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29T

HE

ST

AB

ILIT

Y O

F C

OU

ET

TE

FL

OW

Cha

p. V

II

fi of the two integrals included in 14 is positive definite for p, ? 0; but

the

seco

nd is

com

plex

. How

ever

, the

rea

l par

t of 1

4 is

pos

itive

def

inite

for

p, ?

0; i

n fa

ct,

Ii /dV vjZ

re(14) = rrP(r) dr -r dr.

'T

For,

exp

andi

ng th

e in

tegr

and

in (

185)

, we

have

~ I I

I rrp(r)I~~ -~r dr = I rp(r)(rl~~IZ + I~IZ) dr- I rP(r) d~~IZ dr;

'T 'T 'T

but I 1 I

I dlv

lZ I

(I )

d1vl

Z I

i dlv

lzrP(r)Tdr=(I-p,) r2-KTdr=(I-p,) r2Tdr. (187)

'T 'T 'T

The

refo

re, t

he r

ight

-han

d si

de o

f (1

86)

is, i

ndee

d, th

e re

al p

art o

f 14

,R

etur

ning

to e

quat

ion

(180

) an

d eq

uatin

g th

e re

al p

arts

of t

his

equa

tion,

we

obta

in

re(a

)(II-

.raz

I3)+

IZ-.

raZ

(a2f

-tre

(14)

) =

O. (

188)

Whe

n p,

? 7

)z, .

r -:

0 a

nd th

e co

effc

ient

of r

e(a)

in e

quat

ion

(188

) is

posi

tive

defi

nite

; and

so

also

are

the

rem

airu

ng te

rms

in th

e eq

uatio

n.T

here

fore

,re

(a)

-: 0

for

p, ?

7)z

, (18

9)an

d th

e fl

ow is

sta

ble;

this

res

ult i

s en

tirel

y to

be

expe

cted

on

phys

ical

grou

nds.

Nev

erth

eles

s, it

app

ears

to b

e th

e on

ly o

ne w

hich

can

be

esta

blis

hed

by g

ener

al a

naly

tical

arg

umen

ts. I

n pa

rtic

ular

, it d

oes

not

seem

that

one

can

ded

uce

the

gene

ral v

alid

ity o

f th

e pr

inci

ple

of th

eex

chan

ge o

f st

abilt

ies

for

this

pro

blem

. For

exa

mpl

e, b

y eq

uatin

g th

eim

agin

ary

part

s of

equ

atio

n (1

80),

we

obta

in (

cf. e

quat

ion

(176

))i

im(a

)(II+

.raz

I3)

= -

2TaZ

im I

~ d

;r*

dr, (

190)

'T

and

no g

ener

al c

oncl

usio

ns c

an b

e dr

awn

from

this

equ

atio

n; w

hen

p, -

: 0,

even

13

is n

ot p

ositi

ve d

efin

ite!

71. T

he s

olut

ion

for

the

case

of

a na

rrow

gap

whe

n th

e m

argi

nal

stat

e is

sta

tiona

ryH

the

gap

Rz-

Ri b

etw

een

the

two

cylin

ders

is s

mal

l com

pare

d to

thr

mea

n ra

dius

.HR

z+R

i), w

e ne

ed n

ot (

as in

§ 6

8 (b

)) d

istin

guis

h

'0 r

' s: c

' ¡, /

1 /lL

j/? I

I r-

€ k

l- A

- )

YI!#d/J':¡'~ ~C" P?0 17YtJ/)D/MP'Á-ø'iì¿

f'7/ ~?I y f ._(1

85)

(186

)

§ 71 THE STABILITY OF COUETTE FLOW 299

between D and D* in equations (161) and (162); and we ca alo replace

(A+

Blr

Z)

whi

ch o

ccur

s on

the

righ

t-ha

nd s

ide

of e

qua.

tion

(161

) by

( r-Ri J

01 1-(I-p,) Rz-Ri . (191)

In r

ewri

ting

equa

tions

(16

1) a

nd (

162)

in th

e fr

amew

o:ro

f~~p

roxi

-mations, it wil be convenient to measure radial ~.:f the

surf

ace

of th

e in

ner

cylin

der

in th

e un

it d

= R

z-R

i. 'l~

;l-""

"",;:

;;; .

~ =

(r-

Ri)

ld, k

= a

id, a

nd 0

' = ix

/v. (

192)

we

have

to c

onsi

der

the

equa

tions

20 dZ

(DZ-aZ-a)(DZ-aZ)u = --aZ(I-(I-p,)ij-

v 2AdZ

and

(DZ

-aZ

-a)v

= -

'1.

v

By

the

furt

her

tran

sfor

mat

ion

20 dZaz

'1 -+ I '1,

v

the

equa

tions

bec

ome

(DZ

-aZ

-a)(

DZ

-aZ

)u =

(l+

tX~

)v

(DZ

-aZ

-a)v

= ~

Taz

u,

T = _ 4AOi d4

vZ

and

whe

re, n

ow,

and

tX =

-(I

-p,)

.E

quat

ions

(19

6) a

nd (

197)

mus

t be

cons

ider

ed to

geth

er w

ith th

e bo

un:

conditions r d

'1 =

Du

= v

= 0

for.

