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11/28/2018
1
11/28/2018 PHY 711 Fall 2018 -- Lecture 36 1
PHY 711 Classical Mechanics and Mathematical Methods
10-10:50 AM MWF Olin 103
Plan for Lecture 36
Viscous fluids – Chap. 12 in F & W
1. Navier-Stokes equation
2. Derivation of Stoke’s law
3. Effects of viscosity on sound waves
11/28/2018 PHY 711 Fall 2018 -- Lecture 36 22
11/28/2018 PHY 711 Fall 2018 -- Lecture 36 3
Comments on presentation schedule:
11/28/2018
2
11/28/2018 PHY 711 Fall 2018 -- Lecture 36 4
11/28/2018 PHY 711 Fall 2018 -- Lecture 36 5
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
Fluid / (m2/s) (Pa s)
Water 1.00 x 10-6 1 x 10-3
Air 14.9 x 10-6 0.018 x 10-3
Ethyl alcohol 1.52 x 10-6 1.2 x 10-3
Glycerine 1183 x 10-6 1490 x 10-3
Typical viscosities at 20o C and 1 atm:
11/28/2018 PHY 711 Fall 2018 -- Lecture 36 6
Viscous effects in incompressible fluids
RuF
u
R
D
6
:osity with viscmediumin speedat moving
radius of sphere aon drag viscousof analysis Stokes'
Plan:1. Consider the general effects of viscosity on fluid
equations2. Consider the solution to the linearized equations
for the case of steady-state flow of a sphere of radius R
3. Infer the drag force needed to maintain the steady-state flow
u
FD
11/28/2018
3
11/28/2018 PHY 711 Fall 2018 -- Lecture 36 7
RuF
u
R
D
6
:osity with viscmediumin speedat moving
radius of sphere aon drag viscousof analysis Stokes'
tm
R
eR
Ftu
udt
dumuRF
Fm
6
16
)(
0)0( with 6
: forceconstant with mass of particle
ofmotion on force drag of Effects
u
FD
F
11/28/2018 PHY 711 Fall 2018 -- Lecture 36 8
tm
R
eR
Ftu
udt
dumuRF
Fm
6
16
)(
0)0( with 6
: forceconstant with mass of particle
ofmotion on force drag of Effects
t
u
u
FD
F
11/28/2018 PHY 711 Fall 2018 -- Lecture 36 9
0
6
0
Effects of drag force on motion of particle of mass
with an initial velocity with (0) and no external force
6
( )R
tm
m
u U
duR u m
dt
u t U e
u
u
FD
t
11/28/2018
4
11/28/2018 PHY 711 Fall 2018 -- Lecture 36 10
Recall: PHY 711 -- Assignment #17 Nov. 2, 2018
Determine the form of the velocity potential for an incompressible fluid representing uniform velocity in the zdirection at large distances from a spherical obstruction of radius a. Find the form of the velocity potential and the velocity field for all r > a. Assume that for r = a, the velocity in the radial direction is 0 but the velocity in the azimuthal direction is not necessarily 0.
2
3
20
0
, cos2
ar v r
r
Note: in this case, no net pressure is exerted on the sphere.
