o n is

ur

N o w if t h e t u b e is t h i n , s o t h a t

R0 — Ri + 5,

w e c a n e x p a n d t h e e x p r e s s i o n in t h e d e n o m i n a t o r a n d g e t

PRt = Scree = T m , (14 .B.13)

w h e r e w e h a v e d e f i n e d Tm a s t h e membrane tension. T h i s r e s u l t ° h o u l d

l o o k v e r y f ami l i a r . I t is e x a c t l y t h e r e l a t i o n b e t w e e n t h e p r e s s u r e a n d

r a d i u s of c u r v a t u r e w h i c h w a s o b t a i n e d in C h a p t e r 5 f o r a fluid w i t h

s u r f a c e t e n s i o n . T h u s , a v e r y t h i n m e m b r a n e c a n b e t h o u g h t of in t h e

s a m e w a y a s s u r f a c e t e n s i o n — a s a c o m p o n e n t of t h e s y s t e m w h i c h t e n d s

t o o p p o s e i n c r e a s e s in s u r f a c e a r e a .

I t is m o r e u s u a l t o b e c o n c e r n e d w i t h t i m e - d e p e n d e n t p r e s s u r e w h e n

d e a l i n g w i t h b l o o d flow. A f t e r al l , t h e m o v i n g f o r c e b e h i n d t h e flow is t h e

p e r i o d i c p u m p i n g of t h e h e a r t . H o w w o u l d t h e a b o v e a n a l y s i s b e c h a n g e d

if t h e p r e s s u r e , i n s t e a d of b e i n g c o n s t a n t , w e r e t i m e d e p e n d e n t ? I n t h i s

c a s e , w e w o u l d h a v e t o g u e s s a t a s o l u t i o n of E q . (14 .B.5) of t h e f o r m

(14 .B.10)

(14 .B.11)

(14 .B.8)

If w e n o w c o m b i n e o u r de f in i t ion of t h e s t r e s s t e n s o r [ E q . (14 .B.4) ] w i t h

E q . (14 .B .7 ) , w e find

(14 .B.9)

T h e r e is a n i n t e r e s t i n g s ide l igh t t o t h i s r e s u l t . S u p p o s e w e n o w a s k fo r

t h e v a l u e of t h e s t r e s s w h i c h is e x e r t e d a x i a l l y a r o u n d t h e t u b e a t i t s o u t e r

b o u n d a r y . T h i s is j u s t

(14 .B.12)

Response of Elastic Arterial Walls to Pressure 255

ur=R(r)T(t), (14 .B.14)

w h e r e R is a f u n c t i o n of r a l o n e , a n d T a f u n c t i o n of t. I n s e r t i n g t h i s i n t o

E q . ( 14 .B .5 ) , w e g e t

w h e r e o)n = lirnlr, a n d r is t h e p e r i o d of t h e p u l s e .

(14 .B .15)

(14 .B.16)

w h e r e A is a c o n s t a n t . T h e s o l u t i o n f o r T is t h e n

T(t)=T0eiKt,

w h i l e t h e e q u a t i o n f o r R i s j u s t

(14 .B.17)

w h i c h is r a t h e r c o m p l i c a t e d . H o w e v e r , w e c a n m a k e a v e r y r e a s o n a b l e

a p p r o x i m a t i o n if w e p u t in s o m e n u m b e r s w h i c h a r e t y p i c a l of b l o o d flow.

A t y p i c a l v a l u e f o r A, t h e f r e q u e n c y a s s o c i a t e d w i t h t h e p r e s s u r e , m i g h t

b e 7 r a d / s e c w h i l e v a l u e s of t h e o t h e r p a r a m e t e r s m i g h t b e p ~ 1.1 g / c c ,

R ~ 1 c m , a n d E ~ 10 6 d y n e s / c m . T h u s , w e s e e t h a t

s o t h a t t h e s e c o n d t e r m in p a r e n t h e s e s in E q . (14 .B .17) c a n b e d r o p p e d . I n

t h i s c a s e , E q . (14 .B .17) r e d u c e s t o t h e e q u a t i o n f o r ur w h i c h w e h a d in t h e

p r e v i o u s c a s e , s o t h a t E q s . (14 .B.8) a n d (14 .B .9) a r e a g a i n v a l i d . T h u s , t h e

m o s t g e n e r a l s o l u t i o n f o r ur wi l l b e

(14 .B .18)

w h e r e w e u n d e r s t a n d t h a t w e h a v e t a k e n a n a r b i t r a r y s u m of all p o s -

s ib le s o l u t i o n s , a n d t h e s u m m a t i o n is u n d e r s t o o d t o e x t e n d o v e r all

a l l o w e d v a l u e s of A. T h e s e v a l u e s wi l l b e d e t e r m i n e d b y t h e b o u n d a r y

c o n d i t i o n s .

E q u a t i o n (14 .B .6) still d e s c r i b e s t h e b o u n d a r y c o n d i t i o n a t t h e i n n e r

r a d i u s , b u t n o w t h e p r e s s u r e is a f u n c t i o n of t, a n d n o t a c o n s t a n t . I n

g e n e r a l , t h e p r e s s u r e wi l l b e s o m e t i m e - d e p e n d e n t f u n c t i o n . If w e F o u r i e r

a n a l y z e i t , w e c a n w r i t e


If w e n o w i m p o s e t h e b o u n d a r y c o n d i t i o n a t t h e i n n e r s u r f a c e , w e h a v e

(14 .B.19)

I n o r d e r t o s a t i s fy E q . (14 .B.19) f o r a n y v a l u e of t, w e m u s t h a v e

(14 .B.20)

s o t h a t , f o r o u r final s o l u t i o n , w e h a v e

(14 .B.21)

w h i c h is i d e n t i c a l t o E q . (14 .B .11) , e x c e p t t h a t n o w t h e p r e s s u r e is

u n d e r s t o o d t o b e t i m e d e p e n d e n t .

O n e i n t e r e s t i n g r e s u l t c a n b e s e e n i m m e d i a t e l y f r o m t h i s e q u a t i o n .

W h e n w e a r e d e a l i n g w i t h p h y s i o l o g i c a l s y s t e m s , w e o f t e n c a n n o t

m e a s u r e q u a n t i t i e s of d i r e c t i n t e r e s t , b u t m u s t i n fe r t h e m f r o m i n d i r e c t

m e a s u r e m e n t s . F o r e x a m p l e , it is o f t e n n o t c o n v e n i e n t t o m e a s u r e

p r e s s u r e i n s i d e of a n a r t e r y d i r e c t l y ( a l t h o u g h t h i s c a n b e e a s i l y d o n e ) a n d

o n e m i g h t w i s h t o k n o w t h e p r e s s u r e j u s t f r o m o b s e r v i n g t h e o u t e r w a l l of

t h e a r t e r y . H o w wil l it m o v e a s t h e p r e s s u r e is a p p l i e d ? If w e de f ine

ur(r = R0) a s t h e d i s t a n c e t h e o u t e r w a l l wi l l m o v e , t h e n

I n o t h e r w o r d s , f o r a p e r f e c t l y e l a s t i c a r t e r y w a l l , t h e o u t e r s u r f a c e wil l

m o v e in phase w i t h t h e i n t e r n a l p r e s s u r e . F o r a v i s c o e l a s t i c m a t e r i a l ,

h o w e v e r , t h i s wi l l n o t b e t h e c a s e , s i n c e t h e r e wi l l b e a t i m e lag w h i l e t h e

w a l l r e s p o n d s t o t h e c h a n g e s in p r e s s u r e w h i c h wi l l , in t u r n , b e r e f l e c t e d

b y a p h a s e l ag .

W i t h t h i s i n t r o d u c t i o n t o t h e b e h a v i o r of a r t e r i a l w a l l s , w e wil l t u r n t o

t h e p r o b l e m of d e s c r i b i n g t h e flow of b l o o d in a n a r t e r y .

C. BLOOD FLOW IN AN ARTERY

W e a r e n o w in a p o s i t i o n t o w r i t e d o w n t h e e q u a t i o n s w h i c h g o v e r n t h e

flow of b l o o d in a n a r t e r y . F o r t h e e q u a t i o n s w h i c h d e s c r i b e t h e b l o o d

i tself , of c o u r s e , w e h a v e t h e N a v i e r - S t o k e s e q u a t i o n

(14 .C.1)

Blood Flow in an Artery 257

a n d t h e e q u a t i o n of c o n t i n u i t y

w h e r e p w is t h e d e n s i t y of t h e m a t e r i a l s in t h e w a l l . p w wi l l , in g e n e r a l ,

s a t i s fy a c o n t i n u i t y e q u a t i o n l i ke E q . (14 .C .2 ) .

T h e s e e q u a t i o n s m u s t b e s o l v e d s u b j e c t t o b o u n d a r y c o n d i t i o n s w h i c h

w e sha l l d i s c u s s in d e t a i l l a t e r . A s t h e y s t a n d , t h e y a r e e x t r e m e l y difficult

t o s o l v e . T h e y a r e b a d l y n o n l i n e a r , a n d in E q . (14 .C .1 ) , t h e coef f ic ien t 17 i s ,

in g e n e r a l , a f u n c t i o n of t h e v e l o c i t y . N e v e r t h e l e s s , if w e w a n t t o f ind a

s i m p l e s o l u t i o n t o t h e p r o b l e m , w e wi l l h a v e t o m a k e s o m e a p p r o x i m a -

t i o n s . T h e first of t h e s e wi l l b e t o a s s u m e t h a t b l o o d is a c l a s s i c a l

i n c o m p r e s s i b l e N e w t o n i a n fluid, s o t h a t

TJ = c o n s t . (14 .C.5)

a n d

V • v = 0. (14 .C.6)

S e c o n d , w e wi l l a s s u m e t h a t t h e n o n l i n e a r t e r m in t h e N a v i e r - S t o k e s

e q u a t i o n c a n b e d r o p p e d . T h i s c o r r e s p o n d s t o a s s u m i n g t h a t t h e v i s c o u s

t e r m s a r e q u i t e l a r g e , s o t h a t

( v V ) v < ^ V 2 v . (14 .C.7) P

F i n a l l y , t h r o u g h o u t t h e s e c t i o n , w e wi l l a s s u m e t h a t t h e r e is c o m p l e t e

a z i m u t h a l s y m m e t r y , s o t h a t n o t h i n g d e p e n d s o n t h e c o o r d i n a t e a n g l e .

T h e e q u a t i o n s of m o t i o n f o r t h e fluid t h e n b e c o m e ( s e e P r o b l e m 14.1)

(14 .C.3)

(14 .C.4)

a n d

(14.C.8)

| e + V . ( p v ) = 0. (14 .C.2)

F o r t h e a r t e r i a l w a l l s , t h e e q u a t i o n s a r e s i m p l y t h e o b v i o u s g e n e r a l i z a -

t i o n s of E q . (14 .B .3 ) , s o t h a t

N o w E q . (14 .C.16) is of t h e f a m i l i a r f o r m w h o s e g e n e r a l s o l u t i o n is t h e

s u m of a p a r t i c u l a r a n d a h o m o g e n e o u s s o l u t i o n . T h e h o m o g e n e o u s


(14.C.9)

F o l l o w i n g t h e s t a n d a r d p r o c e d u r e o u t l i n e d in E q . (14 .B .14) , w e a s s u m e

t h a t t h e s o l u t i o n is of a s e p a r a b l e f o r m

P=R(r)Z(z), (14 .C.10)

a n d find t h a t

(14 .C.11)

w h e r e k is a n a r b i t r a r y c o n s t a n t . W e t h e n s e e i m m e d i a t e l y t h a t

Z(z) = e i k \ (14 .C .12)

a n d a r e lef t w i t h a n e q u a t i o n f o r R of t h e f o r m

(14 .C.13)

w h i c h is j u s t B e s s e l ' s e q u a t i o n of o r d e r z e r o ( th i s c a n b e s e e n b y w r i t i n g

- k 2 = (ikf).

T h u s , t h e p r e s s u r e is g i v e n b y

P ( r , z, t) = AJ0(ikr)eikzeito\ (14 .C.14)

w h e r e A is a n a r b i t r a r y c o n s t a n t , a n d w e h a v e f o l l o w e d t h e p r o c e d u r e

o u t l i n e d in S e c t i o n 14.B a n d a s s u m e d t h a t all t i m e d e p e n d e n c e s a r e of t h e

f o r m eitdt.

W e c a n t h e n p u t t h i s s o l u t i o n f o r P b a c k i n t o E q . (14 .C.8) a n d s o l v e f o r

vz. If w e a s s u m e a f o r m f o r vz s u c h a s

vz = vz(r)eikzei(°\ (14 .C.15)

t h e n w e find t h e e q u a t i o n g o v e r n i n g vz(r) t o b e

(14 .C.16)

(14 .C.17)

w h e r e w e h a v e de f ined

T h i s l e a d s u s t o a n e q u a t i o n f o r t h e p r e s s u r e ( s e e P r o b l e m 14.2) w h i c h

is of t h e f o r m


T h u s , w e h a v e s o l v e d E q s . (14 .C .1) a n d (14 .C .2) w h i c h d e s c r i b e t h e

m o t i o n of t h e fluid in t h e a r t e r y , s u b j e c t t o t h e a p p r o x i m a t i o n s in E q s .

(14 .C .5 ) , ( 14 .C .6 ) , a n d (14 .C .7 ) . L e t u s e x a m i n e t h e e q u a t i o n s of m o t i o n

f o r t h e w a l l s b e f o r e w e s t a r t d i s c u s s i n g t h e b o u n d a r y c o n d i t i o n s . A s w e

d i d f o r t h e fluid, w e wi l l m a k e s o m e a p p r o x i m a t i o n s t o s imp l i fy o u r w o r k .

W e sha l l a s s u m e t h a t w e a r e d e a l i n g w i t h a p u r e l y e l a s t i c so l i d w h i c h

o b e y s H o o k e ' s l a w , a n d t h a t t h e so l i d is i n c o m p r e s s i b l e , s o t h a t

V • u = 0 . (14 .C .20)

I n t h i s c a s e , c o m p o n e n t s of t h e s t r e s s t e n s o r w h i c h w e n e e d a r e g i v e n

b y

(14 .C.21)

(14 .C.19)

I t m u s t b e e m p h a s i z e d t h a t in t h i s e x p r e s s i o n , o n l y C is u n k n o w n . T h e

c o n s t a n t A wi l l b e d e t e r m i n e d b y t h e b o u n d a r y c o n d i t i o n s o n t h e

p r e s s u r e a t z = 0.

F o l l o w i n g s i m i l a r s t e p s , it is s h o w n in P r o b l e m 14.3 t h a t t h e r a d i a l

v e l o c i t y is j u s t

(14 .C.18)

s o t h a t t h e m o s t g e n e r a l f o r m f o r vz is j u s t

t h e n B c a n b e d e t e r m i n e d b y p l u g g i n g b a c k i n t o E q . (14 .C .8) t o b e

s o l u t i o n , in a n a l o g y t o t h e s o l u t i o n t o E q . (14 .C.9) is s i m p l y

vz

h = J0(iyr).

If w e g u e s s t h a t t h e f o r m of t h e p a r t i c u l a r s o l u t i o n is

vz

p = BJ0(ii<r),


(14 .C.22)

(14 .C.23)

T h e s e e q u a t i o n s a r e still v e r y c o m p l i c a t e d b e c a u s e t h e y a r e c o u p l e d .

A l t h o u g h t h e y c o u l d b e s o l v e d n u m e r i c a l l y , w e wi l l l o o k fo r f u r t h e r

a p p r o x i m a t i o n s w h i c h m i g h t g i v e u s a n e a s y s o l u t i o n .

