11/02/2012phy 711 fall 2012 -- lecture 281 phy 711 classical mechanics and mathematical methods...
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PHY 711 Fall 2012 -- Lecture 28 111/02/2012
PHY 711 Classical Mechanics and Mathematical Methods
10-10:50 AM MWF Olin 103
Plan for Lecture 28:
Introduction to hydrodynamics
1. Correction – Euler formulation revisited
2. Euler’s equation for fluid dynamics
3. Bernoulli’s integrals
PHY 711 Fall 2012 -- Lecture 28 211/02/2012
PHY 711 Fall 2012 -- Lecture 28 311/02/2012
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’
', :'at Particle
, :at Particle
ttt
tt
vr
r
PHY 711 Fall 2012 -- Lecture 28 411/02/2012
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 2012 -- Lecture 28 511/02/2012
Continuity equation:
equation continuity of
form ealternativ 0
:Consider
0
0
v
v
vv
v
dt
dtdt
dt
t
PHY 711 Fall 2012 -- Lecture 28 611/02/2012
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 2012 -- Lecture 28 711/02/2012
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
PHY 711 Fall 2012 -- Lecture 28 811/02/2012
Examples of Bernoulli’s theorem for constant r
0 :equation Continuity
0 :flowsteady For
02
21
vt
Ct
vUp
1
2222
12
2212
11
1
1
21
21
0
vUp
vUp
v
ghUU
ppp atm
PHY 711 Fall 2012 -- Lecture 28 911/02/2012
Examples of Bernoulli’s theorem -- continued
222
12
2212
11
1
22111
21
21
) 0
vUp
vUp
AvA(vv
ghUU
ppp atm
ghv 22
1
2
3
Examples of Bernoulli’s theorem -- continued
232
13
3212
11
1
133 ;0
vUp
vUp
gyUUp
m
mg
py
gypv
atm
atm
3.10
8.91000
10013.1
2/25
3
PHY 711 Fall 2012 -- Lecture 28 1011/02/2012
Examples of Bernoulli’s theorem -- continued
constant221 vU
p
21
222
12
2212
11
1
21
21
21
equation continuity
vUp
vUp
avAv
UU
pppA
Fp atmatm
PHY 711 Fall 2012 -- Lecture 28 1111/02/2012
Examples of Bernoulli’s theorem -- continued
constant221 vU
p
21
22
22
2
1
/2
12
Aa
AFv
A
a v
A
F
PHY 711 Fall 2012 -- Lecture 28 1211/02/2012
Examples of Bernoulli’s theorem – continued Approximate explanation of airplane lift
Cross section view of airplane wing http://en.wikipedia.org/wiki/Lift_%28force%29
1
2
22
212
112
222
12
2212
11
1
21
vvpp
vUp
vUp
UU
PHY 711 Fall 2012 -- Lecture 28 1311/02/2012
Some details on the velocity potential
0
0 :flow alIrrotation
0
constant :fluid ibleincompressFor
0
0
:equation Continuity
2
vv
v
vv
v
t
t
PHY 711 Fall 2012 -- Lecture 28 1411/02/2012
z
a
b
Example – uniform flow
0
0
2
2
2
2
2
2
2
zyx
zv ˆ
:solution Possible
o
o
v
zv
PHY 711 Fall 2012 -- Lecture 28 1511/02/2012
Example – flow around a cylinder
v0 Z v0 Z
0
02
arr
to be continued…
PHY 711 Fall 2012 -- Lecture 28 1611/02/2012
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 2012 -- Lecture 28 1711/02/2012
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 2012 -- Lecture 28 1811/02/2012
Solution of Euler’s equation for fluids – isentropic (continued)
pp
p
p
dQ
dQ
:gRearrangin
:Consider2
0
20
PHY 711 Fall 2012 -- Lecture 28 1911/02/2012
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