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
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PHY 711 Fall 2014 -- Lecture 38
1
Final grades due 12/17/2014
11/24/2014
11/22/2013
PHY 711 Fall 2013
2014 -- Lecture 34
38
2
11/24/2014
PHY 711 Fall 2014 -- Lecture 38
3
Newton-Euler equations for viscous fluids
Navier-Stokes equation
v
1
 2
1
1 
  v    v  f  p   v           v 
t



3 
Continuity condition

    v  0
t
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PHY 711 Fall 2014 -- Lecture 38
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Example – steady flow of an incompressible fluid in a long
pipe with a circular cross section of radius R
Navier-Stokes equation
v
1
 2
1
1 
  v    v  f  p   v           v 
t



3 
Continuity condition

    v  0
t
Incompressible fluid    v  0
v
Steady flow

0
t
Irrotational flow
  v  0
No applied force
 f =0
Neglect non-linear terms    v 2   0
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PHY 711 Fall 2014 -- Lecture 38
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Example – steady flow of an incompressible fluid in a long
pipe with a circular cross section of radius R -- continued
Navier-Stokes equation becomes:
1
 2
0   p   v


Assume that v  r, t   vz  r  zˆ
p
  2vz (r ) (independent of z )
z
p
p
Suppose that

z
L
p
2
  vz ( r )  
L
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PHY 711 Fall 2014 -- Lecture 38
R
v(r)
L
6
Example – steady flow of an incompressible fluid in a long
pipe with a circular cross section of radius R -- continued
p
 vz ( r )  
L
1 d dvz (r )
p
r

r dr
dr
L
2
R
pr 2
vz ( r )  
 C1 ln(r )  C 2
4 L
 C1  0
v(r)
L
pR 2
vz ( R )  0  
 C2
4 L
p
2
2
vz ( r ) 
R

r


4 L
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PHY 711 Fall 2014 -- Lecture 38
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Example – steady flow of an incompressible fluid in a long
pipe with a circular cross section of radius R -- continued
p
2
2
vz ( r ) 
R

r


4 L
Mass flow rate through the pipe:
R
dM
p  R 4
 2  rdrvz (r ) 
0
dt
8 L
R
v(r)
L
Poiseuille formula;
Method for measuring 
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PHY 711 Fall 2014 -- Lecture 38
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Newton-Euler equations for viscous fluids – effects on sound
Recall full equations:
Navier-Stokes equation
v
1
 2
1
1 
  v    v  f  p   v           v 
t



3 
Continuity condition

    v  0
t
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PHY 711 Fall 2014 -- Lecture 38
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Newton-Euler equations for viscous fluids – effects on sound
Without viscosity terms:
v
1

  v    v  f  p
    v  0
t

t
Assume: v  0   v
f =0
p  p0   p  p0  c 2
 v
c2
Linearized equations:
  
t
0
 v   v 0 e i  k r   t 
Let
  v 0 
c 20
0
k
  0  

  0    v   0
t
   0 ei  k r t 
 0   0k   v 0  0
0
kˆ   v 0
k  2

c
0
c
Pure longitudinal harmonic wave solutions
2
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2
PHY 711 Fall 2014 -- Lecture 38
10
Newton-Euler equations for viscous fluids – effects on sound
Recall full equations:
Navier-Stokes equation
v
1
 2
1
1 
  v    v  f  p   v           v 
t



3 
Continuity condition

    v  0
t
Assume: v  0   v
  0  
f =0
 p 
p  p0   p  p0  c      s
 s  
2
 p 
where c  

   s
2
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PHY 711 Fall 2014 -- Lecture 38
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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
v
1
 2
1
1 
  v    v  f  p   v           v 
t



3 

s
    v  0
T  kth 2T
t
t
Assume: v  0   v
f =0
 p 
p  p0   p  p0  c      s
 s  
2
  0  
 p 
where c  




s
2
 T 
 T 
T  T0   T  T0  
     s

 s  
   s
s  s0   s
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PHY 711 Fall 2014 -- Lecture 38
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Newton-Euler equations for viscous fluids – effects on sound
Linearized equations (with the help of various
thermodynamic relationships):
 T 
 v
c2
 2
1 
1 
    0 
 s    v            v 

t
0
0
0 
3 
   s

  0    v   0
t
c p  T  2
 s
2
   s 
 


t
T0    s
Here:  
Let
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cp
kth

c p 0
cv
 v   v 0 e i  k r   t 
  0 ei  k r t 
PHY 711 Fall 2014 -- Lecture 38
 s   s0 ei  k r t 
13
Newton-Euler equations for viscous fluids – effects on sound
Linearized plane wave solutions:
 v 0  
c 20
0
 T 
i k 2
i 
1 
k  0 

s
k


v



  k k   v0 
0
0


0
0 
3 
   s
0
 0 k   v 0  0
t
ic p  T  2
2
 s0  i k  s0 
k  0


T0    s
Longitudinal solutions:
2
 2

k
4

2 2
2 2  T 
 s0  0
  c k  i
       0   0 k 

0  3

   s

ic p k 2  T 
2




i

k
 s0  0


0


T0    s
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PHY 711 Fall 2014 -- Lecture 38
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Newton-Euler equations for viscous fluids – effects on sound
Linearized plane wave longitudinal solutions:
Longitudinal solutions:
2
 2

k
4

2 2
2 2  T 
 s0  0
  c k  i
       0   0 k 

0  3

   s

ic p k 2  T 
2




i

k
 s0  0


0


T0    s
Approximate solution:
k

c
 i
2 2

c

 4

p 0   T 
where   3      
  
5
2c  0  3
2
T
c


s
0
2
  0e
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2
 kˆ r i  k r   t 
e
PHY 711 Fall 2014 -- Lecture 38
15
Brief introduction to elastic continua
r 1’
r1
deformation
reference
r1 '  r1  u(r1 )
r2 '  r2  u(r2 )
r2 ' r1 '  r2  r1    r2  r1     u(r1 )
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PHY 711 Fall 2014 -- Lecture 38
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Brief introduction to elastic continua
Deformation components:
ui 1  ui u j
 

x j 2  x j xi
 òij  Oij
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 1  ui u j 

  

 2  x j xi 
PHY 711 Fall 2014 -- Lecture 38
17
To be continued ….
Have a great Thanksgiving Holiday.
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PHY 711 Fall 2014 -- Lecture 38
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