W e k n o w f r o m S e c t i o n 14 .B t h a t w h e n a p u l s e m o v e s d o w n t h e a r t e r y ,

t h e a r t e r y wi l l e x p a n d a n d c o n t r a c t . I t s e e m s r e a s o n a b l e t o a s s u m e t h a t

m o s t of t h i s m o t i o n is in t h e r a d i a l d i r e c t i o n , a n d t h a t t h e s t r e t c h i n g of t h e

a r t e r y in t h e z - d i r e c t i o n is p r o b a b l y l e s s p r o n o u n c e d . W e wil l t a k e t h i s

p h y s i c a l i d e a t o i t s e x t r e m e , a n d a s s u m e t h a t

uz ur. (14 .C.24)

I t m u s t b e n o t e d t h a t t h i s a p p r o x i m a t i o n , w h i l e it d o e s s imp l i fy t h e

e q u a t i o n s of m o t i o n , d o e s a c e r t a i n a m o u n t of v i o l e n c e t o o u r in i t ia l

a s s u m p t i o n s , s i n c e it a s s u m e s t h a t t h e r e is a n a n i s o t r o p y in t h e a r t e r i a l

w a l l w h i c h r e s t r i c t s m o t i o n in t h e z - d i r e c t i o n . B e c a u s e of t h i s , w e wil l

d r o p c o n s i d e r a t i o n s of uz f r o m t h i s p o i n t o n .

W i t h E q . (14 .C .24) , t h e e q u a t i o n of m o t i o n b e c o m e s

W i t h t h e s e s o l u t i o n s , w e m u s t t u r n t o t h e b o u n d a r y c o n d i t i o n s . W e wil l

s t a r t w i t h t h o s e w h i c h m u s t b e i m p o s e d o n t h e fluid. W e k n o w f r o m

s y m m e t r y t h a t a t r = 0 w e m u s t h a v e

s o t h a t t h e e q u a t i o n s of m o t i o n a r e

(14 .C.25)

w h i c h h a s t h e s o l u t i o n

ur(r, z, 0 = FJx(iYr)eikzeiu>\ (14 .C.26)


(14 .C.27)


Fig. 14.4. The arterial wall during pulsatile movement.

(14 .C.28)

s i n c e t h e fluid m a y n o t f low a w a y f r o m t h e c e n t e r . T h e s e c o n d i t i o n s a r e

a u t o m a t i c a l l y sa t i s f ied b y E q s . (14 .C.18) a n d (14 .C .19) . T h e o t h e r

b o u n d a r y c o n d i t i o n s c o n c e r n t h e i n n e r s u r f a c e of t h e a r t e r y ( s e e F i g .

14.4).

T h e g e n e r a l b o u n d a r y c o n d i t i o n h e r e is t h a t t h e r e l a t i v e m o t i o n

b e t w e e n t h e fluid a n d t h e w a l l m u s t v a n i s h a t t h i s s u r f a c e in k e e p i n g w i t h

o u r i d e a s a b o u t t h e n a t u r e of v i s c o s i t y . T h u s , a t r = r i n , w e m u s t h a v e

(14 .C .29a )

(14 .C .29b )

A t t h i s s u r f a c e , w e m u s t a l s o h a v e a s i t u a t i o n w h e r e t h e s t r e s s e s a r e

c o n t i n u o u s . T h e r e a d e r m a y c o m p a r e t h i s t o t h e b o u n d a r y c o n d i t i o n

w h i c h w a s i m p o s e d in t h e d e r i v a t i o n of L o v e w a v e s in S e c t i o n 1 2 . F . T h e

s h e e r s t r e s s a l o n g a s u r f a c e o n t h e i n n e r f a c e of t h e a r t e r y e x e r t e d b y t h e

wa l l is j u s t

s o t h a t w e m u s t h a v e

(14 .C .32)

V r ( r = 0) = 0,

(14 .C .30)

w h i l e in P r o b l e m 14.4, w e s h o w t h a t t h e s t r e s s e x e r t e d b y t h e fluid is j u s t

(14 .C.31)

r0ut(z, f)


a t r = r i n . A s imi l a r a r g u m e n t f o r s t r e s s e s in t h e r a d i a l d i r e c t i o n y i e l d s t h e

r e s u l t t h a t

A t t h e o u t e r b o u n d a r y , t h e s t r e s s e s e x e r t e d b y t h e a r t e r i a l w a l l m u s t b e

c o n t i n u o u s w i t h t h o s e e x e r t e d b y t h e s u r r o u n d i n g m e d i u m , a n d m u s t

v a n i s h if t h a t m e d i u m is a v a c u u m ( s e e P r o b l e m 14.5).

W e n o w c o m e t o t h e t h i r d i m p o r t a n t s e t of c o m p l i c a t i o n s in o u r t h e o r y .

I n g e n e r a l , t h e d e f o r m a t i o n of t h e a r t e r y is n o t s m a l l c o m p a r e d t o i t s

r a d i u s , s o t h e full t i m e d e p e n d e n c e of r i n a n d r o u t m u s t b e i n c l u d e d in

w r i t i n g d o w n t h e b o u n d a r y c o n d i t i o n s . I n a d d i t i o n , t h e g e n e r a l s h a p e of

a r t e r i e s n e e d n o t b e c y l i n d e r s of c o n s t a n t c r o s s s e c t i o n . I n f a c t , t h e a o r t a ,

t h e a r t e r y l e a d i n g a w a y f r o m t h e h e a r t , h a s a g e n e r a l s h a p e l i ke t h a t

s h o w n in F i g . 14.5 . T h e r e f o r e , in o r d e r t o a p p l y t h e b o u n d a r y c o n d i t i o n s

e a s i l y , w e sha l l a s s u m e t h a t f o r t h e a r t e r y in q u e s t i o n ,

w h e r e a is c o n s t a n t , a n d t h a t t h e d e f o r m a t i o n s a r e s m a l l , s o t h a t

W e a r e n o w in a p o s i t i o n t o a p p l y t h e b o u n d a r y c o n d i t i o n s . I n g e n e r a l ,

w e w i s h t o t r e a t t h e c o n s t a n t A w h i c h a p p e a r s in E q . (14 .C.14) a s g i v e n b y

t h e in i t ia l c o n d i t i o n s ( a s w e d i d in S e c t i o n 14 .B) , s o t h a t w e w a n t t o

d e t e r m i n e t h e c o n s t a n t s C , D , a n d F f r o m E q s . (14 .C .18 ) , (14 .C .19) , a n d

(14 .C .26) .

F r o m t h e c o n d i t i o n i n E q . ( 1 4 . C . 2 9 a ) , t o g e t h e r w i t h o u r a p p r o x i m a t i o n

of i g n o r i n g uz, w e h a v e

(14 .C.33)

r f a = a + £

(14 .C.34)

vz(r = a ) = 0, (14 .C.35)

heart

Fig. 14.5. A typical shape of an aorta.


w h i c h g i v e s

(14 .C.36)

a n d d e t e r m i n e s o n e of t h e t h r e e c o n s t a n t s .

E q u a t i o n (14 .C .29b ) g i v e s

AJi(ika) + DJi(iya) = ia>FJi(Ta), (14 .C.37)

w h i c h g i v e s o n e r e l a t i o n b e t w e e n t h e o t h e r t w o c o n s t a n t s . T h e t h i r d

r e l a t i o n c o m e s f r o m t h e s t r e s s c o n d i t i o n in E q . (14 .C .33) .

w h e r e t h e p r i m e d e n o t e s d i f f e r e n t i a t i o n of t h e B e s s e l f u n c t i o n w i t h

r e s p e c t t o i t s a r g u m e n t .

T h u s , b y m a k i n g a l a r g e n u m b e r of a p p r o x i m a t i o n s , w e w e r e a b l e t o

c o m e t o a s o l u t i o n of t h e g e n e r a l p r o b l e m . I t s h o u l d b e o b v i o u s t o t h e

r e a d e r t h a t n o n e of t h e s e a p p r o x i m a t i o n s s t a n d s o n v e r y f i rm p h y s i c a l

g r o u n d s , s o t h a t in r e a l i s t i c w o r k , t h e y w o u l d h a v e t o b e e x a m i n e d v e r y

c a r e f u l l y . H o w e v e r , it is h o p e d t h a t w o r k i n g t h r o u g h t h e s i m p l e s t p o s s i b l e

c a s e of a r t e r i a l f low h a s d e m o n s t r a t e d t h e t e c h n i q u e s w h i c h m u s t b e

e m p l o y e d t o l ink u p t h e s t u d i e s of fluid m e c h a n i c s w h i c h w e r e t r e a t e d in

t h e first p a r t of t h e t e x t a n d t h e s t u d i e s of e l a s t i c so l i d s w h i c h w e r e

d i s c u s s e d in t h e s e c o n d . T h e w o r k i n g o u t of a n a c t u a l e x a m p l e is lef t t o

P r o b l e m 14.6.

T h e p r o b l e m of t h e f low in t h e c i r c u l a t o r y s y s t e m is r e c e i v i n g a g r e a t

d e a l of a t t e n t i o n in c u r r e n t r e s e a r c h . M o s t of t h e w o r k t h a t is d o n e is

l og ica l ly q u i t e s i m i l a r t o w h a t h a s b e e n d i s c u s s e d in t h i s s e c t i o n . T h e

g e n e r a l t e c h n i q u e is t o m a k e a s m a n y a p p r o x i m a t i o n s a s o n e c a n , t r y i n g

a l w a y s t o t r e a t t h e q u a n t i t i e s of i n t e r e s t a s e x a c t l y a s p o s s i b l e . W e

m e n t i o n a f e w e x a m p l e s t o i l l u s t r a t e t h i s p o i n t .

(1) Arteriosclerosis. T h i s is t h e p r o b l e m of t h e f low of b l o o d t h r o u g h

a n a r t e r y w h i c h c a n b e p a r t i a l l y o b s t r u c t e d b y d e p o s i t s . I n t h i s c a s e , t h e

a s s u m p t i o n t h a t a r t e r i a l w a l l s w e r e a l m o s t c i r c u l a r c y l i n d e r s of c o n s t a n t

c r o s s s e c t i o n s [ E q . (14 .C .34) ] w o u l d n o l o n g e r b e u s e f u l , s i n c e w e w i s h t o

s t u d y t h e e f fec t of c h a n g e s in a r t e r i a l c r o s s s e c t i o n o n t h e f low. W h a t is

u s u a l l y d o n e in t h i s c a s e is t o a s s u m e t h a t t h e w a l l s a r e r ig id , b u t of a

d e f o r m e d s h a p e , a n d t h e n t r y t o p r o c e e d a s r e a l i s t i c a l l y a s p o s s i b l e .

J\(ika) + iyDJ\(iya) = 2iEFYJ\(Ya) (14 .C.38)


(2) The Entry Problem. T h i s i s t h e p r o b l e m w h i c h is c o n c e r n e d w i t h

t h e w a y t h e v e l o c i t y prof i le d e v e l o p s f r o m t h e p o i n t a t w h i c h t h e b l o o d

e n t e r s (e .g . , a t t h e h e a r t ) u n t i l it is fu l ly d e v e l o p e d . I n t h i s c a s e , t h e

p r o c e d u r e of d r o p p i n g t h e n o n l i n e a r t e r m s [ E q . (14 .C.6) ] wi l l n o t b e

u s e f u l , s i n c e w e a r e t r y i n g t o e x a m i n e c h a n g e s in t h e v e l o c i t y i tself . I n

t h i s p r o b l e m , o n e u s u a l l y k e e p s t h e n o n l i n e a r t e r m s , a n d k e e p s t h e

a p p r o x i m a t i o n t h a t t h e a r t e r i a l w a l l s a r e r ig id a n d of u n i f o r m c r o s s

s e c t i o n .

T h e p o i n t of t h e s e e x a m p l e s is t h a t e v e n t h o u g h t h e g e n e r a l p r o b l e m of

flow in t h e c i r c u l a t o r y s y s t e m is t o o c o m p l i c a t e d t o s o l v e w i t h p r e s e n t

t e c h n i q u e s , a g r e a t d e a l of p r o g r e s s c a n b e m a d e in i s o l a t i n g i n d i v i d u a l

a s p e c t s of t h e p r o b l e m a n d s o l v i n g t h e m . I n e a c h c a s e , t h e s i m p l i f y i n g

a s s u m p t i o n s h a v e t o b e c h o s e n in s u c h a w a y a s t o r e t a i n a r e a l i s t i c

d e s c r i p t i o n of t h e p h e n o m e n o n w e a r e t r y i n g t o d e s c r i b e , a n d t r e a t o t h e r

a s p e c t s of t h e p r o b l e m a s r e a l i s t i c a l l y a s p o s s i b l e .

I t s h o u l d b e r e a s s u r i n g t o t h e s t u d e n t t h a t e v e n t h o u g h t h e s c i e n c e of

h y d r o d y n a m i c s w a s d e v e l o p e d o v e r a c e n t u r y a g o , t h e r e a r e still

i m p o r t a n t p r o b l e m s w a i t i n g t o b e s o l v e d .

D. THE URINARY DROP SPECTROMETER

A n o t h e r , m o r e s p e c u l a t i v e a p p l i c a t i o n of t h e t e c h n i q u e s w h i c h w e h a v e

l e a r n e d in t h i s t e x t t o a n a r e a of m e d i c i n e is t h e u r i n a r y d r o p

s p e c t r o m e t e r . T h i s is a n i n s t r u m e n t w h o s e f u n c t i o n is t o p r o v i d e e a r l y

d i a g n o s e s of a b n o r m a l i t i e s in t h e u r i n a r y t r a c t .

I n F i g . 14.6 is p r e s e n t e d a s impl i f ied s k e t c h of t h e l o w e r u r i n a r y t r a c t .

T h e u r i n e f r o m t h e k i d n e y s is s t o r e d in t h e b l a d d e r , a n d p a s s e s t o t h e

kidneys

Fig. 14.6. A schematic diagram of a urinary system.

The Urinary Drop Spectrometer 265

o u t s i d e t h r o u g h a d e f o r m a b l e t u b e c a l l e d t h e u r e t h r a . S i n c e t h e u r e t h r a i s

o p e n t o t h e o u t s i d e , it i s c o n s t a n t l y b e i n g i n v a d e d b y b a c t e r i a . U r i n a t i o n

p e r f o r m s t h e i m p o r t a n t f u n c t i o n of w a s h i n g t h e s e b a c t e r i a o u t .

C l e a r l y , o b s t r u c t i o n s o r i m p e d i m e n t s t o t h e flow wil l g i v e t h e b a c t e r i a a

c h a n c e t o c a u s e i n f e c t i o n s in t h e u r e t h r a , w h i c h wi l l , in t u r n , w e a k e n t h e

t i s s u e a n d m a k e t h e s y s t e m m o r e s u s c e p t i b l e t o i n f e c t i o n a t a l a t e r d a t e .

O v e r t h e c o u r s e of y e a r s , t h e s e i n f e c t i o n s c a n p r o g r e s s t o t h e b l a d d e r , a n d

e v e n t h e k i d n e y s . F o r t h i s r e a s o n ( a s w e l l a s f o r m a n y o t h e r s w h i c h a r e

e q u a l l y c o m p e l l i n g ) , it is i m p o r t a n t t o b e a b l e t o d e v e l o p a d i a g n o s t i c

t e c h n i q u e f o r d e t e c t i n g t h e s e s m a l l o b s t r u c t i o n s a n d i m p e d i m e n t s before

t h e y h a v e a c h a n c e t o c a u s e a g r e a t d e a l of d a m a g e .

