Download Chapter 16: Fluid Dynamics

Physics I
Chap 16. Fluid Dynamics
Prof. WAN, Xin
[email protected]
http://zimp.zju.edu.cn/~xinwan/
Definitions


Aerodynamics (gases in motion)
Hydrodynamics (liquids in motion)
– Blaise Pascal
– Daniel Bernoulli, Hydrodynamica (1738)
– Leonhard Euler
– Lagrange, d’Alembert, Laplace,
von Helmholtz

Airplane, petroleum, weather, traffic
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The Naïve Approach
N particles ri(t), vi(t); interaction V(ri-rj)
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Euler’s Solution
For fluid at a point at a time:
  x, y, z, t  ,


Field
v  x, y, z, t 
State of the fluid: described by
parameters p, T.
Laws of mechanics applied to
particles, not to points in space.
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Ideal Fluids



Steady: velocity, density and pressure
not change in time; no turbulence
Incompressible: constant density
Nonviscous: no internal friction
between adjacent layers
Irrotational: no particle rotation
about center of mass
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Ferris Wheel at the
Concorde Square,
Paris

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Streamlines
Paths of particles
P
Q
vP



R
vQ
vR
PQR
v tangent to the streamline
No crossing of streamlines
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Mass Flux

Tube of flow: bundle of streamlines
Q
P
v1
A1
v2
A2
 m1
 m1  1 A1v1 t  mass flux
 1 A1v1
 t1
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Conservation of Mass
IF: no sources and no sinks/drains
1 A1v1  2 A2 v2  constant
A1v1  A2v2  constant, for incompressible fluid

– Narrower tube == larger speed, fast
– Wider tube == smaller speed, slow

Example of equation of continuity.
Also conservation of charge in E&M
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Who Accelerates the Fluid?
Acceleration due to pressure difference.
Bernoulli’s Principle = Conservation of energy
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Conservation of Energy
Steady, incompressible, nonviscous, irrotational
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Bernoulli’s Equation
kinetic E, potential E, external work
 m   A1 x1   A2 x2
1
1
2
p1 A1 x1  p2 A2 x2   mv2   mgy2   mv12   mgy1
2
2
1 2
1 2
p1   v1   gy1  p2   v2   gy2
2
2
1 2
p   v   gy  constant
2
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BEq in Everyday Life
Open a faucet, the
stream of water gets
narrower as it falls.
V1
V2
A1
A2
Velocity increases due to
gravity as water flow down,
thus, the area must get narrower.
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Q & A on Bernoulli’s Eq.
A bucket full of water.
One hole and
one pipe, both
open at bottom.
Out of which water flows faster?
Same. It only depends on depth.
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Bend it like Beckham
Dynamic lift
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Beckham, Applied Physicist
Distance 25 m
Initial v = 25 m/s
Flight time 1s
Spin at 10 rev/s
Lift force ~ 4 N
Ball mass ~ 400 g
a = 10 m/s2
A swing of 5 m!
~ 5m
Goal!!
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Measuring Pressure…

E. Torricelli: Mercury Barometer
Patm   gh
p=0
patm
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h
Patm
 h
g
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U-Tube Manometer
pA  1 gh1  patm  2 gh2
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The Venturi Meter
Speed changes as
diameter changes.
Can be used to
measure the speed
of the fluid flow.
1 2
1 2
p1   v1  p2   v2 ,
2
2
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v1 A1  v2 A2
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The Pitot Tube
1 2
pa   va  pb
2
pb  pa   gh
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Viscous Fluid Flow
Laminar flow:


Following streamlines
Fluids at low speeds
Turbulent flow:


Random or irreproducible
Fluids at high speeds
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Laminar Flow
Fluid flows layer by layer with varying v.
A
y
F, v
A
F = h A dv/dy
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h: coefficient of viscosity
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Cylindrical Pipes
r
V(r)
2R
P 2 2
v r  
(R  r )
4h L
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Dimensional Analysis
Goal: vc ∝ habDc
D
Dimensions:
 vc: LT-1
 h: ML-1T-1
 : ML-3
 D: L
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v
(F = h A dv/dy)
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Reynolds Number

a = 1, b = -1, c = -1
vc ~ h / (D)

vc = R h / (D)

Cylindrical pipes: Rc ~ 2000

 Dv
R
h
– For water, vc = 10 cm/s
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A Remarkable Family
Jakob Bernoulli (1654-1705)
Johann Bernoulli
(1667-1748), brother
of Jokob
Daniel Bernoulli (1700-1782),
son of Johann; discovered
Bernoulli’s Principle
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Leonhard Euler (1707-83)
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
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Born in Basel on April 15, 1707
Studied under Johann Bernoulli
Master’s degree (1724)
– Comparing natural philosophy
of Descartes and of Newton
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

Petersburg Academy of Sciences (1727)
Berlin Academy of Sciences (1741)
Petersburg Academy of Sciences (1766)
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Achievements of Euler
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
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Mathematics: calculus, differential
equations, analytic and differential
geometry, number theory, calculus of
variations, …
Physics: hydrodynamics; theories of
heat, light, and sound, …
Others: analytical mechanics,
astronomy, optical instruments, …
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Homework
Fluid Statics:
Page 348-349
Problems: 4, 14
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Fluid Dynamics:
Page 369
Problems: 4, 14
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Zoom into a Fluid
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Almond Blossom
Almond Blossom
by Vincent van Gogh,
Rijksmuseum,
Amsterdam
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Brainteaser
Q: What is greater than God, the
dead eat it, and the living would die
if they ate it!?
Hint: For the theologically minded,
please give a very short answer.
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