11/03/2014phy 711 fall 2014 -- lecture 291 phy 711 classical mechanics and mathematical methods...
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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