T h e u r i n a r y d r o p s p e c t r o m e t e r i s s u c h a t e c h n i q u e . I t w o r k s o n t h e

f o l l o w i n g p r i n c i p l e : T h e s t r e a m of u r i n e p a s s e s t h r o u g h t h e u r e t h r a d u r i n g

t h e p r o c e s s of u r i n a t i o n , a n d flows a r o u n d t h e o b s t r u c t i o n . I n f o r m a t i o n

a b o u t t h e o b s t r u c t i o n is t h e n c o n t a i n e d in t h e s t r e a m , w h i c h e m e r g e s

a n d b r e a k s i n t o d r o p s . I t is a r e a s o n a b l e a s s u m p t i o n t h a t s o m e of t h e

i n f o r m a t i o n a b o u t t h e o b s t r u c t i o n i s t r a n s m i t t e d t o t h e s e d r o p s . If w e t h e n

a r r a n g e t h i n g s s o t h a t t h e d r o p s i n t e r r u p t a l igh t b e a m b e t w e e n a l igh t

s o u r c e a n d a p h o t o t u b e ( s e e F i g . 14.7) , t h e n e a c h d r o p wi l l c o r r e s p o n d t o a

p u l s e in t h e o u t p u t of t h e t u b e . If w e k n e w h o w t o a n a l y z e t h i s s e t of

p u l s e s , w e w o u l d b e a b l e t o g a t h e r i n f o r m a t i o n a b o u t t h e c o n d i t i o n of t h e

u r e t h r a f r o m a n o r m a l u r i n a t i o n . S u c h a t e c h n i q u e , if it w e r e p e r f e c t e d ,

w o u l d b e s o m e t h i n g l i k e a c h e s t X - r a y f o r t h e u r i n a r y s y s t e m — i t c o u l d b e

a r o u t i n e p a r t of a p h y s i c a l e x a m i n a t i o n , a n d c o u l d g i v e e a r l y w a r n i n g of

u r i n a r y t r a c t d i f f icul t ies .

O b v i o u s l y , t h e h y d r o d y n a m i c p r o b l e m s a s s o c i a t e d w i t h t h e t r a n s f e r of

i n f o r m a t i o n a b o u t t h e o b s t r u c t i o n t o t h e d r o p s a r e e x t r e m e l y difficult . T h e

flow in e l a s t i c t u b e s w a s c o n s i d e r e d in t h e p r e v i o u s s e c t i o n . O n c e t h e

s t r e a m e m e r g e s f r o m t h e u r e t h r a , h o w e v e r , a n e n t i r e l y n e w s e t of

c o n s i d e r a t i o n s c o m e s i n t o p l a y . W e t h e n h a v e a c y l i n d r i c a l t u b e of fluid

m o v i n g a l o n g u n d e r t h e i n f l u e n c e of t w o f o r c e s : T h e p r e s s u r e of t h e fluid

a n d t h e s u r f a c e t e n s i o n . S u c h a s y s t e m is c a l l e d a capillary jet. I n t h e n e x t

source /

O-V

O o o phototube

o o Fig. 14.7. The urinary drop spectrometer.


s e c t i o n , w e wi l l l o o k a t t h e s i m p l e s t s u c h j e t — o n e in w h i c h t h e fluid

m o v e s e v e r y w h e r e w i t h c o n s t a n t v e l o c i t y — a n d t r y t o u n d e r s t a n d w h y it

b r e a k s i n t o d r o p s . T h e q u e s t i o n of h o w t h e d r o p s a r e f o r m e d , a n d h o w

t h e y c a n b e r e l a t e d t o u r e t h r a l o b s t r u c t i o n s , is n o t u n d e r s t o o d a t t h e

p r e s e n t t i m e .

E. STABILITY OF A CAPILLARY JET

C o n s i d e r a j e t of i n c o m p r e s s i b l e fluid of d e n s i t y p a n d s u r f a c e t e n s i o n

T m o v i n g w i t h c o n s t a n t v e l o c i t y c t o t h e r i g h t ( s e e F i g . 14.8). W i t h o u t

l o s s of g e n e r a l i t y , w e c a n t a k e c = 0, s i n c e a s i m p l e G a l i l e a n t r a n s f o r m a -

t i o n wil l c h a n g e c t o a n y v a l u e w e c h o o s e . L e t u s f u r t h e r a s s u m e t h a t t h e

fluid is i n v i s c i d (r/ = 0) f o r t h e s a k e of s i m p l i c i t y , a n d t h a t t h e c r o s s

s e c t i o n of t h e u n p e r t u r b e d j e t is a c i r c l e of r a d i u s a.

Fig. 14.8. The unperturbed jet.

T o e x a m i n e t h e q u e s t i o n of s t a b i l i t y , w e wi l l u s e t h e t e c h n i q u e of S e c t i o n

4 . D a n d i n t r o d u c e s m a l l , t i m e - d e p e n d e n t p e r t u r b a t i o n s t o t h e s y s t e m , a n d

s e e u n d e r w h a t c o n d i t i o n s t h e y m i g h t b e e x p e c t e d t o g r o w . W e wil l a l s o

a s s u m e t h a t t h e p e r t u r b a t i o n s w h i c h w e i n t r o d u c e a r e i r r o t a t i o n a l , s o t h a t

w e c a n w r i t e

v = V<£, (14 .E .2 )

w h e r e cp i s t h e v e l o c i t y p o t e n t i a l . A s in S e c t i o n 4 . B , t h e e q u a t i o n fo r t h e

v e l o c i t y p o t e n t i a l is j u s t

V2<f> = 0. (14 .E .3 )

I t s h o u l d b e n o t e d t h a t t h e v e l o c i t i e s r e f e r r e d t o in t h e a b o v e e q u a t i o n s

a r e t h e p e r t u r b i n g v e l o c i t i e s , s i n c e t h e e q u i l i b r i u m v e l o c i t i e s a r e z e r o . L e t

T h e e q u i l i b r i u m fo r s u c h a j e t is c l e a r l y o n e in w h i c h vz = vr = 0 , a n d

t h e p r e s s u r e is a c o n s t a n t g i v e n b y

(14 .E .1 )

Stability of a Capillary Jet 267

u s l o o k a t p e r t u r b a t i o n s of t h e f o r m

cp(r, 0, z, t) = <£i(r, 0 ) c o s kz c o s at, (14 .E .4 )

w i t h t h e u n d e r s t a n d i n g t h a t w e c a n , if w e w i s h , r e g a r d t h i s a s o n e

c o m p o n e n t of a F o u r i e r s e r i e s e x p a n s i o n of a n y a c t u a l p e r t u r b a t i o n . T h e

c o n d i t i o n t h a t t h e j e t b e s t a b l e is t h a t

o - 2 ^ 0 , (14 .E .5 )

s i n c e in t h i s c a s e , t h e r e wil l b e n o g r o w t h of t h e p e r t u r b a t i o n w i t h t i m e .

T h e o t h e r e q u a t i o n s w h i c h w e h a v e a t o u r d i s p o s a l a r e t h e E u l e r

e q u a t i o n in t h e f o r m

(14 .E .6 )

w h e r e w e h a v e d r o p p e d s e c o n d - o r d e r t e r m s in t h e v e l o c i t y , a n d t h e

c o n d i t i o n a t t h e s u r f a c e w h i c h s t a t e s t h a t

(14 .E .7 )

w h e r e Ri a n d R2 a r e t h e p r i n c i p l e r a d i i of c u r v a t u r e .

If w e i n s e r t o u r a s s u m e d f o r m of t h e p e r t u r b a t i o n i n t o E q . ( 1 4 . E . 3 ) , w e

find

(14 .E .8 )

s o t h a t , if w e a s s u m e a s e p a r a b l e f o r m of t h e s o l u t i o n

4>x = fc(r)0(0),

a n d p r o c e e d a s u s u a l , w e find t h a t

(14 .E .9 )


© c o s sd,

w h e r e s is t h e s e p a r a t i o n c o n s t a n t a n a l o g o u s t o t h e c o n s t a n t k in E q .

(14 .C .11) . T h e e q u a t i o n f o r R(r) is t h e n

(14 .E .10 )

w h i c h is j u s t t h e B e s s e l e q u a t i o n [ c o m p a r e w i t h E q . (14 .C .13 ) ] . T h e

s o l u t i o n s wil l b e ( a s s u m i n g t h a t t h e v e l o c i t y p o t e n t i a l r e m a i n s finite a t


r = 0)

R *L(kr),

s o t h a t t h e final e x p r e s s i o n f o r t h e v e l o c i t y p o t e n t i a l is

(f> = AIs(kr) c o s sS c o s kz c o s at. ( 14 .E .11 )

I n t h e s e e x p r e s s i o n s , t h e f u n c t i o n Is(kr) is c a l l e d t h e m o d i f i e d B e s s e l

f u n c t i o n , a n d is i d e n t i c a l t o el7TSl2 Js(ikr).

T h i s r e s u l t , t o g e t h e r w i t h E q s . ( 14 .E .2 ) a n d ( 1 4 . E . 6 ) , c o m p l e t e l y de f ines

all of t h e h y d r o d y n a m i c v a r i a b l e s in t h e p r o b l e m u p t o a c o n s t a n t . T o

p r o c e e d f u r t h e r , it is n e c e s s a r y t o a p p l y t h e b o u n d a r y c o n d i t i o n s .

W h e n t h e p e r t u r b a t i o n s a r e a p p l i e d , t h e s u r f a c e of t h e j e t wi l l b e

d e f o r m e d f r o m a p e r f e c t c i r c u l a r c y l i n d e r ( s e e F i g . 14.9). If w e w r i t e

r = a+£, ( 14 .E .12 )

t h e n £ is a s m a l l p a r a m e t e r r e p r e s e n t i n g t h i s d e v i a t i o n . F r o m t h e

c o n d i t i o n s t h a t a fluid e l e m e n t in t h e s u r f a c e m o v e s w i t h t h e s a m e

v e l o c i t v a s t h e s u r f a c e i tself , w e h a v e

Fig. 14.9. End and side views of the perturbed jet.

(14 .E .13)

a t r = a, w h i l e f r o m P r o b l e m 14.7 a n d E q . ( 1 4 . E . 7 ) , w e h a v e t h e c o n d i t i o n

t h a t

(14 .E .14)

F r o m E q . (14 .E .13 ) , w e h a v e t h e r e s u l t t h a t

kA £ = — Ts(ka) c o s sd c o s kz s in at, (14 .E .15)

a

w h i l e E q . (14 .E .6 ) g i v e s

(kr) c o s sd c o s kz c o s at. ( 14 .E .16)

Stability of a Capillary Jet 269

f o r a n y v a l u e of t h e a r g u m e n t . H e n c e t h e j e t c a n b e u n s t a b l e o n l y if

k 2 a 2 + s 2 - \ < 0 , ( 14 .E .18)

in w h i c h c a s e icrt _i_ — iat

c o s crt = e y êlIm°]t.

F r o m E q . ( 1 4 . E . 9 ) , it is c l e a r t h a t t h e c o n s t a n t s m u s t b e a n i n t e g e r .

O t h e r w i s e , t h e s o l u t i o n f o r © w o u l d n o t b e s ing l e v a l u e d . T h i s m e a n s t h a t

if s h a s any n o n z e r o v a l u e , E q . ( 14 .E .18 ) c a n n e v e r b e sa t i s f i ed , a n d t h e

p e r t u r b a t i o n wil l n o t g r o w in t i m e . T h u s , p e r t u r b a t i o n s l i ke t h a t in F i g s .

14.10(a) a n d 14 .10(b) , in w h i c h t h e j e t is " f l u t e d , " e i t h e r w i t h o r w i t h o u t a

z-dependence, wi l l n o t g r o w w i t h t i m e , b u t wi l l s i m p l y o s c i l l a t e a r o u n d

e q u i l i b r i u m .

Fig. 14.10(a). A "fluted" perturbation of the jet.

Fig. 14.10(b). A "fluted" perturbation with a z-dependence.

If w e n o w c o m b i n e E q s . ( 1 4 . E . 1 4 ) , ( 1 4 . E . 1 5 ) , a n d ( 1 4 . E . 1 6 ) , w e g e t

(14 .E .17)

I t is t h i s e q u a t i o n w h i c h d e t e r m i n e s t h e t i m e d e p e n d e n c e of t h e

p e r t u r b a t i o n , a n d h e n c e t h e s t a b i l i t y of t h e s y s t e m .

I t is a p r o p e r t y of t h e B e s s e l f u n c t i o n s t h a t


If 5 = 0 , h o w e v e r , s o t h a t w e c o n s i d e r o n l y ax i a l l y s y m m e t r i c

p e r t u r b a t i o n s , t h e n E q . (14 .E .18) c a n b e sa t i s f ied p r o v i d e d t h a t

S i n c e k = 27r/A, t h i s m e a n s t h a t t h e p e r t u r b a t i o n s w h o s e w a v e l e n g t h

sa t i s f ies t h e c o n d i t i o n ka < 1 wi l l g r o w e x p o n e n t i a l l y w i t h t i m e . T h i s is

k n o w n a s t h e Rayleigh criterion f o r j e t s t ab i l i t y .

T h u s , w e h a v e s h o w n t h a t t h e c a p i l l a r y j e t is i n d e e d u n s t a b l e , a n d wil l

b r e a k u p i n t o d r o p s a t s o m e t i m e . W e h a v e a l s o s h o w n t h a t t h e

p e r t u r b a t i o n s t o w h i c h t h e j e t is u n s t a b l e a r e t h o s e w h i c h a r e ax i a l l y

s y m m e t r i c a n d w h o s e w a v e l e n g t h is l o n g e r t h a n t h e c i r c u m f e r e n c e of t h e

u n p e r t u r b e d j e t .

E q u a t i o n (14 .E .17) t e l l s u s h o w f a s t e a c h p e r t u r b a t i o n g r o w s w i t h t i m e .

S i n c e or = 0 a t ka = 0 a n d ka = 1, t h e r e m u s t b e a m a x i m u m v a l u e w h i c h a

c a n a t t a i n . N u m e r i c a l a n a l y s i s s h o w s t h a t t h i s o c c u r s w h e n

A first g u e s s a t t h e d r o p s w h i c h w o u l d b e f o r m e d , t h e n , w o u l d b e t o

a s s u m e t h a t t h i s f a s t e s t g r o w i n g p e r t u r b a t i o n o u t s t r i p s all o f t h e o t h e r s ,

a n d t h a t t h e b r e a k u p p r o c e s s is d o m i n a t e d b y t h i s s ing le w a v e l e n g t h

p e r t u r b a t i o n a t l a r g e t i m e s . I n t h i s c a s e , w e w o u l d e x p e c t e q u a l l y s p a c e d

d r o p s of e q u a l m a s s w h e n t h e j e t finally d i s i n t e g r a t e s . D e v i a t i o n s f r o m

t h i s e x p e c t a t i o n w o u l d p r e s u m a b l y b e d u e t o t h e p r e s e n c e of o t h e r

e f f ec t s , a m o n g w h i c h m i g h t b e t h e o b s t r u c t i o n in t h e u r e t h r a t h r o u g h

w h i c h t h e fluid h a s p a s s e d .

PROBLEMS

14.1. Show that in the case of cylindrical symmetry , the Nav ie r -S tokes equat ion can be writ ten as in Eq . (14.C.8).

14.2. Show that the equat ion for the pressure given in Eq . (14.C.9) follows from the Nav ie r -S tokes equat ion, continuity and the approximat ions discussed in the

14.3. Derive Eq . (14.C.9) for the radial velocity of the fluid.

14.4. F rom the definition of the tensor cr'ik in Chapter 8, show that the axial stress exer ted by the fluid at the inner radius is

ka < 1. (14 .E .19)

A -9.2a.

text .

and that the radial s tress is

Problems 271

14.5. Calculate the radial s t ress exer ted by the ar tery in Section 14.C at the outer radius of the vessel . This must be cont inuous with the s tress exer ted by the surrounding medium. Is it possible for the surrounding medium to be a vacuum? What does this tell you about the assumpt ion that uz can be neglected in that case?

