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Published byPhillip O’Neal’ Modified over 9 years ago
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Short Version : 15. Fluid Motion
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Fluid = matter that flows under external forces = liquid & gas. solidliquidgas inter-mol forcesstrongestmediumweakest volumefixed variable shapefixedvariable
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15.1. Density & Pressure Avogadro’s number N A = 6.022 10 23 / 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
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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
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15.2. Hydrostatic Equilibrium Hydrostatic equilibrium : F net = 0 everywhere in fluid Fluid is at rest. F ext 0 gives rise to pressure differences.
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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 )
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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):
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Measuring Pressure Barometer = device for measuring atmospheric pressure vacuum inside tube : For p = 1 atm = 101.3 kPa : Cf. h = 10 m for a water barometer
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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
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Example 15.2. 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
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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
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Example 15.4. 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?
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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.
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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
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Conservation of Mass: The Continuity Equation Flow tube : small region with sides tangent, & end faces perpendicular, to streamlines. flow tubes do not cross streamlines.
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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.
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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
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Example 15.6. 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 :
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Example 15.7. Venturi Flowmeter Find the flow speed in the unconstricted pipe of a Venturi flowmeter. Bernoulli’s eq. Continuity eq.
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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
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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
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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.
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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
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