. = 0

an

1.W

e ar

e pr

imar

ily in

tere

sted

in th

e so

lutio

ns o

f eq

uatio

ns (

196)

(197

) (s

ubje

ct to

the

boun

dary

con

ditio

ns (

200)

) fo

r va

riou

s va

lues

of.

.fo

r w

hich

the

real

par

t of

0' is

zer

o. T

he m

etho

d de

scri

bed

in C

hapt

er I

I.§

31 in

a d

iffe

rent

con

nexi

on is

app

licab

le to

this

pro

blem

. Thu

s, w

Öm

ust o

btai

n so

lutio

ns f

or tw

o ca

ses:

whe

n 0'

is z

ero

and

the

mar

gist

ate

is s

tatio

nary

and

whe

n 0'

is im

agin

ary

and

the

mar

gina

l sta

te is

osci

lato

ry. I

n th

e la

tter

case

for

eac

h va

lue

of a

, ia

mus

t be

dete

rmne

dby

the

cond

ition

that

T is

rea

L. t

In

eith

er c

ase,

we

mus

t fin

d th

e m

ini-

mum

of

T a

s a

func

tion

of a

; and

dep

endi

ng o

n w

hich

oft

he tw

o m

irum

a.is lower, we shall have the onset of

inst

abilt

y as

a s

tatio

nary

sec

onda

ryflow or as overstabilty. Careful experiments on the onset of

inst

abilt

y

t It is, of course, possible that under certain circumstances solutions with thi

pry

do n

ot e

xist

.

I'/JÆ

ÇS

r 19

(1)

TaE STABILITY OF COUETTE FLOW

Cha

p. V

II

and others have failed to reveal any suggestions of over-

stbilty. For this reason, the case a = 0 is the only one which has been

'Cns

ider

ed in

the

liter

atur

e. H

owev

er, a

s no

gen

eral

arg

umen

ts f

or th

e-i

iidity

of

the

prin

cipl

e of

the

exch

ange

of

stab

iltie

s ha

ve b

een

foun

dfo

r th

is p

robl

em, t

he c

ase

of o

vers

tabi

lty r

equi

res

inve

stig

atio

n. W

ereturn to this question in § 72.

Whe

n th

e m

argi

nal s

tate

is s

tatio

nary

, the

equ

atio

ns to

be

solv

ed a

re

(D2_

a2)2

u =

(1+

o:')v

and

(D2_a2)V = - Ta2u,

toge

ther

with

the

boun

dary

con

ditio

ns (

200)

.

(a)

The

sol

utio

n of

the

char

acte

ristic

val

ue p

robl

em fo

r th

e ca

se a

= 0

It c

an b

e re

adily

ver

ifie

d th

at th

e ch

arac

teri

stic

val

ue p

robl

em p

rese

n-te

d by

equ

atio

ns (

200)

-(20

2) is

not

sel

f-ad

join

t in

the

usua

l sen

se. F

orth

is r

easo

n, th

e m

etho

d to

be

desc

ribed

bel

ow w

as p

atte

rned

afte

r th

eon

es w

hich

hav

e be

en f

ound

suc

cess

ful i

n ca

ses

whe

re th

e pr

oble

ms

are

self

-adj

oint

. How

ever

, Rob

erts

has

rec

ently

fou

nd a

var

iatio

nal b

asis

for

the

met

hod;

this

is c

onsi

dere

d in

App

endi

x iv

.T

he m

etho

d of

sol

utio

n w

e sh

all a

dopt

is th

e fo

llow

ing.

. Sin

ce v

is r

equi

red

to v

anis

h at

, =

0 a

nd I,

we

expa

nd it

in a

sin

ese

ries

of

the

form

co

v =

L e

m s

in m

7T'.

(203

)m

=l

Hav

ing

chos

en v

in th

is m

anne

r, w

e ne

xt s

olve

the

equa

tion,

co

(D2_a2)2u = (1+0:') L Cmsinm7T',

m=

l

obta

ined

by

inse

rtin

g (2

03)

in (

201)

, and

arr

ange

that

the

solu

tion

satis

fies

.the

four

rem

aini

ng b

ound

ary

cond

ition

s on

u. W

ith u

det

erm

ined

in this fashion and v given by (203), equation (202) wil

lead

, as

we

shal

lpr

esen

tly s

ee, t

o a

secu

lar

equa

tion

for

T.

The solution of equation (204) is straightforward. The general

solu

tion

can

be w

ritte

n in

the

form

co C

(u=" m A(m)coshaY+B(m)sinhaY+A(m)YcoshaY+

¿ (m27T2+a2)2 1 ~ 1 ~ 2 ~ ~

m=

l

+B

~m),

sin

ha'+

(I+

o:')s

in m

7T' +

~0:

:i7T

2 c

osm

7T'),

m 7

T +

a

where the constants of integration Aim), A~m), Bim), and B~m) are to be

(201

)

(202

)

(204

)

§ 71 THE STABILITY OF COUETTE .FLOW 301

determined by the boundary conditions u = Du = 0 at , = 0 and 1.