14.6. Consider the ar tery in Section 14.C in the case when the wall is rigid. This is the limit F = 0, E -> oo. Calculate the total flow, given by

(Hint: T h e limiting form of the Bessel function for small a rgument is

and compare it to the Poisieulle result.)

14.7. Consider a deformed cylinder, as shown in Fig. 14.11. Le t R be the radius of curva ture at a point , and r the dis tance from the center to that point . Le t As be the arc length along the actual surface (shown as a solid line), and r A 0 the arc length along the surface shown as a dot ted line.

in the limit of s teady flow, given by

(a) Show that

(b) H e n c e show that for small deformat ions ,

(c) H e n c e derive Eq . (14.E.14), given Eq . (14.E.12).

Ar

Fig. 14.11.

Fig. 14.12.

14.9. Calculate the Reynolds number for typical blood flow in a human ar tery, and for typical flow in the ure thra .

14.10. One of the problems discussed in connect ion with the urinary drop spect rometer is the quest ion of whether , in passing through the air, the urine s t ream picks up a static charge. Calculate the effect of a static surface charge density cr on the Rayleigh equat ion (14.E.7).

14.11. The development of the Rayleigh theory assumed that the jet existed in a vacuum. This, of course , is not the case .

(a) Assuming that the jet is proceeding through a stat ionary a tmosphere of density p ' , find the a tmospher ic pressure at the surface of the distorted jet .

(b) H e n c e modify Eq. (14.E.14) to take account of aerodynamic effects. (c) H o w is the Rayleigh equat ion changed by the inclusion of this effect?

14.12. Le t us consider the stability of blood flow in an ar tery. Suppose that in equilibrium flow, the velocity is entirely in the z-direct ion, and is given by a function U(r). Le t us then consider a small per turbat ion whose s t ream function is of the form

ijj(r,z,t) = <t>(r)eii


(b) Show that at B, where the surface is maximally deformed inward, the pressure is

(c) Hence show that if b <a, the film will be unstable unless lira > A.

14.8. Le t us see if we can come to a simple unders tanding of the Rayleigh condit ion for jet instability in Eq . (14.E.14). Consider a film whose surface tension is T, and which is deformed in an axially symmetr ic way as in Fig. 14.12. Le t the equat ion of the surface be given by

r = a + b cos kz,

where k = 277-/A. (a) Show that at the point A, where the surface is maximally deformed outward ,

the pressure is

Problems 273

(a) Find the small per turbat ion velocities vz and vr. (b) Show that we can define a s t ream function for this problem provided that

there is azimuthal symmetry . (c) Show that the equat ion for cp is

where we have defined c = filk and R is the Reynolds number . This is called the Or r -Sommerfe ld equat ion, and is widely used in studying stability.

(d) Wha t are the boundary values for <£? 14.13. Show that if we neglect te rms of order l/R, the Or r -Sommerfe ld equat ion will have a singular point when U = c. If rk is the dis tance to the point where this occurs , show that the per turbat ion velocity in the z-direction must go as

near r = rk. Can you give a reason why this singularity occurs , and how it can be removed? (Hint: Remember the discussion connec ted with boundary layers.) The problem of the transit ion from laminar to turbulent flow is dealt with in great detail in the text by Schlichting included in the references .

14.14. The problem of flow through a constr ic ted tube is, in general , a very difficult one. Suppose that the radius of an artery is given by

where the function / defines the narrowing of the ar tery in some region, (a) Show that if S/z 0 "-k2tb)-U" = (cf)"" -2k2cb'+ k4cb)

uz ~\n(r-rk)

and

(b) H e n c e show that

(c) If we define

V = R-r

R and

vr(r = 0) = w,

a n d a s s u m e a f o r m f o r t h e s o l u t i o n


and dp\dz is determined by demanding consis tency with the equat ion in part (b).

REFERENCES

S. Middleman, Transport Phenomena in the Cardio-vascular System, Wiley Interscience, New York, 1972.

A good description of the physical processes involved in circulation. The text suffers somewhat from a rather inelegant use of mathematics.

H. Schlichting, (op. cit.—see Chapter 8). Contains some useful presentations of flow in tubes and stability criterion.

For a discussion of the capillary jet, see

H. Lamb, (op. cit.—see Chapter 1). N. Bohr, Phil Trans. Roy. Soc, London A209, 281 (1909).

For a discussion of Bessel functions, see

G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge U.P., 1958. Probably the most complete work of this type in existence. Had it been written later, it could have been titled "Everything You Have Always Wanted to Know About Bessel Functions."

M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions, U.S. Department of Commerce.

Chapters 9 and 10 give a complete (but concise) summary of the properties of Bessel and related functions. This is one of the best and most useful reference books on mathematical functions.

J. D. Jackson, Classical Electrodynamics, John Wiley and Sons, New York, 1962. In Chapter 3 there is a good presentation of Bessel's equation and its solutions in the context of a physical problem.

For a discussion of the Urinary Drop Spectrometer, see

G. Aiello, P. La France, R-C. Ritter, and J. S. Trefil, Physics Today, September 1974.

show that applying the boundary condit ions to determine the cons tants yields

where

Appendices

Merely corroborative details to lend an aspect of verisimilitude to what would otherwise be a bald and unconvincing narrative.

GILBERT AND SULLIVAN

The Mikado

INTRODUCTION

T h r o u g h o u t m o s t of t h e t e x t , a n ef for t h a s b e e n m a d e t o m a k e t h e

m a t h e m a t i c a l d e v e l o p m e n t of t h e v a r i o u s t o p i c s s e l f - c o n t a i n e d . I n e v i t a -

b l y , h o w e v e r , t h e r e wi l l b e s t u d e n t s w h o , f o r o n e r e a s o n o r a n o t h e r , h a v e

m i s s e d s o m e of t h e m a t h e m a t i c a l b a c k g r o u n d n e c e s s a r y f o r t h e d i s c u s -

s i o n s . T h e p u r p o s e of t h e s e a p p e n d i c e s is t o p r o v i d e a q u i c k r e f e r e n c e in

t h e m a t h e m a t i c s w h i c h is u s e d t h r o u g h o u t t h e t e x t , p a r t i c u l a r l y f o r

C a r t e s i a n t e n s o r n o t a t i o n , d i f f e ren t i a l e q u a t i o n s , a n d e x p a n s i o n s in s e r i e s .

T h e s e a p p e n d i c e s d o n o t c o n s t i t u t e a t e x t b o o k in m a t h e m a t i c a l p h y s i c s ,

h o w e v e r , a n d a r e i n c l u d e d p r i m a r i l y b e c a u s e in t e a c h i n g t h i s m a t e r i a l I

h a v e f o u n d t h a t s o m e s t u d e n t s c a n bene f i t f r o m a s h o r t p r e s e n t a t i o n of

t h e m a i n m a t h e m a t i c a l t e c h n i q u e s . S t u d e n t s w i s h i n g m o r e d e t a i l , o r

w i s h i n g t o p u r s u e t h e s e t o p i c s f u r t h e r , a r e r e f e r r e d t o t h e f o l l o w i n g

s t a n d a r d t e x t s :

Jon Mathews and R. L. Walker, Mathematical Methods of Physics, W. A. Benjamin, New York, 1970. A readable and concise treatment of mathematical physics, which should be easy for the student to follow.

W. Morse and H. Feschbach, Methods of Theoretical Physics, McGraw-Hill, New York, 1953. A two volume treatise which contains almost anything the average physicist needs in the way of mathematical techniques.

275

276 Appendices

APPENDIX A CARTESIAN TENSOR NOTATION

T h r o u g h o u t t h e t e x t , it is f r e q u e n t l y f o u n d u s e f u l t o u s e t e n s o r r a t h e r

t h a n v e c t o r n o t a t i o n . I n t h i s a p p e n d i x , t h i s t y p e of n o t a t i o n will b e

e x p l a i n e d .

A v e c t o r is u s u a l l y c o n s i d e r e d t o b e a q u a n t i t y w h i c h h a s b o t h

m a g n i t u d e a n d d i r e c t i o n , a n d c a n b e spec i f i ed b y g iv ing i t s l e n g t h a n d t h e

a n g l e s de f in ing i t s d i r e c t i o n s ( s e e F i g . A . l ) . H o w e v e r , a v e c t o r c a n a l s o b e

c o m p l e t e l y spec i f i ed b y l i s t ing i t s t h r e e c o m p o n e n t s ( for o u r p u r p o s e s , w e

t a k e t h e s e t o b e t h e x-, y-, a n d z - c o m p o n e n t s . T h u s , w e c o u l d w r i t e

V = ( V „ Vy, Vz) o r , m o r e s i m p l y ,

V = Vh

w h e r e t h e i n d e x i is u n d e r s t o o d t o r u n f r o m 1 t o 3 , w h e r e V i is t h e

x - c o m p o n e n t of t h e v e c t o r , V2 t h e y - c o m p o n e n t , a n d V3 t h e z - c o m p o n e n t .

T h i s is t h e s i m p l e s t e x a m p l e of C a r t e s i a n t e n s o r n o t a t i o n .

z

Fig. A.l. A vector in three dimensions.

N o w it is w e l l k n o w n t h a t if t h e c o o r d i n a t e s y s t e m is r o t a t e d , t h e

c o m p o n e n t s of t h e v e c t o r in t h e n e w s y s t e m ( w h i c h w e wi l l ca l l V ) a r e

r e l a t e d t o t h e c o o r d i n a t e s in t h e o l d s y s t e m b y t h e r e l a t i o n

V = R V , ( A . l )

w h e r e R is t h e m a t r i x w h i c h d e s c r i b e s t h e c o o r d i n a t e t r a n s f o r m a t i o n .

R e c a l l i n g t h e de f in i t ion of m a t r i x m u l t i p l i c a t i o n , t h e c o m p o n e n t V\ is j u s t

(A.2)

F o r a r o t a t i o n in t w o d i m e n s i o n s , f o r e x a m p l e , a r o t a t i o n t h r o u g h a n a n g l e

0 a b o u t t h e z - a x i s , t h e m a t r i x R h a s t h e f a m i l a r f o r m

Cartesian Tensor Notation 277

( c o s 6 s in 0 0

- s in e cose 0 0 0 1,

(A.3)

P i c t o r i a l l y , w e h a v e F i g . A . 2 .

V V

Fig. A.2. Transformation of vector under rotation.

I t is c u s t o m a r y t o u s e t h e s o - c a l l e d summation convention in w r i t i n g

o u t s u c h q u a n t i t i e s . T h e c o n v e n t i o n c a n b e s t a t e d a s f o l l o w s : W h e n e v e r

a n i n d e x is r e p e a t e d , it is u n d e r s t o o d t h a t t h e r e is a s u m m a t i o n o v e r t h a t

i n d e x , w i t h t h e i n d e x i t se l f r u n n i n g f r o m 1 t o 3 . U s i n g t h e s u m m a t i o n

c o n v e n t i o n , E q . (A .2 ) c a n b e w r i t t e n

U p t o t h i s p o i n t , w e h a v e p r o c e e d e d a s if w e k n e w w h a t a v e c t o r w a s ,

a n d w e r e d e r i v i n g t h e l a w w h i c h t o l d u s h o w t h a t v e c t o r a p p e a r e d in

d i f f e r en t f r a m e s of r e f e r e n c e . W e c a n , h o w e v e r , r e v e r s e t h e l o g i c , a n d u s e

E q . (A.4) a s a definition of a v e c t o r — i . e . w e de f ine a v e c t o r a s a n y

c o l l e c t i o n of t h r e e n u m b e r s w h i c h t r a n s f o r m s a c c o r d i n g t o t h e l a w in E q .

(A .4 ) . T h i s c o n c e p t of de f in ing a n o b j e c t b y t h e w a y in w h i c h it c h a n g e s

u n d e r c o o r d i n a t e t r a n s f o r m a t i o n s i s a f a i r ly r e c e n t d e v e l o p m e n t in

p h y s i c s , a n d h a s b e e n e n o r m o u s l y u s e f u l in f ields a s w i d e l y s e p a r a t e d a s

n u c l e a r p h y s i c s a n d t h e g e n e r a l t h e o r y of r e l a t i v i t y .

If w e t a k e t h i s p o i n t of v i e w , w e s e e t h a t it is p o s s i b l e t o c o n s t r u c t o t h e r

k i n d s of o b j e c t s . F o r e x a m p l e , s u p p o s e w e de f ine a s a s e c o n d - r a n k t e n s o r

a n y o b j e c t w i t h t w o i n d i c e s ( i .e . 9 c o m p o n e n t s ) w h i c h t r a n s f o r m s

a c c o r d i n g t o t h e l a w

A n e x a m p l e of s u c h a n o b j e c t w o u l d b e t h e t e n s o r w h o s e i - j t h c o m p o n e n t is

w h e r e V is a v e c t o r .

T o s e e t h a t t h i s q u a n t i t y sa t i s f ies t h e de f in i t ion of a t e n s o r , w e n o t e t h a t

V'i = (A.4)

Tii — RaRjmTim. (A.5)

278 Appendices

in o n e f r a m e , Ti,- = ViVh w h i l e in t h e f r a m e w h i c h h a s b e e n r o t a t e d

TL = V;V^, (A .6)

b u t Vi' = <RiiVi,

(A .7 ) v : = Rmjvh

s o t h a t t h e t e n s o r in t h e r o t a t e d f r a m e is

TL = RuRmjViVi = RuRmiTib (A.8)

w h e r e T f J i s t h e t e n s o r in t h e o l d f r a m e . O t h e r e x a m p l e s of s u c h t e n s o r s

wil l b e f o u n d in t h e t e x t .

I t m u s t b e n o t e d t h a t n o t e v e r y t w o i n d e x o b j e c t is a t e n s o r , a n d if

s o m e t h i n g is t o b e c a l l e d a t e n s o r , it m u s t b e e x p l i c i t l y ver i f i ed t h a t it

t r a n s f o r m s a c c o r d i n g t o t h e t r a n s f o r m a t i o n l a w of E q . (A .5 ) .

I t s h o u l d a l s o b e n o t e d t h a t in t h e n o m e n c l a t u r e i n t r o d u c e d a b o v e , a

v e c t o r c o u l d b e r e f e r r e d t o a s a first-rank t e n s o r . C l e a r l y , t h i r d - , f o u r t h - ,

a n d h i g h e r - r a n k t e n s o r s c a n b e d e f i n e d in c o m p l e t e a n a l o g y t o t h e

de f in i t ion in E q . (A .5 ) .

T h e g r e a t e s t u s e w h i c h w e sha l l m a k e of t h e C a r t e s i a n t e n s o r n o t a t i o n

wi l l n o t b e c o n c e r n e d w i t h s e c o n d - r a n k t e n s o r s , h o w e v e r , b u t sha l l b e t h e

u t i l i z a t i o n of t h e v e r y c o m p a c t a n d eff icient n o t a t i o n it p r o v i d e s f o r

m a n i p u l a t i n g v e c t o r s a n d v e c t o r o p e r a t o r s . O f t e n o p e r a t i o n s w h i c h

a p p e a r q u i t e c o m p l i c a t e d w h e n w r i t t e n in v e c t o r f o r m a r e s i m p l e t o

a n a l y z e in t e r m s of t e n s o r n o t a t i o n .

L e t u s t h e r e f o r e c a t a l o g u e s e v e r a l c o m m o n v e c t o r o p e r a t i o n s in b o t h

v e c t o r a n d t e n s o r f o r m .