11/28/2018 PHY 711 Fall 2018 -- Lecture 36 11
2
Newton-Euler equation for incompressible fluid,
modified by viscous contribution (Navier-Stokes equation):
Continuity equation: 0
applied
p
t
vv v f v
v
2
Assume steady state: 0
Assume non-linear effects small
Initially set 0; applied
t
p
v
f
v
11/28/2018 PHY 711 Fall 2018 -- Lecture 36 12
2
2
Take curl of both sides of equation:
0
p
p
v
v
2
Note that:
A A A
Assume (with a little insight from Landau):
where ( ) 0r
f r
f r
v u u
11/28/2018
5
11/28/2018 PHY 711 Fall 2018 -- Lecture 36 13
Digression
2
2
Some comment on assumption:
Here
f r
f r
v u u
A A A
A u
v A A
0 or
:note Also22
2
vv
v
p
p
11/28/2018 PHY 711 Fall 2018 -- Lecture 36 14
0)( 0ˆ)( 0ˆ)(
0 0
ˆ)(ˆˆ
ˆ
444
2
2
rfrfrf
rfrfrf
u
rf
zz
vv
zzz
zu
uuv
2 41 2 3
2 41 3
22 4
12 3
( )
2 2 2cos 1 cos 1 4
1sin 1 sin 1 4
r
Cf r C r C r C
rdf C C
v u u Cr dr r r
d f df C Cv u u C
dr r dr r r
11/28/2018 PHY 711 Fall 2018 -- Lecture 36 15
2
4
2 41 2
2
2
3
Some details:
2( ) 0 ( ) 0
( )
d df r f r
dr r dr
Cf r C r C r C
r
2
2
ˆˆ
ˆ ˆ
ˆ ˆ ˆ = ( )
ˆNote that: cos
c
sin
ˆˆ s nos ( ) 1 co
is
u f r
u f r f r
dfu f r
dr
v z z
z
r
z
θ
r
z
z
v θ
11/28/2018
6
11/28/2018 PHY 711 Fall 2018 -- Lecture 36 16
2 41 3
22 4
12 3
1
2 4 2
3
4
2 2 2cos 1 cos 1 4
1sin 1 sin 1 4
To satisfy ( ) : 0
3To satisfy ( ) 0 solve for , ; ,
4 4
r
r
df C Cv u u C
r dr r r
d f df C Cv u u C
dr r dr r r
r C
R RR C C C C
v
v u
v
3
3
3
3
3cos 1
2 2
3sin 1
4 4
R Ru
r r
R Rv u
r r
11/28/2018 PHY 711 Fall 2018 -- Lecture 36 17
32
3 3
3 3
2 2 42
0
3
3ˆ ˆ ˆ( ) = ( )
4 4
3 3cos 1 sin 1
2 2 4 4
Determining pressure:
3ˆ ˆ( ) cos
2
r
R Rf r C r u f r f r
r
R R R Rv u v u
r r r r
Rp u f r f r u
r
p p
v z z z
v z z
2
3cos
2
Ru
r
11/28/2018 PHY 711 Fall 2018 -- Lecture 36 18
0 2
20
3cos
2
Corresponds to:
cos ( ) 4
F = 6
D
D
Rp p u
r
F p R p R
u R
u
FD
11/28/2018
7
11/28/2018 PHY 711 Fall 2018 -- Lecture 36 19
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
Recap --
11/28/2018 PHY 711 Fall 2018 -- Lecture 36 20
Newton-Euler equations for viscous fluids – effects on soundWithout viscosity terms:
1 0p
t t
v
v v f v
0
20 0 0
=0 Ass
ume:
0
s
pp p p p p c
v fv
2
00
0 0
Linearized equations:
0
Let i it t
t
c
e e
t
rk k r
vv
v v
11/28/2018 PHY 711 Fall 2018 -- Lecture 36 21
Sound waves without viscosity -- continued
2
00
0 0
220
0
0 0
0
Linearized equations: 0
Let
0
i it t
c
e e
t
c
t
c
t
t
r rk k
vv
v v
vv k
v 0 0 0
22 0 0
20
0
ˆ k
c c
k
k
v
v
Pure longitudinal harmonic wave solutions
11/28/2018
8
11/28/2018 PHY 711 Fall 2018 -- Lecture 36 22
Newton-Euler equations for viscous fluids – effects on soundRecall 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
Ass
( , )
where
ume:
0
s
pp s p p p c s
s
pc
v fv
viscosity causes heat transfer
11/28/2018 PHY 711 Fall 2018 -- Lecture 36 23
Newton-Euler equations for viscous fluids – effects on soundNote 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
