11/03/2014phy 711 fall 2014 -- lecture 291 phy 711 classical mechanics and mathematical methods...
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![Page 1: 11/03/2014PHY 711 Fall 2014 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29: Chap. 9 of](https://reader036.vdocuments.us/reader036/viewer/2022081516/56649f435503460f94c636ba/html5/thumbnails/1.jpg)
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
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PHY 711 Fall 2014 -- Lecture 29 211/03/2014
![Page 3: 11/03/2014PHY 711 Fall 2014 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29: Chap. 9 of](https://reader036.vdocuments.us/reader036/viewer/2022081516/56649f435503460f94c636ba/html5/thumbnails/3.jpg)
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
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r
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PHY 711 Fall 2014 -- Lecture 29 411/03/2014
Euler analysis -- continued
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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
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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
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fvvv 2
21
![Page 7: 11/03/2014PHY 711 Fall 2014 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29: Chap. 9 of](https://reader036.vdocuments.us/reader036/viewer/2022081516/56649f435503460f94c636ba/html5/thumbnails/7.jpg)
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
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vUp
tCtt
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vUp
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rrv
For incompressible fluid
![Page 8: 11/03/2014PHY 711 Fall 2014 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29: Chap. 9 of](https://reader036.vdocuments.us/reader036/viewer/2022081516/56649f435503460f94c636ba/html5/thumbnails/8.jpg)
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
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:amics thermodynof lawFirst
amics thermodynlittleA
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PHY 711 Fall 2014 -- Lecture 29 911/03/2014
dM
dV
dVV
MdVM
V
M
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: variableand fixedFor
:density mass of In terms
Solution of Euler’s equation for fluids – isentropic (continued)
20
2
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int
Let : variablesintensivein In terms
pd
pd
dp
MpdVdWMddE
ME
dQ
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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
![Page 11: 11/03/2014PHY 711 Fall 2014 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29: Chap. 9 of](https://reader036.vdocuments.us/reader036/viewer/2022081516/56649f435503460f94c636ba/html5/thumbnails/11.jpg)
PHY 711 Fall 2014 -- Lecture 29 1111/03/2014
0221
221
tvU
p
pUv
t
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vt applied
fvvv 2
21
Solution of Euler’s equation for fluids – isentropic (continued)
U
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applied
fvv 0
![Page 12: 11/03/2014PHY 711 Fall 2014 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29: Chap. 9 of](https://reader036.vdocuments.us/reader036/viewer/2022081516/56649f435503460f94c636ba/html5/thumbnails/12.jpg)
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
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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
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![Page 14: 11/03/2014PHY 711 Fall 2014 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29: Chap. 9 of](https://reader036.vdocuments.us/reader036/viewer/2022081516/56649f435503460f94c636ba/html5/thumbnails/14.jpg)
PHY 711 Fall 2014 -- Lecture 29 1411/03/2014
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PHY 711 Fall 2014 -- Lecture 29 1511/03/2014
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![Page 16: 11/03/2014PHY 711 Fall 2014 -- Lecture 291 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 29: Chap. 9 of](https://reader036.vdocuments.us/reader036/viewer/2022081516/56649f435503460f94c636ba/html5/thumbnails/16.jpg)
PHY 711 Fall 2014 -- Lecture 29 1611/03/2014
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PHY 711 Fall 2014 -- Lecture 29 1711/03/2014
Analysis of wave velocity in an ideal gas:
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pc
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moleculeeach of mass average
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:gas idealfor state ofEquation
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PHY 711 Fall 2014 -- Lecture 29 1811/03/2014
pf
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PHY 711 Fall 2014 -- Lecture 29 1911/03/2014
p
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PHY 711 Fall 2014 -- Lecture 29 2011/03/2014
pp
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PHY 711 Fall 2014 -- Lecture 29 2111/03/2014
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