(A) Inner Product

T h e i n n e r p r o d u c t b e t w e e n t w o v e c t o r s i s j u s t

A • B = AXBX + AyBy + AZBZ = AiBt. (A .9 )

T h i s c a n a l s o b e w r i t t e n a s

A • B = AiBjdtj, (A . 10)

w h e r e 8ih t h e K r o n e c k e r d e l t a , is d e f i n e d b y

( A . 11)

The Gravitational Potential Inside of a Uniform Ellipsoid 279

( A . 12)

w h e r e i, j , a n d k a r e u n i t v e c t o r s in t h e x-, y-, a n d z - d i r e c t i o n s . I n t e n s o r

n o t a t i o n , t h i s b e c o m e s

(A.13)

(C) Divergence

T h e d i v e r g e n c e of a v e c t o r is

( A . 14)


(D) Cross Product

T h e c r o s s p r o d u c t of a v e c t o r c a n b e w r i t t e n

A x B)i = €iJkAjBk, ( A . 15)

w h e r e eiik is d e f i n e d b y

(+ 1 i, j , k c y c l i c

- 1 i, j , k a n t i - c y c l i c . ( A . 16)

0 a n y 2 i n d i c e s e q u a l

(E) Curl

T h e c u r l is a s p e c i a l c a s e of t h e c r o s s p r o d u c t a n d is w r i t t e n

( A . 17

APPENDIX B THE GRAVITATIONAL POTENTIAL INSIDE OF A UNIFORM ELLIPSOID

I n t h i s a p p e n d i x , w e wi l l w o r k t h r o u g h t h e s t r a i g h t f o r w a r d b u t t e d i o u s

d e r i v a t i o n of t h e g r a v i t a t i o n a l p o t e n t i a l i n s i d e of a u n i f o r m e l l i p s o i d . T h i s

q u a n t i t y is n e c e s s a r y f o r t h e s t u d y of c l a s s i c a l s t e l l a r s t r u c t u r e in C h a p t e r

2 .

(B) Gradient

T h e g r a d i e n t of a f u n c t i o n / is d e f i n e d t o b e

280 Appendices

C o n s i d e r a p o i n t P , w h o s e c o o r d i n a t e s a r e xP, y P , a n d z P i n s i d e a n

e l l i p s o i d w h o s e s u r f a c e is d e s c r i b e d b y t h e e q u a t i o n

( B . l )

W e b e g i n b y c h a n g i n g c o o r d i n a t e s ( s e e F i g . B . l ) t o

x = xP + r s in 0 c o s </>,

y = y P + r s in 0 s in <£, (B.2)

z = Zp + r c o s 0.

y

X

Fig. B . l . Coordinates given in Eq. (B.2).

w h e r e rx is t h e d i s t a n c e f r o m P t o t h e b o u n d a r y of t h e e l l i p s o i d f o r a g i v e n

c h o i c e of 6 a n d cf> ( c l e a r l y , rx wil l b e a f u n c t i o n of b o t h a n g u l a r v a r i a b l e s

a n d of P ) .

T o s o l v e f o r r u it is n e c e s s a r y o n l y t o p u t t h e v a l u e s of e x p r e s s i o n s f o r

x, y, a n d z f r o m E q . (B .2) i n t o t h e e q u a t i o n d e s c r i b i n g t h e b o u n d a r y , E q .

( B . l ) . W e q u i c k l y find

A r 1

2 + 2J3r 1 + C = 0 , (B.5)

(B.4)

T h e v o l u m e e l e m e n t in t h e n e w v a r i a b l e s i s t h e u s u a l o n e f o r s p h e r i c a l

c o o r d i n a t e s

dV = r2 drd(cos 0) d<f>, (B .3)

s o t h a t t h e g r a v i t a t i o n a l p o t e n t i a l a t P is n o w j u s t

The Gravitational Potential Inside of a Uniform Ellipsoid 281

w h e r e

I n p r i n c i p l e , w e c o u l d n o w s i m p l y s u b s t i t u t e t h e d e f i n i t i o n s of A , B , a n d

C f r o m E q . (B .6) i n t o t h i s i n t e g r a l a n d c a r r y o u t t h e i n t e g r a t i o n s .

H o w e v e r , w e c a n n o t e s e v e r a l s y m m e t r i e s in t h e i n t e g r a n d w h i c h g r e a t l y

s imp l i fy t h e r e s u l t .

F i r s t , w e n o t e t h a t if w e le t

4> - > 7T + </>,

<(> - > 77 - 0.

A a n d C r e m a i n u n c h a n g e d , b u t B g o e s t o — J B . T h u s , in i n t e g r a t i n g o v e r

t h e c o m p l e t e so l id a n g l e , t e r m s l i n e a r in B wi l l g i v e a z e r o i n t e g r a l . T h u s ,

t h e t e r m i n v o l v i n g t h e r a d i c a l in t h e i n t e g r a n d a b o v e c a n b e d r o p p e d .

S i m i l a r l y , in c a l c u l a t i n g t h e t e r m i n v o l v i n g B2, w e e x p e c t t h a t t h e r e wi l l

b e t e r m s p r o p o r t i o n a l t o xP

2, yP

2, a n d z P

2 , a n d , in a d d i t i o n , c r o s s t e r m s

p r o p o r t i o n a l t o x P , y P , e t c . A r g u m e n t s s i m i l a r t o t h a t in t h e p r e c e d i n g

p a r a g r a p h c a n b e e v o k e d t o s h o w t h a t t h e c r o s s t e r m s d o n o t c o n t r i b u t e

t h e final r e s u l t . C o n s i d e r a s a n e x a m p l e t h e t e r m

T h i s t e r m wil l c h a n g e s ign u n d e r t h e t r a n s f o r m a t i o n cp-> — cp, a n d h e n c e

wil l v a n i s h w h e n i n t e g r a t e d o v e r t h e so l id a n g l e . S i m i l a r a r g u m e n t s c a n b e

m a d e f o r t h e o t h e r c r o s s t e r m s .

(B.8)

s in 0 c o s cp s in cp.

(B .6)

I t is a s i m p l e e x e r c i s e t o s h o w t h a t f o r p o i n t s i n s i d e t h e b o d y , t h e

c o r r e c t c h o i c e of s i g n s in t h e q u a d r a t i c f o r m u l a g i v e s

(B.7)

s o t h a t a f t e r p e r f o r m i n g t h e i n t e g r a l o v e r t h e r - c o o r d i n a t e

(B .14)

w h i c h c a n b e p u t i n t o a s o m e w h a t m o r e f a m i l i a r f o r m b y c h a n g i n g

v a r i a b l e s t o A, w h e r e

282 Appendices

(B.9)

T h e s e i n t e g r a l s a r e still r a t h e r c o m p l i c a t e d , b u t t h e r e is a t r i c k w h i c h

wi l l a l l o w u s t o p u t t h e m i n t o m u c h s i m p l e r f o r m . L e t u s de f ine t h e

q u a n t i t y

(B.10)

T h e n it is s i m p l e t o s h o w t h a t

( B . l l )

S i m i l a r e x p r e s s i o n s c a n b e w r i t t e n f o r dW/db a n d dWIdc. If w e p u t all

of t h e s e i n t o t h e a b o v e i n t e g r a l , a n d r e c a l l t h e de f in i t ion of C, w e find

(B.12)

= axP

2 + j8y P

2 + yzp2 +

w h i c h is t h e g e n e r a l f o r m w h i c h w e u s e d in C h a p t e r 2 . T o g e t o u r final

r e s u l t , w e h a v e o n l y t o e v a l u a t e W.

A c t u a l l y , t h i s c a n n o t b e d o n e in c l o s e d f o r m f o r a n a r b i t r a r y e l l i p so id ,

b u t w e c a n c a r r y o u t o n e of t h e a n g u l a r i n t e g r a l s in t h e de f in i t ion of W b y

makine the substitution

(B .13 )

I t is t h e n p o s s i b l e t o c a r r y o u t t h e i n t e g r a l o v e r </> b y w r i t i n g

W e a r e t h e n lef t w i t h

The Critical Frequency 283

t o g i v e

(B .15)

APPENDIX C THE CRITICAL FREQUENCY

I n C h a p t e r 2 , w e s a w t h a t f o r a M a c l a u r i n e l l i p s o i d , it w a s i m p o s s i b l e t o

a c h i e v e e q u i l i b r i u m if t h e f r e q u e n c y of r o t a t i o n e x c e e d e d a c e r t a i n v a l u e ,

o n t h e o r d e r of t h e c r i t i c a l f r e q u e n c y

a>c

2 = 2iTPG. ( C . l )

I n t h i s a p p e n d i x , w e wi l l s h o w t h a t t h e c r i t i c a l f r e q u e n c y is t h e u p p e r l imi t

o n t h e f r e q u e n c y of r o t a t i o n f o r a n y i n c o m p r e s s i b l e b o d y .

C o n s i d e r a n a r b i t r a r y v o l u m e V in t h e r o t a t i n g fluid, s u r r o u n d e d b y a

s u r f a c e S. If <\> a n d ij/ a r e a n y t w o f u n c t i o n s , t h e n Green's theorem t e l l s u s t h a t

(C .2)

w h e r e d/dn r e p r e s e n t s t h e d e r i v a t i v e of t h e f u n c t i o n a l o n g t h e o u t w a r d

n o r m a l t o S.

N o w le t u s t a k e t h e c a s e

<f> = \ a n d

ifj = P.

T h e n E q . (C .2) b e c o m e s

f V2PdV= f ?rdS. (C .3) J v Js dn

If w e s u b s t i t u t e in t h e l e f t - h a n d i n t e g r a l t h e e x p r e s s i o n f o r P w h i c h w e

o b t a i n e d b y i n t e g r a t i n g t h e E u l e r e q u a t i o n ( 2 . A . 4 ) , w e h a v e

(C.4) = 2 p V ( c o 2 - 2 7 r p G ) ,

w h e r e V is t h e t o t a l v o l u m e a n d w e h a v e u s e d t h e e x p r e s s i o n

V 2 H = 47rpG (C.5)

t o e l i m i n a t e t h e g r a v i t a t i o n a l p o t e n t i a l .

284 Appendices

T h e p r e s s u r e o n t h e s u r f a c e of o u r b o d y is c o n s t a n t , a n d f o r p u r p o s e s

of e x p l a n a t i o n , w e c a n t a k e it t o b e z e r o . If co > coc, t h e n t h e i n t e g r a l of

/ dP/dnds o v e r a n y s u r f a c e in t h e fluid m u s t b e p o s i t i v e , w h i c h m e a n s

t h a t t h e p r e s s u r e m u s t b e i n c r e a s i n g a s w e g o f r o m t h e i n t e r i o r of t h e fluid

t o w a r d t h e s u r f a c e . T h u s , t h e p r e s s u r e f o r c e s a c t i n w a r d , in t h e s a m e

d i r e c t i o n a s g r a v i t y . E q u i l i b r i u m in s u c h a c a s e is c l e a r l y i m p o s s i b l e , s i n c e

t h e f o r c e s in t h e z - d i r e c t i o n o n a n y e l e m e n t of fluid wi l l n o t c a n c e l e a c h

o t h e r ( w e t a l k a b o u t t h e z - d i r e c t i o n b e c a u s e t h e c e n t r i f u g a l f o r c e h a s n o

z - c o m p o n e n t ) .

O n t h e o t h e r h a n d , if co < coc, t h e p r e s s u r e m u s t d e c r e a s e a s w e m o v e

f r o m t h e c e n t e r of t h e fluid t o t h e s u r f a c e , a n d it is p o s s i b l e f o r

e q u i l i b r i u m t o b e a c h i e v e d . W h e t h e r o r n o t t h i s p o s s i b i l i t y is a c t u a l l y

r e a l i z e d d e p e n d s , of c o u r s e , o n t h e s h a p e of t h e fluid m a s s .

T h u s , w e s e e t h a t o n v e r y g e n e r a l g r o u n d s , n o fluid m a s s c a n b e in

e q u i l i b r i u m if it is s p i n n i n g w i t h a f r e q u e n c y g r e a t e r t h a n coc, w h i c h is

w h a t w e s e t o u t t o p r o v e .

APPENDIX D EXPANSION IN ORTHOGONAL POLYNOMIALS

T h r o u g h o u t t h e t e x t , w e h a v e u s e d t h e i d e a of e x p a n d i n g a r b i t r a r y

f u n c t i o n s in t e r m s of o t h e r , s i m p l e r f u n c t i o n s . I n t h i s a p p e n d i x , w e wil l

d i s c u s s t h i s i d e a in d e t a i l , a l t h o u g h f o r a r i g o r o u s p r o o f of t h e t h i n g s w e

s a y , t h e r e a d e r wil l h a v e t o c o n s u l t a m a t h e m a t i c s t e x t b o o k .

T h e i d e a of e x p a n s i o n is a c t u a l l y a f a m i l i a r o n e . C o n s i d e r a v e c t o r V in

a C a r t e s i a n c o o r d i n a t e s y s t e m a s s h o w n in F i g . D . l . W e k n o w t h a t w e c a n

a l w a y s e x p a n d t h i s v e c t o r in t e r m s of t h e t h r e e b a s i s v e c t o r s , i, j , a n d k

V = Vj+Vy] + V£

w h e r e t h e c o m p o n e n t s Vt a r e g i v e n b y

vx = V • I, V y = V • / , } vz =

(D.l)

V

Fig. D.l. A vector in three dimensions.

Expansion in Orthogonal Polynomials 285

T h e b a s i s v e c t o r s h a v e t w o i m p o r t a n t p r o p e r t i e s . F i r s t , t h e y a r e

o r t h o g o n a l t o e a c h o t h e r , s o t h a t

i'j = i ' k = j ' k = 0 ,

a n d s e c o n d , t h e y a r e n o r m a l i z e d , s o t h a t

i ' i = ] ' ] = k ' k = \.

(D .2 )

(D .3 )

A s e t of v e c t o r s w h i c h h a s t h e s e p r o p e r t i e s i s c a l l e d a n orthonormal set

of v e c t o r s .

W e c a n u s e a s l igh t ly d i f f e ren t n o t a t i o n in w r i t i n g d o w n t h e s e f a c t s

a b o u t e x p a n d i n g a v e c t o r in t e r m s of i t s c o m p o n e n t s . If w e d e n o t e b y ft

t h e b a s i s v e c t o r in t h e i t h - d i r e c t i o n , t h e n t h e r e q u i r e m e n t of o r t h o n o r m a l -

i ty t a k e s t h e f o r m ft • ft — 8ij,

w h i l e t h e e x p a n s i o n of t h e v e c t o r V c a n b e w r i t t e n

(D.4)

(D .5 )

I n w h a t f o l l o w s , w e wi l l ca l l t h e c o n s t a n t at t h e coef f ic ien t of e x p a n s i o n .

N o w t h e r e is n o t h i n g in t h e a b o v e d e v e l o p m e n t w h i c h f o r c e s u s t o

c o n f i n e o u r a t t e n t i o n t o t h r e e - d i m e n s i o n a l s p a c e s . If w e c o n s i d e r e d a

v e c t o r V in a n N - d i m e n s i o n a l s p a c e , a n d d e f i n e d a s e t of b a s i s v e c t o r s ft

a s in E q . ( D . 4 ) , b u t n o w le t t h e i n d e x i r u n u p t o N r a t h e r t h a n j u s t t o 3 ,

t h e n w e c o u l d e x p a n d t h e n e w v e c t o r a s

V = 2(V-ft)ft (D .6 )

b y s i m p l e a n a l o g y .

C o n s i d e r n o w a f u n c t i o n f(x) d e f i n e d o n s o m e i n t e r v a l in x, s a y f r o m

z e r o t o L ( s e e F i g . D . 2 ) . L e t u s sp l i t t h e i n t e r v a l u p i n t o N e q u a l s p a c e s

Fig. D.2. The representation of a function by a vector.

