Physics I Chap 16.Fluid Dynamics Prof. WAN, Xin

Definitions F Aerodynamics (gases in motion) F Hydrodynamics (liquids in motion) –Blaise Pascal –Daniel Bernoulli, Hydrodynamica (1738) –Leonhard Euler –Lagrange, d’Alembert, Laplace, von Helmholtz F Airplane, petroleum, weather, traffic

The Naïve Approach N particles r i (t), v i (t); interaction V(r i -r j )

Euler’s Solution For fluid at a point at a time: F State of the fluid: described by parameters p, T. F Laws of mechanics applied to particles, not to points in space. Field

Ideal Fluids F Steady: velocity, density and pressure not change in time; no turbulence F Incompressible: constant density F Nonviscous: no internal friction between adjacent layers F Irrotational: no particle rotation about center of mass

Viscous Fluid Flow Laminar flow: F Following streamlines F Fluids at low speeds Turbulent flow: F Random or irreproducible F Fluids at high speeds

Streamlines Paths of particles F P  Q  R F v tangent to the streamline F No crossing of streamlines P Q R

Mass Flux F Tube of flow: bundle of streamlines A1A1 A2A2 P Q

Conservation of Mass F IF: no sources and no sinks/drains –Narrower tube == larger speed, fast –Wider tube == smaller speed, slow F 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

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

Beckham, Applied Physicist ~ 5m 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/s 2 A swing of 5 m! Goal!!

Measuring Pressure… F E. Torricelli: Mercury Barometer h p atm p=0

U-Tube Manometer

The Venturi Meter Speed changes as diameter changes. Can be used to measure the speed of the fluid flow.

The Pitot Tube

A Remarkable Family Jakob Bernoulli ( ) Johann Bernoulli ( ), brother of Jokob Daniel Bernoulli ( ), son of Johann; discovered Bernoulli’s Principle

Leonhard Euler ( ) F Born in Basel on April 15, 1707 F Studied under Johann Bernoulli F Master’s degree (1724) –Comparing natural philosophy of Descartes and of Newton F Petersburg Academy of Sciences (1727) F Berlin Academy of Sciences (1741) F Petersburg Academy of Sciences (1766)

Achievements of Euler F Mathematics: calculus, differential equations, analytic and differential geometry, number theory, calculus of variations, … F Physics: hydrodynamics; theories of heat, light, and sound, … F Others: analytical mechanics, astronomy, optical instruments, …

Viscous Fluid Flow Laminar flow: F Following streamlines F Fluids at low speeds Turbulent flow: F Random or irreproducible F Fluids at high speeds

Dimensional Analysis Goal: v c ∝  a  b D c Dimensions: F v c : LT -1   : ML -1 T -1 (F =  A dv/dy)   : ML -3 F D: L v D

Reynolds Number F a = 1, b = -1, c = -1  v c ~  / (  D)  v c = R  / (  D) F Cylindrical pipes: R c ~ 2000 –For water, v c = 10 cm/s

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