The

se la

tter

cond

ition

s le

ad to

the

equa

tions

:

Aim

) =

4mm

x -,m

27T

2+a2

aBim)+A~m) = -m7T,

Aim

) co

sha+

Bim

) si

nha+

A~

m)

cosh

a+B

~m

) si

nha

= (-I )m+1 4m7T0:

m27

T2+

a2'

Aim

) a

sinh

a+

Bim

) a

cosh

a+

A~

m)(

cosh

a+

a si

nh a

)+

+B

~m

)(si

nha+

acos

ha)

= (

_I)m

+l(l

+o:

)m7T

. (20

6)

On

solv

ing

thes

e eq

uatio

ns, w

e fi

nd th

at

Aim

) =

40:m

7T

m27

T2+

a2'

Bim) = ~7T ta+ßm(sinh

a+a

cosh

a)-Y

m

sinh

a)

,

A~m) = _ m7T tsinh2a+ßma(sinha+acosha)-Ymasinha),

~

B~m) = :7T t(sinha

cosh

a-

a)+

ßm

a2

sinh

a-

Ym

(a

cosh

a-

sinh

a)), (2

07)

whe

re.6

= s

inh2

a-a2

,

ßm =

2 ~

o: ?

((-I

)m+

1+co

sha)

,m 7T +a.

and

Ym

= (

-1)1

+1(

1+0:

)+ 2

~o:

2as

inha

. (20

8)m

7T

+a

Now

sub

stitu

ting

for

v an

d u

from

equ

atio

ns (

203)

and

(20

5) in

equ

a-tio

n (2

02),

we

obta

inco L Cn(n27T2+a2)sin n7T'

n=l

co C

(= Ta2 " m A(m)

cosh

aY+

B(m

) si

nhaY

+A

(m)y

co

shaY

+¿ (m27T2+a2)2 1 ~ 1 ~ 2 ~ ~

m=

l

+B~m)'sinha'+(I+o:')sinm7T'+ ~CX:i7T 2cosm7T'). (209)

m 7

T +

a

Multiplyig equation (209) by sin n7T' and integrating over the ra of

.t. w

e ob

tain

a s

yste

m o

f lin

ear

hom

ogen

eous

equ

atio

ns f

or th

~"!

~

t ,t ~~ t 11

TH

E S

TA

BIL

ITY

OF

CO

UE

TT

E F

LOW

Cha

p. V

II

~.. = Om/(m21T2+a2)2; and the requirement that these constants are

not al zero leads to the secular equation

~(n2.;a2)((I+( _1)"Hl cosha)Aim)+(( _1)n+i sinha )

Bim

) +

+(-I)n+l(cosha- 2a sinaJA(m)+

n21T2+a2 2

+ ((-I)n+lSinha- 2a P+(-I)n+icoshaJJB&m))+

n21T

2+a2

+o:Xnm+l3nm~l(n21T2+a2)3:~TII = 0, (210)

( 0

if m

+n

is e

ven

and

m i=

n,

whe

re X

nm =

ì if

m =

n,

~nm

2(

2; 2

- 2(

21

2))

if m

+n

is o

dd. (

211)

n -m

m 1

T +

a 1T

n-m

On

usin

g th

e fir

st tw

o eq

uatio

ns o

f (20

6), e

quat

ion

(210

) si

mpl

ifies

toth

e fo

rm

II n1T f 4m1T0: (( l)m+n 1)

n21T

2 +

a2\m

21T

2 +

a2

2a (( _i)n+lf A (m) sinh

a+

B(m

) co

sha1

+B

(m))

) +

n21T2+a2 t 2 2 J 2

+o:Xnm+l3nm-l(n21T2+a2)3:~T" = 0; (212)

and on substituting for the constants A&m) and B&m; their explicit solutions

give

n in

(20

7), w

e fi

nd th

at e

quat

ion

(212

) si

mpl

ifie

s gr

eatly

and

we

are

left

with

II 4mn1T20: (( i)m+n I)

(n21

T2+

a2)(

m21

T2+

a2)

( 2 2 2;~~:i2 2)J(SinhaCOSha-a)(1+(l+0:)(-I)m+nJ+

n 1T +a sl a-a \

+

(sin

ha-a

co

sha)

(( -

i)n+

i+(I

+o:

)( _

i)m

+i)

_

t:~nh

~)(

sinh

a+a(

_i)m

+l)(

( _i

)m+

n_i))

+m

1T

+a

+l3nm+o:Xmn-l(n21T2+a2)3:~ITII = O. (213)

§ 71

TH

E S

TA

BIL

ITY

OF

CO

UE

TT

E F

LO

W30

3A

firs

t app

roxi

mat

ion

to th

e so

lutio

n of

equ

atio

n (2

13)

is o

btan

ed b

yse

tting

the

(I, I

)-el

emen

t of

the

mat

rix

equa

to z

e. W

e fi

d

l(1T2+a2)3 T~2 = ìo:+l-

2a1T

2(2+

0:)

.(1T2+a2)2(sinh2 a-a2) ((smh a cosh a-a)+(sa-ccoa)).

(2"1

4)O

n fu

rthe

r si

mpl

ific

atio

n, th

is g

ives

2 (1

T2+

a2)3

. ...