286 Appendices

a n d f o r m a n N - d i m e n s i o n a l v e c t o r

F = ( f 1 , f 2 , . . . , f , ) , (D .7)

w h e r e f N is t h e a v e r a g e v a l u e of t h e f u n c t i o n f(x) in t h e i t h i n t e r v a l

m u l t i p l i e d b y V L / N " . I n e x a c t l y t h e s a m e w a y , w e c o u l d f o r m a v e c t o r

G = ( g l . . . g N ) , (D .8)

f r o m t h e f u n c t i o n g(x) d e f i n e d o n t h e s a m e i n t e r v a l . T h e i n n e r p r o d u c t in

t h e i V - d i m e n s i o n a l s p a c e b e t w e e n t h e v e c t o r s G a n d F is j u s t

(D.9)

S u p p o s e n o w t h a t w e w e r e a b l e t o f ind a s e t of f u n c t i o n s 4>(<x\x) d e f i ned

o n x f r o m z e r o t o L , a s w e r e f(x) a n d g ( x ) , a n d w e r e t o f o r m a v e c t o r <£ ( a )

a s in E q . (D.7) f o r e a c h of t h e s e n e w f u n c t i o n s . S u p p o s e a l s o t h a t t h e

v e c t o r s s o f o r m e d h a d t h e p r o p e r t y t h a t

(D .10)

( I t is i m p o r t a n t t o d i s t i n g u i s h b e t w e e n t h e s u p e r s c r i p t a in 4>iia) a n d t h e

s u b s c r i p t j . T h e s u p e r s c r i p t r e f e r s t o t h e i n d e x w h i c h t e l l s u s w h i c h

f u n c t i o n w e a r e d i s c u s s i n g , w h i l e t h e s u b s c r i p t t e l l s u s w h i c h i n t e r v a l in x

is b e i n g c o n s i d e r e d . ) T h e n t h e v e c t o r s f o r m e d in t h i s w a y w o u l d b e a n

o r t h o n o r m a l s e t of b a s i s v e c t o r s a n d w e c o u l d w r i t e

(DM]

a n d s imi l a r l y f o r G .

I t s h o u l d b e e m p h a s i z e d t h a t u p t o t h i s p o i n t , n o n e w i n f o r m a t i o n h a s

b e e n p r e s e n t e d , a n d w e h a v e o n l y b e e n p r e s e n t i n g c o n s e q u e n c e s ^f t h e

k n o w n p r o p e r t i e s of v e c t o r s . L e t u s a s k w h a t h a p p e n s , h o w e v e r , if w e l e t

N g o t o inf ini ty . I n t h i s c a s e , t h e n u m b e r of c o m p o n e n t s in t h e v e c t o r

de f i ned in E q . (D .7 ) b e c o m e s inf in i te a n d t h e s u m o v e r i n d i c e s in t h e i n n e r

p r o d u c t in E q . (D.9) g e t s c o n v e r t e d t o a n i n t e g r a l , s o t h a t o u r n e w

def in i t ion of t h e i n n e r p r o d u c t b e t w e e n t h e n e w v e c t o r s b e c o m e s

F - G = f(x)g(x) dx. (D .12)

W h a t w e h a v e d o n e is de f ine a n e w v e c t o r s p a c e of inf ini te d i m e n s i o n ,

in w h i c h e a c h f u n c t i o n f(x) i s r e p r e s e n t e d b y a v e c t o r . T h i s i s a n e x a m p l e

Expansion in Orthogonal Polynomials 287

of a Hilbert space. F o r t h e s a k e of h o n e s t y , it m u s t b e p o i n t e d o u t t h a t t h i s

is s i m p l y a n a n a l o g y t h a t w e h a v e d r a w n h e r e , a n d t h e r e a d e r w i s h i n g

m o r e r i g o r in t h e de f in i t ion of t h e s e s p a c e s is r e f e r r e d t o t e x t s in

m a t h e m a t i c s .

S u p p o s e in o u r H i l b e r t s p a c e t h e b a s i s f u n c t i o n s r e t a i n t h e i r o r t h o n o r -

m a l i t y , s o t h a t

f <f>(a)(x)cl>m(x) dx = 8aP. (D .13) Jo

T h e n t h e a n a l o g u e of E q . ( D . l l ) is j u s t

(D .15)

d e f i n e d o n t h e i n t e r v a l 0 =^ x ^ L . S o m e s i m p l e c a l c u l a t i o n s wi l l c o n v i n c e

t h e r e a d e r t h a t

a n d

<̂-> = o. (D .16)

T h u s , t h e s i n e s a n d c o s i n e s f o r m a s e t of b a s i s v e c t o r s in a H i l b e r t

s p a c e , j u s t a s t h e v e c t o r s i, / , a n d k f o r m a complete set—i.e. t h a t t h e r e is

n o v e c t o r in t h e H i l b e r t s p a c e o r t h o g o n a l t o all t h e cfrin) a n d \pin\ j u s t a s

t h e r e is n o v e c t o r in C a r t e s i a n s p a c e o r t h o g o n a l t o f, j , a n d k.

T h i s m e a n s t h a t a n y f u n c t i o n d e f i n e d o n t h e i n t e r v a l 0 ^ x ^ L c a n b e

w r i t t e n in t h e f o r m

(D.17)

(D .14)

T h u s , b y a n a l o g y t o t h e e x p a n s i o n of a n o r d i n a r y v e c t o r in t e r m s of i t s

b a s i s v e c t o r s , w e c a n e x p a n d a n a r b i t r a r y f u n c t i o n in t e r m s of a s e t of

b a s i s f u n c t i o n s w h i c h sa t i s fy E q . ( D . 1 3 ) .

D o s u c h s e t s of b a s i s v e c t o r s e x i s t ? T h e a n s w e r t o t h i s q u e s t i o n is

y e s — t h e r e a r e , in f a c t , m a n y s u c h s e t s . C o n s i d e r , f o r e x a m p l e , t h e s e t of

f u n c t i o n s

288 Appendices

w h e r e

a n d

A n e x p a n s i o n of t h i s t y p e is c a l l e d a Fourier series, a n d p l a y s a n

e x t r e m e l y i m p o r t a n t r o l e in p h y s i c s . T h e r e a d e r wil l s e e , h o w e v e r , t h a t it

i s s i m p l y o n e e x a m p l e of a n e x p a n s i o n of a f u n c t i o n in o r t h o g o n a l

p o l y n o m i a l s , a n d if w e c a n find a n o t h e r s e t of f u n c t i o n s l i ke t h o s e in E q .

(D .15 ) , a l t e r n a t e s e r i e s r e p r e s e n t a t i o n s of t h e f u n c t i o n wil l b e p o s s i b l e ,

j u s t a s a t h r e e - d i m e n s i o n a l v e c t o r c a n b e e x p a n d e d in C a r t e s i a n , s p h e r i -

c a l , o r c y l i n d r i c a l c o o r d i n a t e s . F u r t h e r e x a m p l e s of o r t h o n o r m a l b a s i s

s e t s a r e g i v e n in A p p e n d i x F .

APPENDIX E SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS

T h e r e is n o " r i g h t w a y " o r g e n e r a l m e t h o d t o s o l v i n g d i f fe ren t ia l

e q u a t i o n s . I t is a n a r t , r a t h e r t h a n a s c i e n c e . B y t h i s I m e a n t h a t t h e

s o l u t i o n of d i f fe ren t i a l e q u a t i o n s i n v o l v e s m a k i n g e d u c a t e d g u e s s e s a t

s o l u t i o n s , r a t h e r t h a n p r o c e e d i n g b y log ica l s t e p s f r o m s o m e s e t of first

p r i n c i p l e s . I n t h i s a p p e n d i x , w e wil l r e v i e w t h e m o s t c o m m o n f o r m s of

s o l u t i o n s t o o r d i n a r y l i n e a r e q u a t i o n s , a n d d i s c u s s s o m e i m p o r t a n t

p r o p e r t i e s of t h e s o l u t i o n s .

T h e m o s t g e n e r a l e q u a t i o n of t h i s t y p e is

w h e r e y ( x ) is a f u n c t i o n w h i c h is t o b e d e t e r m i n e d , y ( n ) 0 ) is t h e n t h

d e r i v a t i v e , fo(x)... fn(x) a r e k n o w n f u n c t i o n s of x, a n d g(x), t h e in-

h o m o g e n e o u s t e r m , is a l s o k n o w n .

I n t h e t e x t , w e m o s t o f t e n c o n s i d e r e d e q u a t i o n s of s e c o n d o r d e r , i .e .

e q u a t i o n s w h e r e n = 2 . L e t u s b e g i n b y c o n s i d e r i n g t h e h o m o g e n e o u s

e q u a t i o n of o r d e r 2 , w h i c h is

/ „ ( x ) y ( r ° ( x ) + • • • fo(x)y(x) = g(x), ( E . l )

(E .2 )

T h e g e n e r a l m e t h o d of s o l v i n g s u c h a n e q u a t i o n is t o g u e s s a f o r m of

Solution of Ordinary Differential Equations 289

a r e s o l u t i o n s t o a g e n e r a l rcth-order h o m o g e n e o u s e q u a t i o n , t h e n t h e m o s t

s o l u t i o n , a n d t h e n s e e if t h a t f o r m c a n b e m a d e t o fit t h e e q u a t i o n . F o r

e x a m p l e , w e m i g h t g u e s s a s o l u t i o n f o r y (x) in E q . (E .2 ) t o b e of t h e f o r m

y(x) = Ceax. (E .3 )

L e t u s c o n s i d e r o n l y e q u a t i o n s w h e r e

/ * ( * ) = 1

a n d

/ , (* ) = C „ f2(x) = C2.

T h e n s u b s t i t u t i n g E q . (E .3 ) i n t o E q . (E .2 ) g i v e s a n e q u a t i o n

Aeax[a2+Cxa + C 2 ] = 0, (E .4 )

w h i c h c a n b e s o l v e d f o r a. I n g e n e r a l , t h e r e wil l b e s o l u t i o n s of t h e f o r m

a = p ± y (E .5 )

f r o m t h e q u a d r a t i c f o r m u l a , w h e r e

y = WCl

2-4C2.

T h e c o n s t a n t A c a n n o t b e d e t e r m i n e d f r o m t h e e q u a t i o n , of c o u r s e .

W e a r e n o w in a p o s i t i o n in w h i c h w e h a v e t w o p o s s i b l e s o l u t i o n s of t h e

f o r m ( E . 3 ) . O n e is

y , ( x ) = A x e i p + y ) \

w h i l e t h e o t h e r i s

y2(x) = A2e(e-y)y,

w h e r e Ax a n d A2 a r e a r b i t r a r y c o n s t a n t s .

W h a t is t h e m o s t g e n e r a l s o l u t i o n t o E q . ( E . 2 ) ? If w e s u b s t i t u t e t h e f o r m

y(x) = yx(x) + y2(x) (E .6 )

i n t o E q . (2) , w e s e e t h a t it , t o o , is a s o l u t i o n of t h e e q u a t i o n . I t i s , in f a c t ,

t h e m o s t g e n e r a l s o l u t i o n t o t h e e q u a t i o n ( t h e p r o o f of t h i s is lef t t o

t e x t b o o k s in m a t h e m a t i c s ) . T h e g e n e r a l t h e o r e m ( w h i c h c a n e a s i l y b e

p r o v e d b y s i m p l e s u b s t i t u t i o n ) is t h a t if

</>i, </>2,..., <pn

290 Appendices

T h e r e a r e s e v e r a l t h i n g s w h i c h w e c a n s a y a b o u t t h i s e q u a t i o n . F i r s t of

al l , s u p p o s e t h a t yP(x) i s a s o l u t i o n of E q . (E .8). T h e n s i m p l e s u b s t i t u t i o n

s h o w s t h a t t . t . , , . / r 7 m

y ( * ) = y p ( * ) + ? » ( * ) (E .9 )

is a l s o a s o l u t i o n of E q . (E.8) p r o v i d e d t h a t yh(x) i s a s o l u t i o n of E q . (E .2).

T h u s , w e s e e t h a t t o a n y p a r t i c u l a r s o l u t i o n of E q . (E .8), w h i c h w e h a v e

c a l l e d yp, w e c a n a d d a n y s o l u t i o n o r c o m b i n a t i o n s of s o l u t i o n s of t h e

h o m o g e n e o u s e q u a t i o n . T h u s , t h e r e a r e j u s t a s m a n y u n d e t e r m i n e d

c o n s t a n t s in t h e i n h o m o g e n e o u s e q u a t i o n a s t h e r e w e r e in t h e h o m o g e -

n e o u s , a n d t h e y , t o o , m u s t b e d e t e r m i n e d f r o m t h e b o u n d a r y c o n d i t i o n s .

H o w c a n t h e p a r t i c u l a r s o l u t i o n yp b e f o u n d ? O n c e a g a i n t h e r e a r e n o

g e n e r a l p r o c e d u r e s , b u t w e h a v e t o m a k e a g u e s s , a n d t h e n s e e if it wi l l

w o r k . F o r e x a m p l e , t a k e t h e e q u a t i o n

g e n e r a l s o l u t i o n wi l l b e

<f> = A i < £ i + A2(j>2 + • " • An(f)n, (E .7 )

w h e r e A„ a r e a r b i t r a r y c o n s t a n t s .

T h e c o n s t a n t s A i a n d A2 c a n n o t , a s w e h a v e s e e n , b e d e t e r m i n e d f r o m

t h e e q u a t i o n a l o n e , b u t m u s t b e d e r i v e d f r o m a d d i t i o n a l i n f o r m a t i o n . T h i s

i n f o r m a t i o n i s u s u a l l y g i v e n in t h e f o r m of b o u n d a r y c o n d i t i o n s . T h e r e a r e

m a n y e x a m p l e s of t h i s in t h e t e x t . F o r e x a m p l e , w e m i g h t b e g i v e n t h e

v a l u e of y(jc) a t t w o p o i n t s , o r t h e v a l u e of y(jc) a n d dy/dx a t a s ing le

p o i n t . A s l o n g a s w e h a v e t w o b o u n d a r y c o n d i t i o n s ( o r n c o n d i t i o n s f o r

t h e n t h - o r d e r e q u a t i o n ) , w e c a n d e t e r m i n e t h e a r b i t r a r y c o n s t a n t s , a n d

t h e r e b y fix t h e s o l u t i o n e x a c t l y . I t m u s t b e e m p h a s i z e d t h a t b o u n d a r y

c o n d i t i o n s a r e g e n e r a l l y g i v e n b y c o n s i d e r a t i o n of t h e p h y s i c s of t h e

s i t u a t i o n , r a t h e r t h a n t h e m a t h e m a t i c s .

L e t u s n o w t u r n o u r a t t e n t i o n t o t h e m o r e g e n e r a l f o r m of E q . (E .2),

n a m e l y t h e i n h o m o g e n e o u s e q u a t i o n of o r d e r 2

(E.8)

( E . 1 0 )

T h e n a r e a s o n a b l e g u e s s m i g h t b e

yp(x) = F,

w h e r e F i s a c o n s t a n t . S u b s t i t u t i n g t h i s g u e s s b a c k i n t o E q . ( E . 1 0 ) , w e find

t h a t it wil l s a t i s fy t h e e q u a t i o n p r o v i d e d t h a t

Solution of Ordinary Differential Equations 291

T h u s , t h e m o s t g e n e r a l s o l u t i o n t o E q . ( E . 1 0 ) i s j u s t

y G ( x ) = F + y ( x ) , ( E . l l )

w h e r e y(x) i s g i v e n in E q . ( E . 6 ) .

