PHY 7 11 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103

PHY 711 Fall 2012 -- Lecture 27 110/31/2012

PHY 711 Classical Mechanics and Mathematical Methods

10-10:50 AM MWF Olin 103

Plan for Lecture 27:

Introduction to hydrodynamics

1. Motivation for topic

2. Newton’s laws for fluids

3. Conservation relations

PHY 711 Fall 2012 -- Lecture 27 210/31/2012

PHY 711 Fall 2012 -- Lecture 27 310/31/2012

PHY 711 Fall 2012 -- Lecture 27 410/31/2012

PHY 711 Fall 2012 -- Lecture 27 510/31/2012

Motivation1. Natural progression from strings, membranes,

fluids; description of 1, 2, and 3 dimensional continua

2. Interesting and technologically important phenomena associated with fluids

Plan3. Newton’s laws for fluids4. Continuity equation5. Stress tensor6. Energy relations7. Bernoulli’s theorem8. Various examples9. Sound waves

PHY 711 Fall 2012 -- Lecture 27 610/31/2012

Newton’s equations for fluids Use Lagrange formulation; following “particles” of fluid

pressureapplied

dt

d

dVm

m

FFF

va

Fa

(x,y,z,t)

p(x,y,z,t)

(x,y,z,t)

v Velocity

Pressure

Density :Variables

PHY 711 Fall 2012 -- Lecture 27 710/31/2012

p(x) p(x+dx)

dVx

p

dxdydzdx

zyxpzydxxp

dydzzyxpzydxxpF xpressure

),,(),,(

),,(),,(

PHY 711 Fall 2012 -- Lecture 27 810/31/2012

pdt

d

pdVdVdt

ddV

m

applied

applied

pressureapplied

fv

fv

FFa

Newton’s equations for fluids -- continued

PHY 711 Fall 2012 -- Lecture 27 910/31/2012

tdt

d

tv

zv

yv

xdt

d

tdt

dz

zdt

dy

ydt

dx

xdt

d

tzyx

zyx

vvv

v

vvvvv

vvvvv

vv ,,,

:on termaccelerati of analysis Detailed

vvvvvvv

2

1

: thatNote

zyx vz

vy

vx

PHY 711 Fall 2012 -- Lecture 27 1010/31/2012

pv

t

pt

pdt

d

applied

applied

applied

fvvv

fv

vvvv

fv

221

2

1

Newton’s equations for fluids -- continued

PHY 711 Fall 2012 -- Lecture 27 1110/31/2012

Continuity equation:

equation continuity of

form ealternativ 0

:Consider

0

0

v

v

vv

v

dt

dtdt

dt

t

PHY 711 Fall 2012 -- Lecture 27 1210/31/2012

Solution of Euler’s equation for fluids

0221

221

tvU

p

pUv

t

fluid ibleincompress (constant) 3.

force applied veconservati .2

flow" alirrotation" 0 .1

:nsrestrictio following heConsider t

Uappliedf

v

v

p

vt applied

fvvv 2

21

PHY 711 Fall 2012 -- Lecture 27 1310/31/2012

Bernoulli’s integral of Euler’s equation

theoremsBernoulli' 0

)(),(),( where

)(

:spaceover gIntegratin

0

221

221

221

tvU

p

tCtt

tCt

vUp

tvU

p

rrv

PHY 711 Fall 2012 -- Lecture 27 1410/31/2012

Examples of Bernoulli’s theorem

constant

0 assuming form; Modified

0

221

221

vUp

t

tvU

p

1

2222

12

2212

11

1

1

21

21

0

vUp

vUp

v

ghUU

ppp atm

PHY 711 Fall 2012 -- Lecture 27 1510/31/2012

Examples of Bernoulli’s theorem -- continued

1

22

221

222

121

11

1

21

21

0

vUp

vUp

v

ghUU

ppp atm

ghv 22

PHY 711 Fall 2012 -- Lecture 27 1610/31/2012

Examples of Bernoulli’s theorem -- continued

constant221 vU

p

21

222

12

2212

11

1

21

21

21

equation continuity

vUp

vUp

avAv

UU

pppA

Fp atmatm

PHY 711 Fall 2012 -- Lecture 27 1710/31/2012

Examples of Bernoulli’s theorem -- continued

constant221 vU

p

21

22

22

2

1

/2

12

Aa

AFv

A

a v

A

F

PHY 711 Fall 2012 -- Lecture 27 1810/31/2012

Examples of Bernoulli’s theorem – continued Approximate explanation of airplane lift

Cross section view of airplane wing http://en.wikipedia.org/wiki/Lift_%28force%29

1

2

22

212

112

222

12

2212

11

1

21

vvpp

vUp

vUp

UU