.. ..

'. .

T=

_ _

(215

)2+

0: a

2p-I

6a1T

2cos

h2la

j((1

T2+

a2)2

(sin

a+a)

J) .

We observe that, apart from the factor2j(2+0:), this expi:~..?

is id

entic

al w

ith w

hat w

as fo

und

in C

hapt

er II

(§

17, e

quat

i (m

Jl_

the Rayleigh number for the simple Bénard problem by the"~f1rlll

met

hod

in th

e fis

t app

roxi

mat

ion

for

the

case

of t

wo

rigid

~c

Con

sequ

ently

, in

this

app

roxi

mat

ion

(cf.

equa

tion

(199

)),

2 34

30 ..

, ,T" = - X 1715 = - and amin = 3'12.....

2+0:

I+p.

We

shal

l see

bel

ow (

§ (b

)) th

at f

or 0

.. p

. .. I

equ

atio

n (2

16)

givalues for the critical Taylor number which do not differ from tI

obtained in the higher approximations by more than one per cent. 1i

reason for this relatively high accuracy of the solution in the fi

appr

oxim

atio

n w

il be

mad

e ap

pare

nt in

§ (

d).

(b)

Num

eric

al r

esul

tsA

met

hod

of s

olvi

ng th

e in

fini

te o

rder

cha

ract

eris

tic e

quat

ion

whi

ch(2

13)

prov

ides

for

T w

ould

be

to s

et th

e de

term

inan

t for

med

by

the

fin rows and columns of the secular matrix equal to zero and let n tae

incr

easi

ngly

larg

er v

alue

s. I

n pr

actic

e, th

e us

eful

ness

of

this

met

hod

wi

depend largely on how rapidly the lowest positive root of the resultin

equa

tion

of o

rder

n te

nds

to it

s lim

it as

n -

+ 0

0. I

t app

ears

that

for

the

prob

lem

on

hand

, the

pro

cess

con

verg

es q

uite

rap

idly

.In

Tab

le X

XX

II th

e va

lues

of

T o

btai

ned

with

the

aid

of e

quat

ion

(213

) in

the

diff

eren

t app

roxi

mat

ions

are

list

ed f

or th

ose

valu

es o

f a

(for

the

assi

gned

p.'s

) at

whi

ch it

was

fou

nd (

by tr

ial a

nd e

rror

) th

at T

atta

ined

its

min

imum

val

ue. F

rom

an

exam

inat

ion

of th

is ta

ble,

itWould appear that for p. ? -1.0, the third approximation provides T

to well within one per cent of the true value. For - 3.0 ~ p. ~ -1,0,

the

calc

ulat

ions

wer

e ca

rrie

d ou

t to

as h

igh

appr

oxim

atio

ns a

s se

med

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(a)

(b)

Figu

re 7

.6.3

. l\1

otio

n in

a r

otat

ing

dish

of

wat

er 4

in. d

eep

due

to s

low

tran

slat

ion

of a

circular cylinder (E) of height i in. fronì right to left across the bottom of the dish,

view

edfr

om a

bove

. In

(a)

the

dye

has

been

rel

ease

d at

a (m

ovin

g) p

oint

A a

bove

the

top

of th

ecy

linde

r an

d di

rect

ly a

head

of

it, a

nd B

(a

divi

ding

poi

nt),

C a

nd D

are

sub

sequ

ent f

)osi

tions

of th

e dy

e; in

(b)

the

poin

t of

rele

ase

(A)

is \v

ithin

the

upw

ard

proj

ectio

n of

the

cylin

der

and the dye remains in the blob D. The flow evidently has a two-dimensional character.

(Fro

m T

aylo

r ,1

923.

)

OY

A/i?

/hie

S //

f ( If

/V -

:-...

./Tn

ö l)J

c. T

l 0 -

l To

FL

U (

0

rJ (

c- d

, k; l

l ~(L

f-,ç

l-/?

Ff( () r'

cÐ~l11? /~c¡ /l'kÇr

/ 9' 6' .7-

556

Flo

w o

f effe

ctiv

ely

invi

scid

flui

d w

ith v

ortic

ity (

7.6

is to say, if the basic rotation in the lateral plane is anti-clockwise, the

Cor

iolis

for

ce te

nds

to tu

rn th

e di

rect

ion

of m

otio

n of

an

elem

ent r

elat

ive

toth

e ro

tatin

g fr

ame

to th

e rig

ht. M

oreo

ver,

the

Cor

iolis

forc

e is

line

ar in

the

velo

city

, and

wil

tend

to c

hang

e th

e di

rect

ion

of th

e co

mpo

nent

of

u in

ala

tera

l pla

ne a

t the

sam

e ra

te fo

r al

l mag

nitu

des

and

dire

ctio

ns o

f tha

tco

mpo

nent

. Thu

s a

mat

eria

l eie

men

t who

se m

otio

n is

dom

inat

ed b

y th

eC

orio

lis f

orce

mov

es o

n a

path

who

se p

roje

ctio

n on

a la

tera

l pla

ne is

a c

ircl

e,th

e w

hole

cir

cle

bein

g co

mpl

eted

in a

tim

e of

ord

er .0

-1. T

he C

orio

lis f

orce

evid

ently

tend

s to

res

tore

an

elem

ent t

o its

initi

al p

ositi

on in

the

late

ral p

lane

.N

ote

that

ther

e is

no

spec

ial s

igni

fican

ce a

bout

the

posi

tion

of th

e ax

is o

fro

tatio

n, s

o fa

r as

the

Cor

iolis

for

ce is

con

cern

ed.