T h e r e is o n e i m p o r t a n t p r o p e r t y of t h e i n h o m o g e n e o u s e q u a t i o n w h i c h

w e h a v e u s e d t h r o u g h o u t t h e t e x t . C o n s i d e r t h e i n h o m o g e n e o u s e q u a t i o n

of t h e f o r m

(E .12 )

a n d l e t ylp b e a s o l u t i o n of

(E .13 )

w h i l e y 2 p is a s o l u t i o n of

(E .14 )

T h u s , b y s u b s t i t u t i o n , w e c a n s e e t h a t t h e m o s t g e n e r a l s o l u t i o n of E q .

(E .12 ) wi l l j u s t b e

y ( x ) = y (x) + y l p ( J C ) + y 2 p ( x ) . ( E . 15)

T h e g e n e r a l i z a t i o n o n t h i s s t a t e m e n t t o a n y n u m b e r of t e r m s o n t h e

r i g h t - h a n d s i d e is o b v i o u s . G i v e n t h e m e t h o d of e x p a n s i o n in o r t h o g o n a l

p o l y n o m i a l s d i s c u s s e d in A p p e n d i x D , w e c a n a l w a y s w r i t e t h e in-

h o m o g e n e o u s t e r m in E q . (E .8 ) a s

g ( * ) = 2 < » n f t . ( * ) ,

w h e r e 6n(x) is s o m e s u i t a b l e s e t of o r t h o g o n a l p o l y n o m i a l s . F o r e x a m p l e ,

if w e w e r e e x p a n d i n g in a F o u r i e r s e r i e s , w e w o u l d h a v e

B y t h e t h e o r e m s t a t e d a b o v e , h o w e v e r , if w e w i s h e d t o s o l v e t h i s

e q u a t i o n , it w o u l d b e suff ic ient t o s o l v e t h e e q u a t i o n

(E .16 )

292 Appendices

T h e g e n e r a l s o l u t i o n t o E q . (E .8 ) w o u l d t h e n b e

y(x) = yn(x) + ^ypn(x), (E .17 ) n

w h e r e ypn(x) is t h e s o l u t i o n t o E q . ( E . 1 6 ) . T h i s i s t h e m a t h e m a t i c a l b a s i s

b e h i n d t h e g e n e r a l p r o c e d u r e w e f o l l o w in t h e t e x t of s o l v i n g f o r o n e

F o u r i e r c o m p o n e n t o n l y , a n d n e g l e c t i n g all o t h e r s .

APPENDIX F THE SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS

A l t h o u g h p a r t i a l d i f fe ren t i a l e q u a t i o n s , l i ke o r d i n a r y e q u a t i o n s , c a n n o t

n e c e s s a r i l y b e s o l v e d b y a p p l y i n g s o m e g e n e r a l m e t h o d , a l a r g e c l a s s of

t h e e q u a t i o n s w h i c h a r e of m o s t i n t e r e s t t o p h y s i c i s t s c a n b e s o l v e d b y t h e

t e c h n i q u e k n o w n a s t h e separation of variables. T h i s t e c h n i q u e i s , l i ke t h e

m e t h o d s d i s c u s s e d in A p p e n d i x D , a g u e s s a t w h a t t h e s o l u t i o n of a n

e q u a t i o n wil l l o o k l i ke . T h e t e c h n i q u e is t o g u e s s a s o l u t i o n , p l u g it i n t o

t h e e q u a t i o n , a n d s e e if it w o r k s . If it d o e s , t h e n w e k n o w f r o m t h e t h e o r y

of d i f fe ren t i a l e q u a t i o n s t h a t t h e s o l u t i o n is u n i q u e .

W e sha l l t a k e a s o u r e x a m p l e t h e e q u a t i o n w h i c h d e s c r i b e s p o t e n t i a l

f low of a n i n c o m p r e s s i b l e fluid,

V 2 < / > = 0 , ( F . l )

w h i c h is c a l l e d L a p l a c e ' s e q u a t i o n . I n C a r t e s i a n c o o r d i n a t e s t h i s b e c o m e s

(F .2 )

T h e e s s e n t i a l s t e p in t h e t e c h n i q u e is t o a s s u m e t h a t t h e s o l u t i o n is of

t h e f o r m

4>=X(x)Y(y)Z(z), (F .3 )

w h e r e X(x) is a f u n c t i o n of x o n l y , Y ( y ) of y, e t c . If w e p u t t h i s a s s u m e d

f o r m i n t o E q . (F .2 ) a n d d i v i d e b y XYZ, w e find

w h e r e t h e p r i m e d e n o t e s d i f f e r e n t i a t i o n w i t h r e s p e c t t o t h e a r g u m e n t of

t h e f u n c t i o n .

N o w w h a t w e h a v e h e r e is a s i t u a t i o n in w h i c h a f u n c t i o n of JC a l o n e

m u s t b e e q u a l t o t h e n e g a t i v e of a f u n c t i o n of y a n d z a l o n e . T h e o n l y w a y

The Solution of Partial Differential Equations 293

t h a t t h i s c a n b e t r u e f o r

s o t h a t

S i m i l a r l y ,

a n d

W e h a v e n o w r e d u c e d t h e p r o b l e m of s o l v i n g a p a r t i a l d i f f e ren t i a l

e q u a t i o n t o t h e p r o b l e m of s o l v i n g t h r e e o r d i n a r y ( a n d in t h i s c a s e

i d e n t i c a l ) e q u a t i o n s . F r o m A p p e n d i x E , w e k n o w t h a t t h e s o l u t i o n t o

t h e s e e q u a t i o n s i s of t h e f o r m

X = A s in ax + B c o s ax, (F .5 )

w i t h s i m i l a r f o r m s f o r Y a n d Z . T h e a c t u a l d e t e r m i n a t i o n of t h e c o n s t a n t s

A a n d B, a s in A p p e n d i x D , i s d o n e b y a p p l y i n g t h e b o u n d a r y c o n d i t i o n s .

T h e d e t e r m i n a t i o n of t h e c o n s t a n t s a a n d j8 is a l s o d o n e b y a p p l y i n g t h e

b o u n d a r y c o n d i t i o n s , b u t in a s o m e w h a t m o r e s u b t l e w a y .

S u p p o s e t h a t t h e fluid w e a r e c o n s i d e r i n g is c o n f i n e d t o a c u b e of s i d e

L . S u p p o s e f u r t h e r t h a t w e k n o w t h a t t h e p o t e n t i a l a l o n g s o m e l i ne of

c o n s t a n t y a n d z i s g i v e n b y a k n o w n f u n c t i o n F(x). T h e n w e m u s t h a v e

4>(x, yl9 z . ) = F{x) = X ( x ) Y ( y , ) Z ( z , ) , (F .6 )

w h e r e yi a n d z, a r e t h e v a l u e s of y a n d z a l o n g t h e p l a n e . S i n c e

Y(yx)Z(zx) is j u s t a c o n s t a n t , w e m u s t h a v e t h a t t h e s o l u t i o n f o r X(x) in

E q . (F .5 ) r e d u c e s t o F(x) in t h i s c a s e .

F r o m A p p e n d i x D a n d t h e de f in i t ion of a F o u r i e r s e r i e s , w e k n o w t h a t

w e c a n d o t h i s b y c h o o s i n g t h e c o n s t a n t s s u c h t h a t

a n d

all v a l u e s of x is f o r t h a t f u n c t i o n t o b e a c o n s t a n t ,

(F .4 )

294 Appendices

s o t h a t , u p t o a n o v e r a l l c o n s t a n t ,

(F .7 )

T h i s i s t h e s o l u t i o n of a t y p i c a l boundary-value problem. I n g e n e r a l ,

Y ( y ) a n d Z ( z ) wi l l a l s o b e g i v e n b y F o u r i e r s e r i e s of s o m e b o u n d a r y -

v a l u e f u n c t i o n s . T h e r e a d e r i s r e f e r r e d t o t h e t e x t s a t t h e e n d of t h e

i n t r o d u c t i o n t o t h e a p p e n d i c e s f o r m o r e d e t a i l e d d i s c u s s i o n of t h i s p o i n t .

T h e m a i n t h i n g t h a t w e w a n t t o e m p h a s i z e is t h a t t h e s o l u t i o n of t h e

L a p l a c e e q u a t i o n is i n t i m a t e l y t i e d t o t h e e x i s t e n c e of o r t h o n o r m a l s e t s of

p o l y n o m i a l s ( in t h i s c a s e t h e s i n e s a n d c o s i n e s ) w h i c h a r e , in f a c t , t h e

s o l u t i o n s t o t h e o r d i n a r y d i f f e ren t i a l e q u a t i o n s w h i c h r e s u l t f r o m a p p l y i n g

t h e t e c h n i q u e of s e p a r a t i o n of v a r i a b l e s .

W e m i g h t g u e s s , t h e n , t h a t o t h e r s e t s of t h e s e p o l y n o m i a l s m i g h t a r i s e

f r o m s o l u t i o n s of t h e e q u a t i o n i n o t h e r c o o r d i n a t e s y s t e m s . F o r e x a m p l e ,

j u s t a s t h e s i n e s a n d c o s i n e s a r e p a r t i c u l a r l y a p p r o p r i a t e f o r e x p a n d i n g

f u n c t i o n s in C a r t e s i a n c o o r d i n a t e s , t h e r e m i g h t b e o t h e r f u n c t i o n s w h i c h

a r e a p p r o p r i a t e f o r e x p a n d i n g f u n c t i o n s in s p h e r i c a l c o o r d i n a t e s .

T h e L a p l a c e e q u a t i o n in s p h e r i c a l c o o r d i n a t e s i s

A s b e f o r e , w e n o t e t h a t t h i s c a n o n l y b e t r u e if t h e f u n c t i o n of cp i s a

c o n s t a n t , w h i c h w e t a k e t o b e

(F .8 )

If w e p r o c e e d a s b e f o r e a n d a s s u m e a s o l u t i o n of t h e f o r m

* = * ( r ) P ( 0 ) Q ( * ) ,

t h e n w e find, u p o n d i v i d i n g b y <Pr2 s i n 2 0,

(F .9 )

s o t h a t Q = e±im*.

C l e a r l y , if w e w i s h t h e f u n c t i o n t o b e s ing l e v a l u e d , s o t h a t

* ( r , « , * ) = * ( r , e , 0 + 2 w ) ,

(F .10 )

w e m u s t h a v e m b e a n i n t e g e r ( t h i s i s a c t u a l l y t h e first a p p l i c a t i o n of t h e


b o u n d a r y c o n d i t i o n s ) . W e a r e t h e n lef t w i t h

O n c e m o r e , t h i s e q u a t i o n c a n b e sa t i s f ied o n l y if t h e f u n c t i o n of R i s a

c o n s t a n t , w h i c h , f o r c o n v e n i e n c e , w e wi l l t a k e t o b e

w h e r e / c a n b e a n y n u m b e r . F r o m t h e m e t h o d s of A p p e n d i x E , a s s u m i n g

a n R of t h e f o r m rq y i e l d s a s a s o l u t i o n

(F.ll)

a n d l e a v e s u s w i t h t h e r e s u l t

(F .12 )

T h i s i s k n o w n a s Legendre's equation. I t i s u s u a l l y w r i t t e n in a f o r m w h e r e t h e c h a n g e of v a r i a b l e s

x = c o s 0

h a s b e e n m a d e , s o t h a t

(F .13 )

L e t u s c o n s i d e r first t h e c a s e of a z i m u t h a l s y m m e t r y , w h e r e m = 0 . T h e e q u a t i o n t o b e s o l v e d is t h e n

( F . 1 4

L e t u s a s s u m e t h a t w e c a n find a s o l u t i o n of t h i s e q u a t i o n of t h e f o r m

(F .15 )

w h e r e t h e coe f f i c i en t s an a r e t o b e d e t e r m i n e d . P r o v i d e d t h a t e v e r y t h i n g

is w e l l b e h a v e d , t h i s i s n o t a l a r g e a s s u m p t i o n , s i n c e i t a m o u n t s t o

e x p a n d i n g t h e s o l u t i o n in a T a y l o r s e r i e s .

If w e i n s e r t t h i s a s s u m e d f o r m of s o l u t i o n i n t o E q . ( F . 1 4 ) , w e find

[/(/ + l ) ] c 0 + 2ca + [ ( / ( / - 1) - 2 ) c , + 6c3]x

+ [ ( / ( / + 1) - 6 ) c 2 + I2c4]x2 + • • • = 0.

296 Appendices

N o w in o r d e r f o r t h e r e t o b e a s o l u t i o n w h i c h is v a l i d f o r e v e r y v a l u e of x,

t h e coef f ic ien t of e a c h p o w e r of x m u s t v a n i s h i d e n t i c a l l y . T h i s m e a n s

t h a t

a n d

w i t h s imi l a r r e l a t i o n s b e t w e e n ci9 c 3 , c 5 , e t c . I n g e n e r a l , w e h a v e

(F .16 )

T h e r e a r e s e v e r a l p o i n t s t o n o t e a b o u t t h i s r e s u l t . F i r s t of a l l , if a n y c„ is

e v e r z e r o , t h e n e v e r y h i g h e r v a l u e of n wi l l a l s o h a v e a v a n i s h i n g

coeff ic ien t . F o r e x a m p l e , if c 6 w e r e z e r o , t h e n E q . (F .16 ) w o u l d g i v e c 8 t o

b e z e r o , a n d a p p l y i n g t h e e q u a t i o n a g a i n w o u l d g i v e cxo = 0 , a n d s o f o r t h .

A s e c o n d p o i n t is t h a t e v e r y t e r m w i t h e v e n n c a n b e r e l a t e d b a c k t o c 0 b y

r e p e a t e d u s e of E q . ( F . 1 6 ) , a n d e v e r y t e r m of o d d n c a n b e r e l a t e d b a c k t o

C i . F u r t h e r m o r e , t h e o d d a n d e v e n t e r m s a r e n o t r e l a t e d t o e a c h o t h e r s o

t h a t E q . (F .15 ) c a n b e w r i t t e n

P(x) = Co 2 a2nx2n + c , 2 b 2 n + x x 2 n + \ (F .17 )

i .e . a s a s u m of e v e n i n d i c e s p l u s a s u m o v e r o d d i n d i c e s . T h e r e is n o t h i n g

in t h e e q u a t i o n , h o w e v e r , t o te l l u s w h a t t o t a k e f o r cx a n d c 0 . B y

c o n v e n t i o n , w e u s u a l l y t a k e e i t h e r c 0 o r cx t o b e z e r o ( s o t h a t t h e s o l u t i o n

is e i t h e r o d d o r e v e n ) , a n d a d j u s t t h e n o n z e r o coef f ic ien t s u c h t h a t

P ( 0 ) = 1 .

T h e p o l y n o m i a l s w h i c h a r e g e n e r a t e d in t h i s w a y a r e c a l l e d t h e

Legendre polynomials. W e n o t e t h a t if / i s a n i n t e g e r , t h e n t h e f a c t o r i — I

in E q . (F .16 ) wil l v a n i s h w h e n i = /, s o t h a t t h e p o l y n o m i a l wi l l b e of o r d e r

/, a n d wil l c o n t a i n n o h i g h e r p o w e r s of x. F o r t h i s r e a s o n , it is c u s t o m a r y

t o d e n o t e t h e L e g e n d r e p o l y n o m i a l b y Pi. T h e first f e w p o l y n o m i a l s a r e

P o = l , Px = x,


(F .18 )

[ s e e , f o r e x a m p l e , t h e t e x t b y M a t h e w s a n d W a l k e r c i t e d in t h e b i b l i o g -

r a p h y ] . T h e r e f o r e , if w e de f ine

(F .19 )

t h e n t h e Ui(x) f o r m a n o r t h o n o r m a l s e t f o r e x p a n s i o n of f u n c t i o n s a s

s e r i e s in c o s 0, j u s t a s t h e s i n e s a n d c o s i n e s d i d f o r e x p a n s i o n in t h e l i n e a r

c o o r d i n a t e .

