PHY 712 Electrodynamics 10-10:50 AM MWF Olin 107 Plan for Lecture 33:
PHY 7 11 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103
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10/31/2012 PHY 711 Fall 2012 -- Lecture 27 1 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 7 11 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 27: Introduction to hydrodynamics Motivation for topic Newton’s laws for fluids Conservation relations. Motivation - PowerPoint PPT Presentation
Transcript of PHY 7 11 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103
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
PHY 711 Fall 2012 -- Lecture 27 210/31/2012
PHY 711 Fall 2012 -- Lecture 27 310/31/2012
PHY 711 Fall 2012 -- Lecture 27 410/31/2012
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
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
PHY 711 Fall 2012 -- Lecture 27 710/31/2012
p(x) p(x+dx)
dVx
p
dxdydzdx
zyxpzydxxp
dydzzyxpzydxxpF xpressure
),,(),,(
),,(),,(
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
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
zyx
vvv
v
vvvvv
vvvvv
vv ,,,
:on termaccelerati of analysis Detailed
vvvvvvv
2
1
: thatNote
zyx vz
vy
vx
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
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
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
Uappliedf
v
v
p
vt applied
fvvv 2
21
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
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
21
0
vUp
vUp
v
ghUU
ppp atm
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
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
vUp
avAv
UU
pppA
Fp atmatm
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
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
vUp
vUp
UU
