PHY 113 A General Physics I 9-9:50 AM MWF Olin 101 Plan for Lecture 28: Chapter 19:
PHY 7 11 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103
description
Transcript of PHY 7 11 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103
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PHY 711 Fall 2012 -- Lecture 27 110/31/2012
PHY 711 Classical Mechanics and Mathematical Methods
10-10:50 AM MWF Olin 103
Plan for Lecture 27:
Introduction to hydrodynamics
1. Motivation for topic
2. Newton’s laws for fluids
3. Conservation relations
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PHY 711 Fall 2012 -- Lecture 27 210/31/2012
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PHY 711 Fall 2012 -- Lecture 27 310/31/2012
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PHY 711 Fall 2012 -- Lecture 27 410/31/2012
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PHY 711 Fall 2012 -- Lecture 27 510/31/2012
Motivation1. Natural progression from strings, membranes,
fluids; description of 1, 2, and 3 dimensional continua
2. Interesting and technologically important phenomena associated with fluids
Plan3. Newton’s laws for fluids4. Continuity equation5. Stress tensor6. Energy relations7. Bernoulli’s theorem8. Various examples9. Sound waves
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PHY 711 Fall 2012 -- Lecture 27 610/31/2012
Newton’s equations for fluids Use Lagrange formulation; following “particles” of fluid
pressureapplied
dt
d
dVm
m
FFF
va
Fa
(x,y,z,t)
p(x,y,z,t)
(x,y,z,t)
v Velocity
Pressure
Density :Variables
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PHY 711 Fall 2012 -- Lecture 27 710/31/2012
p(x) p(x+dx)
dVx
p
dxdydzdx
zyxpzydxxp
dydzzyxpzydxxpF xpressure
),,(),,(
),,(),,(
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PHY 711 Fall 2012 -- Lecture 27 810/31/2012
pdt
d
pdVdVdt
ddV
m
applied
applied
pressureapplied
fv
fv
FFa
Newton’s equations for fluids -- continued
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PHY 711 Fall 2012 -- Lecture 27 910/31/2012
tdt
d
tv
zv
yv
xdt
d
tdt
dz
zdt
dy
ydt
dx
xdt
d
tzyx
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vvv
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vvvvv
vvvvv
vv ,,,
:on termaccelerati of analysis Detailed
vvvvvvv
2
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: thatNote
zyx vz
vy
vx
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PHY 711 Fall 2012 -- Lecture 27 1010/31/2012
pv
t
pt
pdt
d
applied
applied
applied
fvvv
fv
vvvv
fv
221
2
1
Newton’s equations for fluids -- continued
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PHY 711 Fall 2012 -- Lecture 27 1110/31/2012
Continuity equation:
equation continuity of
form ealternativ 0
:Consider
0
0
v
v
vv
v
dt
dtdt
dt
t
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PHY 711 Fall 2012 -- Lecture 27 1210/31/2012
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
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v
v
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vt applied
fvvv 2
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PHY 711 Fall 2012 -- Lecture 27 1310/31/2012
Bernoulli’s integral of Euler’s equation
theoremsBernoulli' 0
)(),(),( where
)(
:spaceover gIntegratin
0
221
221
221
tvU
p
tCtt
tCt
vUp
tvU
p
rrv
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PHY 711 Fall 2012 -- Lecture 27 1410/31/2012
Examples of Bernoulli’s theorem
constant
0 assuming form; Modified
0
221
221
vUp
t
tvU
p
1
2222
12
2212
11
1
1
21
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0
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PHY 711 Fall 2012 -- Lecture 27 1510/31/2012
Examples of Bernoulli’s theorem -- continued
1
22
221
222
121
11
1
21
21
0
vUp
vUp
v
ghUU
ppp atm
ghv 22
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PHY 711 Fall 2012 -- Lecture 27 1610/31/2012
Examples of Bernoulli’s theorem -- continued
constant221 vU
p
21
222
12
2212
11
1
21
21
21
equation continuity
vUp
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avAv
UU
pppA
Fp atmatm
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PHY 711 Fall 2012 -- Lecture 27 1710/31/2012
Examples of Bernoulli’s theorem -- continued
constant221 vU
p
21
22
22
2
1
/2
12
Aa
AFv
A
a v
A
F
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PHY 711 Fall 2012 -- Lecture 27 1810/31/2012
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
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