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
The Naïve Approach
N particles ri(t), vi(t); interaction V(ri-rj)
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.
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
Viscous Fluid Flow
Laminar flow:


Following streamlines
Fluids at low speeds
Turbulent flow:


Random or irreproducible
Fluids at high speeds
Streamlines
Paths of particles
P
Q
vP



R
vQ
vR
PQR
v tangent to the streamline
No crossing of streamlines
Mass Flux

Tube of flow: bundle of streamlines
Q
P
v1
A1
v2
A2
 m1
 m1  1 A1v1 t  mass flux
 1 A1v1
 t1
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
What Accelerates the Fluid?
Acceleration due to pressure difference.
Bernoulli’s Principle = Conservation of energy
Conservation of Energy
Steady, incompressible,
nonviscous, irrotational
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
BEq in Everyday Life
Open a faucet, the
stream of water gets
narrower as it falls.
Velocity increases due to
gravity as water flow down,
thus, the area must get
narrower.
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.
Bend it like Beckham
Dynamic lift
http://www.tudou.com/programs/view/qLaZ-A0Pk_g/
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!!
Measuring Pressure…

E. Torricelli: Mercury Barometer
Patm   gh
p=0
patm
h
Patm
 h
g
U-Tube Manometer
pA  1 gh1  patm  2 gh2
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
v1 A1  v2 A2
The Pitot Tube
1 2
pa   va  pb
2
pb  pa   gh
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
Leonhard Euler (1707-83)



Born in Basel on April 15, 1707
Studied under Johann Bernoulli
Master’s degree (1724)
– Comparing natural philosophy
of Descartes and of Newton



Petersburg Academy of Sciences (1727)
Berlin Academy of Sciences (1741)
Petersburg Academy of Sciences (1766)
Achievements of Euler



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, …
Viscous Fluid Flow
Laminar flow:


Following streamlines
Fluids at low speeds
Turbulent flow:


Random or irreproducible
Fluids at high speeds
Dimensional Analysis
Goal: vc ∝ habDc
D
Dimensions:
 vc: LT-1
 h: ML-1T-1
 : ML-3
 D: L
(F = h A dv/dy)
v
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
Homework
CHAP. 16 Exercises
7, 10 (P367)
17, 21, 23 (P368)