11/28/2018 PHY 711 Fall 2018 -- Lecture 36 24
Newton-Euler equations for viscous fluids –linearized equations
2
2
0 0
1 1 1
3
1 1 1
3
pt
pt
vv v f v
vv
v
v
2
00 0
1
s s
p p Ts s
s
c
2
2 20
Digression -- from the first law of thermod
/
ynamics:
1
s ss
p T p pTds
s s sd
sd
00
1
s
p T
s
11/28/2018
9
11/28/2018 PHY 711 Fall 2018 -- Lecture 36 25
Linearized equations -- continued
0
Further relationships:
v
TT
cs
heat capacity at constant volume
0
0
0
t
t
v
v
2
2 2
0 0
th
th
s
sT k T
t
ks T Ts
t T s
th
pc
k
11/28/2018 PHY 711 Fall 2018 -- Lecture 36 26
Linearized equations -- continued
2 2
0
where p p
vs
c cs Ts
t T c
0
Further relationships:
v
TT
cs
2
2 2
0 0
th
th
s
sT k T
t
ks T Ts
t T s
th
pc
k
11/28/2018 PHY 711 Fall 2018 -- Lecture 36 27
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
11/28/2018
10
11/28/2018 PHY 711 Fall 2018 -- Lecture 36 28
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 hydrodynamic equations
220
0
It can be shown that
1 where (thermal expansion)
pp ps
T cT Vc
TVc c
T
0 0 0 L et i t t i tie e es s r rk k rkv v
11/28/2018 PHY 711 Fall 2018 -- Lecture 36 29
2 2 2
0 00 0 0 0
0 0 0
0 0 0
22 2
0 0 00
1
3
0
p
T c i k is
c
i cs i s
c
kk
v k k k kv v
k v
Linearized hydrodynamic equations; plane wave solutions:
In the absense of thermal expansion, 0
2 2
00 0 0
0 0 0
0 0 0
20 0
1
3
0
c i k i
s i sk
v k k k vv
k v
Entropy and mechanical modes are independent
11/28/2018 PHY 711 Fall 2018 -- Lecture 36 30
2 2
00 0 0
0 0 0
0 0 0
20 0
1
3
0
c i k i
s i sk
v k k k vv
k v
In the absense of thermal expa 0 --nsion cont, in ued:
0 0 0 i t t ti ie e es s k k kr r rv v
ˆ
0
ˆ/ ( / ) 2 where Entropy wave: i te es s
k r k r
( )( )ˆ ˆ/00 0 0
21
0
/
20
2
3
Density wave:
142
3 3
4
ci cte e
k i i ic c cc c
k r k rk v
0
ˆ
0
0
ˆ/ ( / )Transverse wave: 0
where 2
tie e
k r k rv vk v
11/28/2018
11
11/28/2018 PHY 711 Fall 2018 -- Lecture 36 31
0 0
1/ 220
0
00
ˆ ˆ/ ( / )
0)
0
0
Transverse modes ( :
0
whe
2re i t
s
i
e e
i kk
k r k r
v k
v k
v v
Linearized hydrodynamic equations; full plane wave solutions:
2 2 2
0 00 0 0 0
0 0 0
0 0 0
22 2
0 0 00
1
3
0
p
T c i k is
c
i cs i s
c
kk
v k k k kv v
k v
11/28/2018 PHY 711 Fall 2018 -- Lecture 36 32
2 222 2 2 0 0
0 0
0
22 2
0 00
Longitudinal solutions: ( :
4
3
0
0)
0p
T c kkc k s
c
i ck i k
i
s
v k
Linearized hydrodynamic equations; full plane wave solutions:
2 2 2
0 00 0 0 0
0 0 0
0 0 0
22 2
0 0 00
1
3
0
p
T c i k is
c
i cs i s
c
kk
v k k k kv v
k v
11/28/2018 PHY 711 Fall 2018 -- Lecture 36 33
2 220
30
4where
Approximate solution:
2 3 2
p
k ic
T
c c c
ˆ( / )ˆ
0
i c ctee
k rk r
Linearized hydrodynamic equations; full plane wave solutions:
2 222 2 2 0 0
0 00
22 2
0 00
Longitudinal solutions: ( :
4
3
0
0)
0p
T c kkc k s
c
i ck i k
i
s
v k
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