PHY 711 Fall 2014 -- Lecture 29 111/03/2014

PHY 711 Classical Mechanics and Mathematical Methods

10-10:50 AM MWF Olin 103

Plan for Lecture 29: Chap. 9 of F&W

Introduction to hydrodynamics

1. Details of Euler formulation of hydrodynamic equations

2. Bernoulli integrals for irrotational flow

3. Sound equations

PHY 711 Fall 2014 -- Lecture 29 211/03/2014

PHY 711 Fall 2014 -- Lecture 29 311/03/2014

Newton’s equations for fluids Use Euler formulation; properties described in terms of stationary spatial grid

(x,y,z,t)

p(x,y,z,t)

(x,y,z,t)

v Velocity

Pressure

Density :Variables

t

t’

ttt

ttt

tt

'

', :'at Particle

, :at Particle

vr

r

PHY 711 Fall 2014 -- Lecture 29 411/03/2014

Euler analysis -- continued

ft

f

dt

df

t

tfttf

t

tftf

dt

df

tf

tttttt

tt

t

v

rvrrr

r

vr

r

),(),(),()',(

:),(For

' here w', :'at Particle

, :at Particle

lim0

PHY 711 Fall 2014 -- Lecture 29 511/03/2014

Continuity equation:

equation continuity of

form ealternativ 0

:Consider

0

0

v

v

vv

v

dt

dtdt

dt

t

PHY 711 Fall 2014 -- Lecture 29 611/03/2014

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 2014 -- Lecture 29 711/03/2014

Bernoulli’s integral of Euler’s equation for constant r

theoremsBernoulli'

)(),(),( where

)(

:spaceover gIntegratin

0

02

21

221

221

Ct

vUp

tCtt

tCt

vUp

tvU

p

rrv

For incompressible fluid

PHY 711 Fall 2014 -- Lecture 29 811/03/2014

Solution of Euler’s equation for fluids -- isentropic

fluid isentropic (constant) 3.

force applied veconservati .2

flow" alirrotation" 0 .1

:nsrestrictio following heConsider t

Uappliedf

v

v

p

vt applied

fvvv 2

21

pdVdWdE

dQ

dWdQdE

int

int

0 :conditions isentropicFor

:amics thermodynof lawFirst

amics thermodynlittleA

PHY 711 Fall 2014 -- Lecture 29 911/03/2014

dM

dV

dVV

MdVM

V

M

pdVdWdE

2

2

int

: variableand fixedFor

:density mass of In terms

Solution of Euler’s equation for fluids – isentropic (continued)

20

2

2int

int

Let : variablesintensivein In terms

pd

pd

dp

MpdVdWMddE

ME

dQ

PHY 711 Fall 2014 -- Lecture 29 1011/03/2014

Solution of Euler’s equation for fluids – isentropic (continued)

pp

p

p

dQ

dQ

:gRearrangin

:Consider2

0

20

PHY 711 Fall 2014 -- Lecture 29 1111/03/2014

0221

221

tvU

p

pUv

t

p

vt applied

fvvv 2

21

Solution of Euler’s equation for fluids – isentropic (continued)

U

pp

applied

fvv 0

PHY 711 Fall 2014 -- Lecture 29 1211/03/2014

Summary of Bernoulli’s results

0221

t

vUp

For isentropic fluid with internal energy density e

0221

t

vUp

For incompressible fluid

PHY 711 Fall 2014 -- Lecture 29 1311/03/2014

Application of fluid equations to the case of air in equilibrium plus small perturbation

0 :equation Continuity

:motion ofequation Euler -Newton

v

fvvv

t

p

t applied

0

0

:mequilibriuNear

0

0

applied

ppp

f

vv

PHY 711 Fall 2014 -- Lecture 29 1411/03/2014

0 0

:onperturbatiin order lowest toEquations

0

0

vv

vfvv

v

tt

p

t

p

t applied

0 0

0

:potential velocity theof In terms

200

00

tt

p

t

p

t

v

v

v

PHY 711 Fall 2014 -- Lecture 29 1511/03/2014

2

00

0

),(

entropy (constant) thedenotes re whe),(

:density theof in terms pressure Expressing

cp

p

spp

sppspp

s

0 0

0

constant 0

222

22

0

0

2

2

2

0

2

0

ctt

t

c

t

ct

p

t

PHY 711 Fall 2014 -- Lecture 29 1611/03/2014

vs

pc

ct

2

222

2

Here,

0

:airfor equation Wave

0 0

:surface Free

ˆ ˆ ˆ

:y at velocit moving ˆ normal with surface leImpenetrab

:aluesBoundary v

0

t

Φp

nvnVn

Vn

0

0

:have also we that,Note

222

2

222

2

pct

p

ct

PHY 711 Fall 2014 -- Lecture 29 1711/03/2014

Analysis of wave velocity in an ideal gas:

s

pc

2

moleculeeach of mass average

/1038.1

:gas idealfor state ofEquation

0

23

00

0

M

kJk

TM

kT

M

k

V

Mp

M

MNNkTpV

PHY 711 Fall 2014 -- Lecture 29 1811/03/2014

pf

TM

kfMNkT

fE

22

2

:gas idealfor energy Internal

0

1

1

2

1

2

2

:ratioheat specific of In terms

2

2

00

0

γ

f

C

C

M

Mk

M

Mkf

T

Vp

T

E

dT

dQC

M

Mkf

T

E

dT

dQC

dWdQdE

C

C

f

f

V

p

pppp

VVV

V

p

PHY 711 Fall 2014 -- Lecture 29 1911/03/2014

p

TM

kMNkTE

1

1

1

1

1

1

:gas idealfor energy Internal

0

pp

pppp

dp

dVM

pd

s

sss

22

2

11

1

1

1

:conditions isentropicunder gas idealfor energy Internal

PHY 711 Fall 2014 -- Lecture 29 2011/03/2014

pp

p

pd

p

dp

s

00

:state ofequation adiabaticor Isentropic

:derivation eAlternativ

m/s 340 /3.1

10013.15.1

sound of speed Linearized

03

52

0

0

0

,,

20

00

cmkg

Pac

ppc

ps

PHY 711 Fall 2014 -- Lecture 29 2111/03/2014

1

0

20

0

0

0

02

00

2

/

/

:gas idealfor sound of speed of dependenceDensity

cppp

c

p

p

ppc

s