Short Version : 15. Fluid Motion

Fluid = matter that flows under external forces = liquid & gas. solidliquidgas inter-mol forcesstrongestmediumweakest volumefixed variable shapefixedvariable

15.1. Density & Pressure Avogadro’s number N A =  / mol. 1 mole = amount of substance containing N A basic elements. ( with N A = number of atoms in 12 g of 12 C ). Fluid: average position of molecules not fixed. Macroscopic viewpoint: deformable continuum. Density  = mass / vol, [  ] = kg / m 3. Incompressible = density unchanged under pressure Liquid is nearly incompressible (molecules in contact). Gas is compressible. dV fluid point dV  0 thousands of molecules

Pressure Pressure = normal force per unit area Pressure is a scalar. The pressure at a point in a fluid is the magnitude of the radial force per unit area acting on a fluid point at that position. AnAn F Fluid point

15.2. Hydrostatic Equilibrium Hydrostatic equilibrium : F net = 0 everywhere in fluid Fluid is at rest. F ext  0 gives rise to pressure differences.

x Let f be the force density within the fluid : Force experienced by the fluid element: ( f is the force per unit volume experienced by a small fluid element due to pressure differences )

Hydrostatic Equilibrium with Gravity Fluid element: area A, thickness dh, mass dm. Net pressure force on fluid element: Gravitational force on fluid element: Hydrostatic Equilibrium :  Liquid (~incompressible):

Measuring Pressure Barometer = device for measuring atmospheric pressure vacuum inside tube : For p = 1 atm = kPa : Cf. h = 10 m for a water barometer

Manometer Manometer = U-shaped tube filled with liquid to measure pressure differences. Gauge pressure = excess pressure above atmospheric. Used in tires, sport equipments, etc. E.g., tire gauge pressure = 30 psi  tire pressure = 44.7 psi Pascal’s law: An external pressure applied to a fluid in a closed vessel is uniformly transmitted throughout the fluid. equal p

Example Hydraulic Lift In a hydraulic lift, a large piston supports a car. The total mass of car & piston is 3200 kg. What force must be applied to the smaller piston to support the car?  Pascal’s law

15.3. Archimedes’ Principle & Buoyancy Archimedes’ Principle: The buoyancy force on an object is equal to the weight of the fluid it displaces. Buoyancy force: Upward force felt by an object in a fluid Neutral buoyancy : average density of object is the same as that of fluid. fluid element in equilibrium F b unchanged after replacement

Example Tip of the Iceberg Average density of a typical iceberg is 0.86 that of seawater. What fraction of an iceberg’s volume is submerged?

Center of Buoyancy Buoyancy force acts at the center of buoyancy (CB), which coincides with the CM of the displaced water. CM must be lower than CB to be stable.

15.4. Fluid Dynamics Moving fluid is described by its flow velocity v( r, t ). Streamlines = Lines with tangents everywhere parallel to v( r, t ). Spacing of streamlines is inversely proportional to the flow speed. Steady flow: Small particles (e.g., dyes) in fluid move along streamlines. e.g., calm river. Example of unsteady flow: blood in arteries ( pumped by heart ). Fluid dynamics: Newton’s law + diffusing viscosity  Navier-Stokes equations slowfast

Conservation of Mass: The Continuity Equation Flow tube : small region with sides tangent, & end faces perpendicular, to streamlines.  flow tubes do not cross streamlines.

Steady flow Conservation of mass: Mass entering tube: Mass leaving tube: Equation of continuity for steady flow : Mass flow rate = [  v A ] = kg / s Volume flow rate = Liquid: [ v A ] = m 3 / s Liquid : flows faster in constricted area. Gas with v < v s ound : flows faster in constricted area. Gas with v > v sound : flows slower in constricted area.

Conservation of Energy: Bernoulli’s Equation Same fluid element enters & leaves tube: Work done by pressure upon its entering tube: Work done by pressure upon its leaving tube: Work done by gravity during the trip: W-E theorem:  Incompressible fluid:  Bernoulli’s Equation Viscosity & other works neglected

Example Draining a Tank A large open tank is filled to height h with liquid of density . Find the speed of liquid emerging from a small hole at the base of the tank. At top surface : At hole : 

Example Venturi Flowmeter Find the flow speed in the unconstricted pipe of a Venturi flowmeter.  Bernoulli’s eq. Continuity eq. 

Bernoulli Effect A ping-pong ball supported by downward-flowing air. High-velocity flow is inside the narrow part of the funnel. Bernoulli Effect: p   v  Example: Prairie dog’s hole Dirt mound forces wind to accelerate over hole  low pressure above hole  natural ventilation

Flight & Lift Aerodynamic lift Top view on a curved ball : spin Blade pushes down on air Air pushes up (3 rd law) Faster flow, lower P : uplift. Top view on a straight ball : no spin

Application: Wind Energy A chunk of air, of speed v & density , passing thru a turbine of area A in time  t, has kinetic energy  available power per unit area = Better analysis  For Present tech gives 80% of this.

15.6. Viscosity & Turbulence Smooth flow becomes turbulent. Viscosity: friction due to momentum transfer between adjacent fluid layers or between fluid & wall. B.C.:v = 0 at wall drag on moving object. provide 3 rd law force on propellers. stabilize flow. flow with no viscosity flow with viscosity