Physics IPhysics I

Chap 16.Chap 16. Fluid DynamicsFluid Dynamics

Prof. WAN, Xin

xinwan@zju.edu.cnhttp://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

The Naïve ApproachThe Naïve Approach

N particles ri(t), vi(t); interaction V(ri-rj)

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

Conservation of EnergyConservation of Energy

Steady, incompressible, nonviscous, irrotational

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!!

Measuring Pressure…Measuring Pressure…

E. Torricelli: Mercury Barometer

h patm

p=0 atm

atm

P ghPh

g

U-Tube ManometerU-Tube Manometer

1 1 2 2A atmp gh p gh

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

The Pitot TubeThe Pitot Tube

212a a b

b a

p v p

p p gh

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

HomeworkHomework

CHAP. 16 Exercises 7, 10 (P367) 17, 21, 23 (P368)