PHY 711 Fall 2014 -- Lecture 38 111/24/2014

PHY 711 Classical Mechanics and Mathematical Methods

10-10:50 AM MWF Olin 103

Plan for Lecture 38

1. Chapter 12: Example of solution to Navier-Stokes equation

2. Chapter 13: Physics of elastic continua

PHY 711 Fall 2014 -- Lecture 38 211/24/201411/22/2013 PHY 711 Fall 2013 -- Lecture 34 2

Final grades due 12/17/2014

PHY 711 Fall 2014 -- Lecture 38 311/24/2014

Please sign u

p

PHY 711 Fall 2014 -- Lecture 38 411/24/2014

Newton-Euler equations for viscous fluids

2

Navier-Stokes equation

1 1 1

3

Continuity condition

0

pt

t

vv v f v

v

v

PHY 711 Fall 2014 -- Lecture 38 511/24/2014

Example – steady flow of an incompressible fluid in a long pipe with a circular cross section of radius R

2

Navier-Stokes equation

1 1 1

3

Continuity condition

0

pt

t

vv v f v

v

v

2

0

Steady flow 0

Irrotational flow 0

No applied force

Incompressible flu

=0

Neglect non-linear terms

id

0

t

v

v

v

v

f

PHY 711 Fall 2014 -- Lecture 38 611/24/2014

Example – steady flow of an incompressible fluid in a long pipe with a circular cross section of radius R -- continued

2

Navier-Stokes equation becomes:

10 p

v R

v(r)

2

2

( ) (independent of )

Suppo

ˆAssu

se that

( )

me that , z

z

z

pv r z

zp p

z Lp

v rL

t v r

v r z L

PHY 711 Fall 2014 -- Lecture 38 711/24/2014

Example – steady flow of an incompressible fluid in a long pipe with a circular cross section of radius R -- continued

R

v(r)

2

2

1 2

2

1 2

( )

1 ( )

( ) ln( ) C4

0 ( ) 0 C4

z

z

z

z

pv r

L

d dv r pr

r dr dr L

prv r C r

L

pRv R

LC

2 2( )4z

pv r R r

L

L

PHY 711 Fall 2014 -- Lecture 38 811/24/2014

Example – steady flow of an incompressible fluid in a long pipe with a circular cross section of radius R -- continued

R

v(r)

2 2

4

0

( )4

Mass flow rate through the pipe:

2 ( )8

z

R

z

pv r R r

L

dM p Rrdrv r

dt L

L

Poiseuille formula; Method for measuring h

PHY 711 Fall 2014 -- Lecture 38 911/24/2014

Newton-Euler equations for viscous fluids – effects on sound Recall full equations:

2

Navier-Stokes equation

1 1 1

3

Continuity condition

0

pt

t

vv v f v

v

v

PHY 711 Fall 2014 -- Lecture 38 1011/24/2014

Newton-Euler equations for viscous fluids – effects on sound Without viscosity terms:

1 0p

t t

v

v v f v

0

20 0

=0

Assume: 0

p p p p c

v fv

2

00

0 0

20

0 0 0 00

22 0 0

20

Linearized equations: 0

Let

0

ˆ

i tit

c

e e

kc

t

c

c

t

k kr r

vv

v v

k

v v

v

k k

Pure longitudinal harmonic wave solutions

PHY 711 Fall 2014 -- Lecture 38 1111/24/2014

Newton-Euler equations for viscous fluids – effects on sound Recall full equations:

2

Navier-Stokes equation

1 1 1

3

Continuity condition

0

pt

t

vv v f v

v

v

0

20 0

2

=0

where

Assume: 0

s

pp p p p c s

s

pc

v fv

PHY 711 Fall 2014 -- Lecture 38 1211/24/2014

Newton-Euler equations for viscous fluids – effects on sound Note that pressure now depends both on density and entropy so that entropy must be coupled into the equations

2

2

1 1 1

3

0 th

pt

sT k T

t t

vv

v v f v

v

0

2 20 0

0 0

0

=0

where

Assume:

0

s

s

p pp p p p c s c

s

T TT T T T s

s

s s s

v fv

PHY 711 Fall 2014 -- Lecture 38 1311/24/2014

Newton-Euler equations for viscous fluids – effects on sound

22

00 0 0

0

2 2

0

1 1

3

0

p

s

s

Ts

t

tcs T

st T

c

vv

v

v

Linearized equations (with the help of various thermodynamic relationships):

0

Here: p th

v p

c k

c c

0 0 0 L et i t t i tie e es s r rk k rkv v

PHY 711 Fall 2014 -- Lecture 38 1411/24/2014

Newton-Euler equations for viscous fluids – effects on sound

2 2

00 0 0 0 0

0 0 0

00 0

2 20 0 0

0

1

3

0

s

p

s

T i k is

tic T

s i s kT

c

k

vv k k k k v

k v

Linearized plane wave solutions:

22 2 2 2 2

0 0 00

22

0 00

Longitudinal solutions

4

3

0

0

:

p

s

s

k Tc k ki s

ic k Ti k s

T

PHY 711 Fall 2014 -- Lecture 38 1511/24/2014

Newton-Euler equations for viscous fluids – effects on soundLinearized plane wave longitudinal solutions:

22 2 2 2 2

0 0 00

22

0 00

Longitudinal solutions

4

3

0

0

:

p

s

s

k Tc k ki s

ic k Ti k s

T

22 220

3 50 0

4where

Approximate sol

2

ution:

3

2p

s

c T

c T

i

c

kc

ˆ

0i te e k rk r

PHY 711 Fall 2014 -- Lecture 38 1611/24/2014

Brief introduction to elastic continua

reference deformation

r1 r1’

2 2

2 1 2

1 1 1 2

1 2 1 1

( ) (' '

' '

)

( )

ur r r r

r r r r r

r r

u rr

u

PHY 711 Fall 2014 -- Lecture 38 1711/24/2014

Brief introduction to elastic continua

Deformation components:

1 1

2 2

i i i

j j j

ij ij

j j

i i

O

u uu u u

x x x x x

ò

PHY 711 Fall 2014 -- Lecture 38 1811/24/2014

To be continued ….

Have a great Thanksgiving Holiday.