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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.