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Physics IPhysics I
Chap 16.Chap 16. Fluid DynamicsFluid Dynamics
Prof. WAN, Xin
[email protected]://zimp.zju.edu.cn/~xinwan/
DefinitionsDefinitions
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
Euler’s SolutionEuler’s Solution
For fluid at a point at a time:
State of the fluid: described by parameters p, T.
Laws of mechanics applied to particles, not to points in space.
, , , , , , ,x y z t v x y z t
Field
Ideal FluidsIdeal 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 FlowViscous Fluid Flow
Laminar flow: Following streamlines Fluids at low speeds
Turbulent flow: Random or irreproducible Fluids at high speeds
StreamlinesStreamlines
Paths of particles
P Q R v tangent to the streamline No crossing of streamlines
Pv Rv
Qv
PQ
R
Mass FluxMass Flux
Tube of flow: bundle of streamlines
2v
A1 A2
1vP
Q
11 1 1 1 1 1 1
1
mass flux mm A v t A vt
Conservation of MassConservation of Mass
IF: no sources and no sinks/drains
– Narrower tube == larger speed, fast– Wider tube == smaller speed, slow
Example of equation of continuity. Also conservation of charge in E&M
1 1 1 2 2 2
1 1 2 2
constantco fornstant incompressible fluid,
A v A vA v A v
What Accelerates the Fluid?What Accelerates the Fluid?
Acceleration due to pressure difference.Bernoulli’s Principle = Conservation of energy
Bernoulli’s EquationBernoulli’s Equation
1 1 2 2
2 21 1 1 2 2 2 2 2 1 1
2 21 1 1 2 2 2
2
1 12 2
1 12 2
1 constant2
m A x A x
p A x p A x mv mgy mv mgy
p v gy p v gy
p v gy
kinetic E, potential E, external work
BEq in Everyday LifeBEq 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.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 BeckhamBend it like Beckham
Dynamic lift
http://www.tudou.com/programs/view/qLaZ-A0Pk_g/
Beckham, Applied PhysicistBeckham, Applied Physicist~ 5mDistance 25 m
Initial v = 25 m/sFlight time 1sSpin at 10 rev/sLift force ~ 4 NBall mass ~ 400 g a = 10 m/s2 A swing of 5 m!
Goal!!Goal!!
The Venturi MeterThe Venturi Meter
Speed changes as diameter changes. Can be used to measure the speed of the fluid flow.
2 21 1 2 2 1 1 2 2
1 1 ,2 2
p v p v v A v A
A Remarkable FamilyA Remarkable Family
Jakob Bernoulli (1654-1705)
Johann Bernoulli (1667-1748), brother of Jokob
Daniel Bernoulli (1700-1782),son of Johann; discoveredBernoulli’s Principle
Leonhard Euler (1707-83)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 EulerAchievements 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 FlowViscous Fluid Flow
Laminar flow: Following streamlines Fluids at low speeds
Turbulent flow: Random or irreproducible Fluids at high speeds
Dimensional AnalysisDimensional Analysis
Goal: vc ∝ abDc
Dimensions: vc: LT-1 : ML-1T-1 (F = A dv/dy) : ML-3 D: L
vD
Reynolds NumberReynolds Number
a = 1, b = -1, c = -1 vc ~ / (D)
vc = R / (D)
Cylindrical pipes: Rc ~ 2000
– For water, vc = 10 cm/s
DvR