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Transcript

Hydrostatics: Fluids at Rest Fluid Mechanics • applying Newtonian principles to fluids • hydrostatics—the study of stationary fluids in which all forces are in equilibrium Fluid Mechanics • hydrodynamics—the study of fluids in motion Density • • • • abbreviation: ρ mass per unit volume g/cm³ is commonly used SI unit: kg/m³ Density • specific gravity: density relative to water • dimensionless number • numerically equal to the density of the substance in g/cm³ Units of Pressure • Pressure is defined as the force exerted perpendicular to a unit area. • When a fluid is at rest, the pressure is uniform throughout the fluid in all directions. Units of Pressure • At the boundaries of a fluid, the container exerts a pressure on the fluid identical to the pressure the fluid exerts on the container. Units of Pressure • SI unit: Pascal (Pa) • Earliest: atmosphere (atm) • 1 atm = 1.013 × 105 Pa • torr • bars and millibars (mb) • 1 atm = 1.013 bar = 1013 mb Units of Pressure • gauge pressure (Pg) often used with piping systems • absolute pressure (P) Incompressible Fluids • pressure changes with depth • density is usually assumed to be constant throughout depth • y = d2 = d1 + Δd • ΣF = 0 N Incompressible Fluids • ΣFy = Fd1 + Fd2 + Fw = 0 N • to calculate the pressure at any depth d: Pd = Pref + ρgd Incompressible Fluids Pd = Pref + ρgd • d is expressed as a negative scalar distance • g = -9.81 m/s² • Pref is atmospheric pressure if the liquid’s container is open to the atmosphere Compressible Fluids • usually referring to gases, since their density is not constant with height/depth - P = Pref e ρref Pref |g|h Compressible Fluids • must remember that temperature also affects the pressure of a gas Hydraulic Devices • Pascal’s principle: the external pressure applied to a completely enclosed incompressible fluid is distributed in all directions throughout the fluid Hydraulic Devices • machines that transmit forces via enclosed liquids • small input forces can generate large output forces Hydraulic Devices • note the crosssectional areas of each • Fout = nFin Hydraulic Devices • note the distance each piston travels Pressure Indicators • manometer • barometer • first instrument to accurately measure atmospheric pressure • used mercury Buoyancy • famous problem: Archimedes and the crown • What happens when an object is placed in a fluid? Buoyancy • for object in fluid: • Fw-o: gravitational force on object in fluid • Fb: buoyant force on object • Fb = ρ|g|V Buoyancy • Fb = ρ|g|V • ρ is the density of the displaced fluid Buoyancy • Archimedes’ principle: any system that is submerged or floats in a fluid is acted on by an upward buoyant force equal in magnitude to the weight of the fluid it displaces Buoyancy • If the buoyant force is equal to the system’s weight, the forces are balanced and no acceleration occurs. • requires object and fluid to have equal density Buoyancy • If the weight of a system is greater than that of the displaced fluid, its density is greater than the fluid’s. • Since weight exceeds the buoyant force, the object will sink. Buoyancy • If the weight of a system is less than that of the displaced fluid, its density is less than the fluid’s. • Since buoyant force is greater than weight, the object will accelerate up. Buoyancy • When the object rises to the surface of the liquid, its volume remaining beneath the surface changes the buoyant force until they are in equilibrium. Buoyancy • This is also true with gases. • The density of a gas changes with altitude and temperature. • The object may respond to a change in pressure. Center of Buoyancy • Every object submerged in a fluid has both a center of mass and a center of buoyancy. • These are the same for objects of uniform density that are completely submerged. Center of Buoyancy • defined: the center of mass of the fluid that would occupy the submerged space that the object occupies Center of Buoyancy • If the center of mass and center of buoyancy are not the same, the object will experience a torque and rotate. • The center of buoyancy will be directly above the center of gravity. Hydrometer • instrument used to measure density • has many uses Hydrodynamics: Fluids in Motion Ideal Fluids • assumptions: • the fluid flows smoothly • the velocity of the fluid does not change with time at a fixed location in the fluid path Ideal Fluids • assumptions: • the density of the fluid is constant (incompressible) • friction has no effect on fluid flow Ideal Fluids • Streamlines • not a physical reality • laminar • turbulent • flow tube Ideal Fluids • The rate of volume and mass flow into a segment of a flow tube equals the rate of volume and mass flow out of the flow tube segment. Flow Continuity • equation of flow continuity: A1v1 = A2v2 • requires tubes with smaller cross-sectional areas to have higher fluid velocities Bernoulli’s Principle • background equations: ΔK = ½ρΔVv22 – ½ρΔVv12 Equation 17.12 ΔU = ρΔV|g|h2 – ρΔV|g|h1 Equation 17.13 Bernoulli’s Principle • background equations: Wncf = ΔK + ΔU Equation 17.14 Wncf = P1ΔV – P2ΔV Equation 17.15 Bernoulli’s Principle • Bernoulli’s Equation: P1 + ½ρv12 + ρ|g|h1 = P2 + ½ρv22 + ρ|g|h2 Bernoulli’s Principle • if the velocity does not change: v1 = v2 P1 + ½ρv12 + ρ|g|h1 = P1 + ρ|g|h1 = P2 + ρ|g|h2 P2 + ½ρv22 + ρ|g|h2 Bernoulli’s Principle • if the elevation of the fluid does not change: h1 = h2 P1 + ½ρv12 2+ ρ|g|h1 = 2 P1 + ½ρv1 = P2 + ½ρv2 P2 + ½ρv22 + ρ|g|h2 Bernoulli’s Principle • A faster-flowing fluid will have streamlines that are closer together. • A lower-pressure fluid will have streamlines that are closer together. Lift • airfoil: any device that generates lift as air flows along its surface • hydrofoil: object that creates lift in liquid Theories of Lift • Bernoulli principle • Conadă effect Real Fluids • viscosity: a measure of the resistance of fluid to a flow • caused by cohesive forces between particles of a fluid • a type of internal friction • coefficient of viscosity (η) Real Fluids • lower coefficients of viscosity indicate that the fluids flow more easily • viscosity is sometimes referred to as the “thickness” of a fluid Real Fluids • particles closest to the walls move more slowly than those farther from the walls