zC

ompo

nent

of

u in

late

ral

7"'ti~ W w"""" -

"", Coriolis force

Dir

ectio

n of

rotation of a

xes

"

Figu

re 7

.6. i

. To

show

the

dire

ctio

n of

the

Cor

iolis

for

ce w

hich

act

s in

a r

otat

ing

refe

renc

esy

stem

. The

(y,

z)-

plan

e is

nor

mal

to th

e ax

is o

f rot

atio

n.

Sinc

e th

e m

otio

ns o

f di

ffer

ent e

lem

ents

of

the

flui

d no

rmal

ly e

xerc

ise

ast

rong

mut

ual i

nflu

ence

thro

ugh

the

actio

n of

pre

ssur

e gr

adie

nts,

it is

desi

rabl

e to

con

side

r al

so th

e co

llect

ive

effe

ct o

f C

orio

lis f

orce

s on

dif

fere

ntel

emen

ts. S

uppo

se th

at r

elat

ive

to th

e ro

tatin

g ax

es th

ere

is g

ener

ated

am

otio

n w

hich

lead

s to

a n

on-z

ero

and

posi

tive

valu

e of

the

rate

of

expa

nsio

nin

the

late

ral p

lane

, tha

t is,

to a

pos

itive

val

ue o

f

ôv/ôy+ ôw/ôz

(in

term

s of

the

co-o

rdin

ate

syst

em o

f fi

gure

7.6

.1)

over

a c

erta

in r

egio

n of

the

flui

d. T

he a

rea

encl

osed

by

the

proj

ectió

n on

a la

tera

l pla

ne o

f a

clos

edm

ater

ial c

urve

in th

is r

egio

n w

il th

en b

e in

crea

sing

. The

eff

ect o

f th

e C

orio

lisfo

rce

acco

mpa

nyin

g th

is g

ener

al o

utw

ard

mov

emen

t of

the

proj

ecte

d cu

rve

is to

gen

erat

e a

tang

entia

l mot

ion

of th

e m

ater

ial c

urve

, whi

ch m

akes

ane

gativ

e co

ntri

butio

n to

the

circ

ulat

ion

roun

d it.

Thi

s ch

ange

in th

e ci

rcul

a-tio

n of

the

mot

ion

rela

tive

to r

otat

ing

axes

is o

f co

urse

sim

ply

the

chan

gere

quir

ed to

kee

p th

e ci

rcul

atio

n re

lativ

e to

an

abso

lute

fra

me

cons

tant

dur

ing

a m

ovem

ent l

eadi

ng to

incr

ease

of

the

proj

ecte

d ar

ea o

n a

late

ral p

lane

. Now

the

new

tang

entia

l mot

ion

of th

e m

ater

ial c

urve

itse

lf le

ads

to a

Cor

iolis

forc

ein

a d

irec

tion

norm

al to

the

curv

e; a

nd in

asm

uch

as th

e ne

w ta

ngen

tial

mot

ion

mak

es a

net

neg

ativ

e co

ntri

butio

n to

the

circ

ulat

ion,

the

asso

ciat

edC

orio

lis f

orce

is d

irec

ted

mos

tly in

war

ds a

nd s

o te

nds

to p

rodu

ce a

red

uctio

n

7.6) Flow systems rotating as a whole 557

in th

e ar

ea e

nclo

sed

by th

e pr

ojec

ted

curv

e. I

n ot

her

wor

ds, a

t pla

ces

in th

eflu

id w

here

ther

e is

a p

ositi

ve v

alue

of ô

vjôy

+ ô

wjô

z th

e ef

fect

of C

orio

lisfo

rces

is to

tend

to p

rodu

ce a

neg

ativ

e va

lue,

and

vic

e ve

rsa.

Thu

s th

e ne

tef

fect

of

Cor

iolis

for

ces

is to

opp

ose

disp

lace

men

ts o

f fl

uid

elem

ents

whi

chto

geth

er le

ad to

cha

nge

of th

e ar

ea e

nclo

sed

by th

e pr

ojec

tion

of a

mat

eria

lcu

rve

on a

late

ral p

lane

, tha

t is,

to a

non

-zer

o ex

pans

ion

in a

late

ral p

lane

.T

he e

xten

t to

whi

ch th

e re

stor

ing

effe

ct o

f Cor

iolis

forc

es r

estr

icts

the

disp

lace

men

t of

flui

d el

emen

ts e

vide

ntly

dep

ends

on

the

rela

tive

mag

nitu

des

of C

orio

lis f

orce

s an

d ot

her

forc

es a

ctin

g on

the

flui

d; a

nd in

the

pres

ent

cont

ext t

hese

oth

er f

orce

s ar

e in

ertia

for

ces.