A m o r e u s e f u l s e t of f u n c t i o n s c a n b e g e n e r a t e d if w e c o n s i d e r t h e c a s e

of n o n a z i m u t h a l s y m m e t r y . I t is s t r a i g h t f o r w a r d , b u t r e l a t i v e l y t e d i o u s t o

s h o w t h a t t h e g e n e r a l s o l u t i o n t o E q . (F .12 ) is g i v e n b y

(F .20 )

T h i s i s c a l l e d t h e associated Legendre function, a n d h a s t h e p r o p e r t y

[ a n a l o g o u s t o E q . (F .18) ] t h a t

(F .21 )

T h e s o l u t i o n t o t h e L a p l a c e e q u a t i o n m u s t t h e n b e of t h e f o r m

<$> = R(r)crPr(x)eim*.

T h e a n g u l a r p a r t of t h i s f u n c t i o n , c o n t a i n i n g t h e d e p e n d e n c e of t h e

s o l u t i o n o n t h e a n g l e s 6 a n d cp, i s e x t r e m e l y i m p o r t a n t , a n d is g i v e n t h e

n a m e of spherical harmonic. I t is w r i t t e n

(F .22 )

where we have inserted a factor of l/V27rto normalize the function eim<f>.

T h e spherical harmonics have the property that

(F .23 ) Yim(0, <j>)YVmiO, <f>) d ( c o s 0 ) d<f> = 8U. 8m„

a n d h i g h e r o r d e r s c a n b e w o r k e d o u t f r o m t h e r e c u r s i o n r e l a t i o n in E q .

( F . 1 6 ) .

H a v i n g s o l v e d L a p l a c e ' s e q u a t i o n in s p h e r i c a l c o o r d i n a t e s , w e n o w a s k

o u r s e l v e s w h e t h e r t h e s o l u t i o n s in t h i s c a s e f o r m a n o r t h o n o r m a l s e t , a s

d i d t h e s i n e s a n d c o s i n e s in t h e C a r t e s i a n c a s e . I t i s r e l a t i v e l y s i m p l e t o

s h o w t h a t r 0

298 Appendices

i .e . t h e y a r e a n o r t h o n o r m a l s e t of f u n c t i o n s . U n l i k e t h e s i n e s a n d c o s i n e s

o r t h e L e g e n d r e p o l y n o m i a l s , h o w e v e r , t h e y a r e t h e b a s i s v e c t o r s in a

s p a c e of f u n c t i o n s of t w o v a r i a b l e s , r a t h e r t h a n o n e . T h e e x t e n s i o n of t h e

i d e a of A p p e n d i x D t o t h i s c a s e s h o u l d b e o b v i o u s .

T h i s m e a n s t h a t , j u s t a s w e c o u l d e x p a n d a n y f u n c t i o n d e f i n e d o n t h e

i n t e r v a l 0 ^ x ^ L i n a F o u r i e r s e r i e s , w e c a n e x p a n d a n y f u n c t i o n d e f i n e d

o n t h e i n t e r v a l 0 =^ </> ^2IT 0 = ^ 0 =^ 7r in a s e r i e s i n v o l v i n g s p h e r i c a l

h a r m o n i c s . S u c h a s e r i e s w o u l d t a k e t h e f o r m

*) = 2 flimYim(0, (F .24 ) i l,m w h e r e

/(»', <f>')Ylm(Of, <y) d ( c o s 0 ' ) d<t>. (F .25 )

S u c h e x p a n s i o n s a r e e x t r e m e l y i m p o r t a n t in p r o b l e m s d e a l i n g w i t h

s p h e r i c a l g e o m e t r i e s , s u c h a s p r o b l e m s r e l a t i n g t o m o t i o n s o n t h e s u r f a c e

of t h e e a r t h o r d e f o r m a t i o n s of a n u c l e u s .

T h e r e r e m a i n s a t h i r d s e t of c o o r d i n a t e s w h i c h w e u s e d in t h e t e x t , a n d

t h i s w a s t h e c y l i n d r i c a l . L a p l a c e ' s e q u a t i o n in c y l i n d r i c a l c o o r d i n a t e s i s

(F .26)

If w e p r o c e e d a s in E q . ( F . 6 ) , a n d a s s u m e t h a t t h e s o l u t i o n is s e p a r a b l e , s o

t h a t )Z(z),

t h e n t r a c i n g t h e s t e p s f r o m E q . (F .8 ) t o E q . (F .12 ) y i e l d s

Z(z) = e ± k z

a n d

Q() = e ± i n \

w h i l e t h e f u n c t i o n R(r) i s d e t e r m i n e d b y t h e e q u a t i o n

(F .27 )

w h e r e w e h a v e s e t x = kr. T h i s i s c a l l e d B e s s e l ' s e q u a t i o n , a n d t h e

s o l u t i o n s t o it a r e c a l l e d Bessel functions.

W e c a n d e t e r m i n e t h e f o r m of t h e B e s s e l f u n c t i o n s j u s t a s w e

d e t e r m i n e d t h e L e g e n d r e p o l y n o m i a l s . A s s u m i n g a p o w e r s e r i e s s o l u t i o n

of t h e f o r m

(F .28 )


w e find, in a n a l o g y t o E q . ( F . 1 6 ) , t h a t

a n d

a = n,

s o t h a t t h e B e s s e l f u n c t i o n is s i m p l y a p o w e r s e r i e s in r, g i v e n b y

w h e r e w e f o l l o w t h e u s u a l c o n v e n t i o n a n d s e t a0 = [2nT(n + 1)]. A n

i m p o r t a n t d i f f e r e n c e in t h i s c a s e is t h a t t h e s e r i e s d o e s n o t t e r m i n a t e , b u t

i n c l u d e s all v a l u e s of n.

T h e f u n c t i o n Jn(x) is c a l l e d t h e Bessel function of order n. I t h a s t h e

g e n e r a l p r o p e r t y t h a t t h e f u n c t i o n o s c i l l a t e s a r o u n d z e r o , a s s h o w n

s c h e m a t i c a l l y in F i g . F . l . W e c a n d e n o t e b y xvn t h e v a l u e of x f o r w h i c h

t h e B e s s e l f u n c t i o n of o r d e r n b e c o m e s z e r o f o r t h e vth t i m e . I t t h e n

f o l l o w s ( s e e t h e t e x t s in t h e b i b l i o g r a p h y ) t h a t t h e B e s s e l f u n c t i o n m u s t

h a v e t h e o r t h o g o n a l i t y r e l a t i o n

w h i c h is a n o r t h o g o n a l i t y c o n d i t i o n s i m i l a r t o E q . (F .18 ) f o r L e g e n d r e

p o l y n o m i a l s .

O b v i o u s l y , t h e B e s s e l f u n c t i o n s c a n b e e x p e c t e d t o p l a y a n i m p o r t a n t

r o l e in p r o b l e m s i n v o l v i n g c y l i n d r i c a l s y m m e t r y , s u c h a s flow of t h e b l o o d

in a n a r t e r y . T h e r e a d e r s h o u l d b e a b l e t o c o n s t r u c t f o r h i m s e l f t h e Bessel

series, w h i c h is t h e a n a l o g u e t o t h e F o u r i e r s e r i e s , in t e r m s of w h i c h

f u n c t i o n s w h o s e a r g u m e n t r u n s f r o m 0 ^ r ^ a c a n b e e x p a n d e d .

T h e r e a r e , of c o u r s e , m a n y m o r e s e t s of o r t h o g o n a l p o l y n o m i a l s w h i c h

a r e of u s e in s p e c i a l i z e d p r o b l e m s . I n t h i s t e x t , o n l y t h e s p h e r i c a l

(F .29 )

Fig. F.l. A typical Bessel function.

300 Appendices

h a r m o n i c s a n d t h e B e s s e l f u n c t i o n s a p p e a r , a n d t h e s t u d e n t wil l b e a b l e t o

h a n d l e a l m o s t all m a t e r i a l w h i c h h e e n c o u n t e r s if h e h a s a g r a s p of t h e s e

b a s i c f u n c t i o n s a n d t h e i d e a s a n d c o n c e p t s w h i c h u n d e r l i e t h e i r u s e .

O n e final p o i n t s h o u l d b e m a d e . W e h a v e m e n t i o n e d t h a t t h e s e n e w

f u n c t i o n s h a v e a p r o p e r t y of o r t h o g o n a l i t y , b u t w e h a v e n o w h e r e s h o w n

t h a t t h e y f o r m a c o m p l e t e s e t of b a s i s f u n c t i o n s . I n f a c t , t h i s is s h o w n in

m o s t t e x t s o n d i f f e ren t i a l e q u a t i o n s , a n d n e e d n o t d i s t u r b t h e r e a d e r

u n d u l y .

Index

Acoustic wave in a fluid, 85-86 in a solid, 221-222

Arms control, 240, 244 Arterial walls

composition, 250 response to pressure, 252ff.

Arteriosclerosis, 263 Artery, 249: see also blood flow

Benard cell: see convection cell Bernoulli equation, 57, 65 Bessel

equation, 258, 267, 298 function, 298-299

Biharmonic equation, 232 Blood

cells, 250 composition, 249 flow, 249ff.

arterial, 256rT. Reynolds number for, 152

Bonneville, 131, 136 Borda's mouthpiece, 66 Bossinesq approximation, 158, 172 Boundary layer, 145 Boundary-layer separation, 151 Boundary-value problem, 294 Breakaway, 241

Buckling, 200 Bulk modulus, 219

Cantilever, 204, 207 Capillary

in blood circulation, 249 jet, 265ff. wave, 81-83

Circulation, 66 Circulatory system, 248ff. Collagen, 250-251 Complex potential, 66 Continental drift, 179, 195, 240 Continuity

equation, 5-8 for plane surface, 71 for spherical surface, 96, 103

Convection cells, 168-170 in the atmosphere, 171, 172 and continental drift, 179

Convective derivative, 2-3 Core (of the earth), 240 Coriolis force, 93 Crust (of the earth), 239

Diffusivity, coefficient of, 157 Disturbing potential, 89

at equator, 90 general form, 97

301

302 Index

Doldrums, 172 Drag, 138, 152

Earth as a fluid, 27 free oscillations, 111 viscosity of, 130-136

Elastic constants, 189, 217-220 Elastic solid, 188 Elastin, 250-251 Entropy flux density, 13 Entry problem, 264 Equation of state, 9, 17

poly tropic, 12 Equilibrium

neutral, 41 stable, 40, 58 systems far from, 118-119 types of, 39 unstable, 41, 58

Euler equation, 4-5, 123 in a galaxy, 30 for potential flow, 57 for rotation, 18-19, 93

Euler theory of struts, 199 Expansion, coefficient of, 158

Fenno-Scandian uplift, 131, 136, 203, 204 First law of thermodynamics, 156 Fission

induced, 119 of a nucleus, 117-119 spontaneous, 116

Fissionability parameter, 116-118 Fluid

classical, 2, 122, 164, 184 incompressible, 7

Fourier series, 288, 291, 293

Galileo, 138 Grashof number, 180 Green's theorem, 283

Hadley cell, 171, 177 Heat equation, 156 Heat transfer equation, 181 Hilbert space, 287 Hooke's law, 188, 190, 209, 217, 218, 230,

250, 259 Horse latitudes, 172 Huygens principle, 235 Hydraulics, 144

Ideal gas law, 9 Incompressible fluid, 7, 84

earth as an incompressible fluid, 27 Inversion, thermal, 164 Irrotational flow, 56

Jacobi ellipsoids, 27

Kronecker delta, 278

Lacolith, 192-195, 205 Lame coefficients, 217-220, 230 Laplace

equation, 75, 292, 294, 298 equation for potential flow, 56 theory of the tides, 90, 102

Legendre equation, 295 function, 296-297

Linearization, 61, 63, 73 Liouville Theorem, 65 Loading

critical, 200 of a solid, 186

Long waves, 68, 74 Longitudinal wave in a solid, 221 Love waves, 227-229, 261 Lubrication, theory of, 153

Mach number, 153 Maclaurin ellipsoids, 22, 28

stability of, 43-47 Mantle, 239 Membrane tension, 254 Micron, 250 Mountain chain, 195-199 Mohorovicic discontinuity, 240

Navier-Stokes equation, 127 Neutral filament, 190 Newton's second law of motion, 2, 5, 123,

220, 252 Newtonian solid, 217, 230, 250

Index 303

Normal modes of oscillation for the earth, 110 for oceans, 104

Orr-Sommerfeld equation, 273

P wave, 223, 235, 238 in nuclear explosions, 244 reflection, 245-256

Pisa, leaning tower, 138 Poisieulle

flow, 127, 152 formula, 130, 271

Poisson equation, 9, 13 ratio, 189, 192, 218 relation, 226

Potential flow, 56 Prandtl, 144

equations, 148 number, 180

Rayleigh criterion for convection, 167, 180 for jet stability, 270, 272

Rayleigh, Lord, 167, 180, 226 Rayleigh wave, 226-227

in a nuclear explosion, 244 Reynolds number, 142

for blood flow, 152, 272 Rheology, 188 Ripple, 83 Roche's limit, 47-48

5 wave, 223, 235, 238 in nuclear explosion, 244 reflection, 245-246

SH wave, 225, 227 SV wave, 225, 227

reflection of SH and SV waves, 245-256

Second sound, 182 Seiche, 84 Seismic radius, 241 Seismic ray parameter, 238 Separation of variables, 292 Shear force

in a fluid, 123

at a fluid surface, 135 in solids, 212, 214, 219

Shear modulus, 219 Shear wave

in a solid, 222 horizontal, 225

Similarity, law of, 144, 152, 180 Slip, 129 Smog, 164 Snell's law, 236, 246 Solar wind, 14 Sound wave in a fluid, 85-86 Specific heat, 156 Spherical harmonics, 297 Stoke's

first problem, 139 formula, 138 second problem, 140

Strain tensor, 210-212, 214, 217 Stream function, 66, 67, 149, 153, 273 Streamline, 65 Stress function, 232 Stress tensor

Maxwell, 215 for a solid, 212-216, 217, 218, 230, 252 for viscosity, 124, 126

Strong interactions, 112 Struts, 200 Summation convention, 277 Superfluid, 182 Surface tension, 79, 254

Tamped explosion, 241 Tangential instability, 59 Tensor

Cartesian, 3, 276ff. Maxwell stress, 215 momentum flux, 8, 13 strain, 210-212 stress, 212-216, 230 viscous, 124

Thermal conductivity coefficient of, 156

Tides, 88 diurnal, 98, 101, 103 equatorial, 89 inverted, 92, 101 monthly, 98

304 Index

Tides (continued) planetary, 102 semi-diurnal, 91-92, 98, 101 solar, 92, 102

Torsional rigidity, 231 Tuning fork, 233

Urethra, 265, 270 Urinary drop spectrometer, 265ff. Urinary system, 264ff.

Velocity field, 49 Velocity potential, 56

for capillary jet, 267 for surface waves, 75

Viscoelastic solid, 250, 256 Viscosity, 122ff.

coefficients of, 126 energy, 137-138 kinematic coefficient of, 127

Vorticity transport equation, 138

Waves in solids body, 220-223 surface, 223-227 thin sheets, 232, 233

Wronskian determinant, 163

Young's modulus, 188, 218, 230

Zonal heating, 172

Documents

J. S. Trefil Auth. Introduction to the Physics of Fluids and Solids 1975