If

U is

a r

epre

sent

ativ

e ve

loci

tym

agni

tude

(re

lativ

e to

rot

atin

g ax

es)

and

L is

a m

easu

re o

f th

e di

stan

ce o

ver

whi

ch u

var

ies

appr

ecia

bly,

the

ratio

of t

he m

agni

tude

s of

the

term

s u.

V'u

and

zSl x

u in

(7.

6.1)

is o

f or

der

UjL

.o.

The value of this ratio, known as the Rossby number in recognition of the

wor

k of

the

Swed

ish

met

eoro

logi

st, p

rovi

des

a co

nven

ient

mea

sure

of

the

impo

rtan

ce o

f Cor

iolis

forc

es. W

hen

Uj L

.o ~

x, C

orio

lis fo

rces

are

like

ly to

caus

e on

ly a

slig

ht m

odif

icat

ion

of th

e fl

ow p

atte

rn; b

ut w

hen

UjL

.o ~

x,

the

tend

ency

for

Cor

iolis

for

ces

to o

ppos

e an

y ex

pans

ion

in a

late

ral p

lane

is li

kely

to b

e do

min

ant.

And

in th

e in

term

edia

te c

ase

whe

n U

j L.o

is o

f or

der

unity

, an

inte

rest

ing

mix

ture

of

effe

cts

is to

be

expe

cted

, som

e hi

nt o

f w

hich

was

pro

vide

d by

the

disc

ussi

on o

f ste

ady

axis

ymm

etric

flow

with

sw

irl in

§ 7

.5.

Steady flow at small Rossby number ~ t !

The

dom

inan

ce o

f Cor

iolis

forc

es in

flow

at v

alue

s of

UjL

.o s

mal

l com

-pa

red

with

uni

ty h

as s

tran

ge c

onse

quen

ces

whe

n th

e fl

ow is

als

o st

eady

rela

tive

to r

otat

ing

axes

, as

was

fir

st p

oint

ed o

ut b

y J.

Pro

udm

an (

xQx6

). I

nst

eady

flo

w a

mat

eria

l ele

men

t of

flui

d m

oves

alo

ng th

e sa

me

stre

amlin

e,w

ithou

t rev

ersa

l of d

irect

ion,

at a

ll tim

es. B

ut th

e st

rong

Cor

iolis

forc

es .

oppo

se a

ny d

ispl

acem

ent o

f fl

uid

elem

ents

lead

ing

to a

non

-zer

o ex

pans

ion

in a

late

ral p

lane

. It f

ollo

ws

that

, in

the

limit

U L

.o ~

0 th

e fo

rmg-

lhe

stre

amlin

es m

ust b

e co

nsis

ent

wit

a ze

ro r

ate

0 ex

pans

ion

in a

lae

can

esta

blis

h th

is r

esul

t for

mal

ly b

y no

ting

that

whe

n ôu

/ôt =

0 a

ndth

e te

rm u

. V'u

is n

eglig

ible

by

com

paris

on w

ith th

e C

orio

lis fo

rce,

the

equa

tion

of m

otio

n (7

.6.1

) be

com

es1

zSl x

U =

--

V'p

, (7.

6.2)

p. x (ÔP ôp ÔP)

that is, p ôx' ôy' ôz = (0,2.oW, -2.oV)

with

the

co-o

rdin

ates

sho

wn

in fi

gure

7.6

. x. E

limin

atio

n of

P th

en g

ives

ôv ôw

ôy + ôz = 0 (7.6.3)

in s

tead

y fl

ow r

elat

ive

to r

otat

ing

axes

, and

, as

a co

nseq

uenc

e of

the

mas

s-

i. E t: j) ~ it ~

l-. y

. , ~

.._ 1

'l -

'2 l

iAJ

~)'r

Flow of effectively inviscid fluid with vorticity (7.6

\ lôu/ôx = o. I. (7.6.4)

The

cur

ious

pro

pert

y of

thes

e ap

prox

imat

e eq

uatio

ns w

hich

hol

d w

hen

U/L

D. ~

1 is

that

the

mot

ion

in th

e la

tera

l or

(y, z

)-pl

ane

is n

ot c

oupl

ed w

ithth

e m

otio

n pa

ralle

l to

the

axis

of r

otat

ion.

Fur

ther

mor

e, n

óne

of th

e flo

wpr

oper

ties

depe

nds

on x

. Pro

udm

an's

theo

rem

is s

omet

imes

sta

ted

as b

eing

that

'slo

w' s

tead

y m

otio

ns r

elat

ive

to r

otat

ing

axes

mus

t be

two-

dim

ensi

onaL

.Si

nce

in th

is b

ook

we

have

reg

arde

d th

e te

rm tw

o-di

men

sion

al m

otio

n as

impl

ying

that

the

velo

city

vec

tor

ever

yher

e lie

s in

a c

erta

in p

lane

, it w

ould

be m

ore

appr

opri

ate

here

to s

ay th

at s

tead

y m

otio

ns a

t sm

all R

ossb

y nu

mbe

rm

ust b

e a

supe

rpos

ition

of

a tw

o-di

men

sion

al m

otio

n in

the

late

ral p

lane

and

an a

xial

mot

ion

whi

ch is

inde

pend

ent o

f x.

The

val

ue o

f the

vel

ocity

com

pone

nt u

par

alle

l to

the

axis

of r

otat

ion

isev

iden

tly d

eter

min

ed b

y th

e bo

unda

ry c

ondi

tions

. It w

il of

ten

happ

en th

atev

ery

line

in th

e fl

uid

para

llel t

o th

e ax

is m

eets

a s

tatio

nary

bou

ndar

y; in

such

cas

es th

e ab

ove

rela

tions

req

uire

u =

0 e

very

whe

re, a

nd o

nly

the

two-

dim

ensi

onal

mot

ion

rem

ains

. The

pho

togr

aphs

on

plat

e 23

(m

ade

by G

. 1.

Tay

lor

man

y ye

ars

befo

re th

e su

bjec

t of

rota

ting

flui

ds h

ad a

ttrac

ted

muc

hno

tice)

of t

he fl

ow in

an

open

flat

dis

h of

wat

er w

hich

is r

otat

ing

show

that

Cor

iolis

for

ces

do in

deed

mak

e th

e m

otio

n tw

o-di

men

sion

al in

thes

e ci

rcum

-st

ance

s. I

n fi

gure

7.6

.2 (

plat

e 23

) a

drop

of

colo

ured

liqu

id h

as b

een

draw

nou

t int

o a

thin

she

et b

y a

'slo

w' m

otio

n im

part

ed to

the

rota

ting

flui

d, a

ndth

e tw

o ph

otog

raph

s, ta

ken

by a

cam

era

plac

ed o

n th

e ax

s of

rot

atio

n of

the

dish

, sho

w th

at th

e sh

eet i

s ev

eryw

here

par

alle

l to

the

axis

and

that

the

com

pone

nt o

f vel

ocity

in a

late

ral p

lane

is in

depe

nden

t of x

. The

flow

rev

eale

dby the streak of dye released from oint A in figure .6. late 2 is more

start mg. e motion re ative to rotatmg axes is ue ere to a portion of a

circ

uIa

cylin

der

E b

eing

dra

wn

slow

ly a

cros

s th

e bo

ttom

of

the

dish

. The

dept

h of

wat

er is

4 in

. and

the

cylin

der

is 1

in. h

igh,

and

in a

non

-rot

atin

gfl

uid

the

wat

er w

ould

pas

s ov

er th

e to

p of

the

mov

ing

cylin

der

as w

ell a

sro

und

the

side

s. H

owev

er, t

he d

ye e

mer

ging

fro

m a

poi

nt il

in. a

bove

the

top

of th

e cy

linde

r an

d di

rect

ly a

head

of

it (f

igur

e 7.

6.3a

) di

vide

s at

poi

nt B

,as

if it

had

met

an

upw

ard

exte

nsio

n of

the

cylin

der,

and

pas

ses

roun

d th

isim

agin

ary

cylin

der

in tw

o sh

eets

, t th

e sh

eet o

n on

e si

de (

D)

even

sho

win

gse

para

tion

and

the

form

atio

n of

edd

ies.

In fi

gure

7.6

.3b

the

dye

is b

eing

're

leas

ed f

rom

a p

oint

just

insi

de th

e cy

lindr

ical

reg

ion

vert

ical

ly a

bove

the

body

, and

col

lect

s in

a b

lob

whi

ch m

oves

with

the

cylin

der.

It s

eem

s th

at th

efl

ow o

utsi

de th

e up

war

d pr

ojec

tion

of th

e cy

linde

r is

app

roxi

mat

ely

the

sam

eas

if th

e cy

linde

r ex

tend

ed f

rom

the

botto

m to

the

top

of th

e la

yer

of w

ater

,an

d th

at v

ertic

ally

abo

ve th

e cy

linde

r th

ere

is a

cyl

indr

ical

col

umn

of w

ater

whi

ch m

oves

with

it. T

hus

the

mot

ion

is tw

o-di

men

sion

al in

the

way

that

isco

nsis

tent

with

tran

slat

ion

of th

e cy

linde

r, e

ven

thou

gh th

e he

ight

of

that

cylin

der

is o

nly

one-

quar

ter

of th

e de

pth

of w

ater

.

t Ano

ther

str

ikin

g ph

otog

raph

of

this

phe

nom

enon

is r

epro

duce

d in

Gre

ensp

an's

boo

k.

558

conservation equation,

7.6) Flow systems rotating as a whole 559

Whe

n th

e fl

uid

is n

ot e

nclo

sed

by s

tatio

nary

bou

ndar

ies

inte

rsec

ted

bylin

es p

aral

lel t

o th

e ax

is o

f ro

tatio

n, th

e va

lue

of th

e ax

ial v

eloc

ity c

ompo

nent

in th

e fl

uid

wil

usua

lly b

e de

term

ined

by

cond

ition

s at

an

inne

r bo

unda

ry.

An

inte

rest

ing

and

fund

amen

tal c

ase

is fl

ow d

ue to

tran

slat

ion

of a

rig

id b

ody,

with

vel

ocity

U p

aral

lel t

o th

e ax

is o

f ro

tatio

n, th

roug

h fl

uid

whi

ch is

un-

boun

ded

in th

at d

irec

tion.

It s

eem

s th

at h

ere

the

abov

e re

quir

emen

ts f

orfl

ow w

ith U

/LD

. -+

0 c

an b

e sa

tisfi

ed o

nly

if a

ll th

e fl

uid

in th

e cy

linde

r ci

r-cu

msc

ribi

ng th

e bo

dy m

oves

par

alle

l to

the

axis

with

the

body

, the

com

-po

nent

of v

eloc

ity in

a la

tera

l pla

ne b

eing

zer

o ev

eryh

ere.

Exp

erim

ents

do

in f

act s

ugge

st th

at a

col

umn

of f

luid

is p

ushe

d ah

ead

of a

bod

y m

ovin

gpa

ralle

l to

the

axs,

alth

ough

the

flow

beh

ind

the

body

see

ms

not t

o be

who

llyin

acc

ord

with

the

abov

e si

mpl

e th

eory

. Fur

ther

ref

eren

ce to

thes

e ex

peri

-m

ents

wil

be m

ade

late

r in

this

sec

tion.

In th

e ca

se o

f bo

dies

mov

ing

eith

er p

aral

lel t

o th

e ax

is o

f ro

tatio

n or

nor

mal

to it

, the

abo

ve th

eory

for

flow

at s

mal

l Ros

sby

num

ber

lead

s to

the

conc

lusi

onth

at a

so-

calle

d 'T

aylo

r co

lum

n' o

f fl

uid

para

llel t

o th

e ax

is a

ccom

pani

es th

ebo

dy. A

t the

edg

e of

the

colu

mn

ther

e ar

e sh

ear

laye

rs w

here

the

vort

icity

isla

rge.

It

is to

be

expe

cted

that

the

appr

oxim

ate

linea

r eq

uatio

n (7

.6.2

)is

not a

pplic

able

eve

ryhe

re in

thes

e la

yers

, alth

ough

the

cons

eque

nces

for

the

who

le f

low

fie

ld a

re n

ot w

ell u

nder

stoo

d.

Prop

agat

ion

of w

aves

in a

rot

atin

g fl

uid

We

have

see

n th

at a

ny d

ispl

acem

ent o

f th

e el

emen

ts o

f a

flui

d in

rig

id-

body

rot

atio

n w

hich

lead

s to

a n

on-z

ero

expa

nsio

n in

a la

tera

l pla

ne is

acco

mpa

nied

by

Cor

iolis

for

ces

whi

ch te

nd to

elim

inat

e th

is e

xpan

sion

.Si

nce

ther

e is

no

diss

ipat

ion

of e

nerg

y in

an

invi

scid

flu

id, i

t fol

low

s th

at a

disp

lace

men

t of

this

kin

d w

hich

is g

iven

to th

e fl

uid

initi

ally

may

set

up

anos

cila

tion.

Thi

s ra

ises

the

poss

ibili

ty th

at a

trai

n of

wav

es c

an p

ropa

gate

thro

ugh

a ro

tatin

g fl

uid,

with

dif

fere

nt p

hase

s of

the

wav

e be

ing

asso

ciat

edw

ith p

ositi

ve a

nd n

egat

ive

valu

es o

f th

e ex

pans

ion

in a

late

ral p

lane

. We

can

exam

ine

this

pos

sibi

lity

by s

eeki

ng s

olut

ions

of

the

equa

tion

gove

rnin

gde

part

ures

fro

m a

sta

te o

f ri

gid-

body

rot

atio

n w

hich

are

per

iodi

c in

tim

ean

d in

cer

tain

spa

tial c

o-or

dina

tes.

We

shal

l con

side

r fi

rst t

he p

hysi

cally

sim

ple

case

of

an a

xisy

mm

etri

c w

ave

mot

ion

with

pro

paga

tion

in th

e di

rect

ion

of th

e ax

is o

f ro

tatio

n. R

elat

ive

toro

tatin

g ax

es, t

he w

ave

mot

ion

is s

uper

impo

sed

on s

tatio

nary

flu

id a

nd s

o,fo

r a

sim

ple

harm

onic

wav

e, a

ll fl

ow q

uant

ities

var

y si

nuso

idal

ly in

tim

e w

ithan

gula

r fr

eque

ncy

ß s

ay (

perio

d 27

T/ß

) an

d si

nuso

idal

ly w

ith r

espe

ct to

the

axial co-ordinate x with wave-number ex say (wavelength 27Tjex). The

gove

rnin

g eq

uatio

n fo

r fl

ow r

elat

ive

to r

otat

ing

axes

is (

7.6.

1), i

n ax

isym

-.' inetric form, and, following

the usual pattern of

inve

stig

atio

n of

wav

e m

otio

ns,

''we

mig

ht p

roce

ed b

y ne

glec

ting

term

s in

this

equ

atio

n ø

f deg

ree

high

er th

an'th

e fi

rst i

n qu

antit

ies

repr

esen

ting

the

depa

rtur

e fr

om th

e un

dist

urbe

d;S

tte. H

owev

er, t

here

is n

o ne

ed to

go

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