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Hydrostatics: Fluids at Rest

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applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium Fluid Mechanics

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hydrodynamics—the study of fluids in motion Fluid Mechanics

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abbreviation: ρ mass per unit volume g/cm³ is commonly used SI unit: kg/m³ Density

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specific gravity: density relative to water dimensionless number numerically equal to the density of the substance in g/cm³ Density

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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

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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

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SI unit: Pascal (Pa) Earliest: atmosphere (atm) 1 atm = 1.013 × 10 5 Pa torr bars and millibars (mb) 1 atm = 1.013 bar = 1013 mb Units of Pressure

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gauge pressure (P g ) often used with piping systems absolute pressure (P) Units of Pressure

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pressure changes with depth density is usually assumed to be constant throughout depth y = d 2 = d 1 + Δd Σ F = 0 N Incompressible Fluids

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Σ F y = F d1 + F d2 + F w = 0 N to calculate the pressure at any depth d: Incompressible Fluids P d = P ref + ρgd

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Incompressible Fluids d is expressed as a negative scalar distance g = -9.81 m/s² P ref is atmospheric pressure if the liquid’s container is open to the atmosphere P d = P ref + ρgd

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usually referring to gases, since their density is not constant with height/depth Compressible Fluids P = P ref e P ref ρ ref - |g|h

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must remember that temperature also affects the pressure of a gas Compressible Fluids

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Pascal’s principle: the external pressure applied to a completely enclosed incompressible fluid is distributed in all directions throughout the fluid Hydraulic Devices

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machines that transmit forces via enclosed liquids small input forces can generate large output forces Hydraulic Devices

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note the cross- sectional areas of each F out = nF in Hydraulic Devices

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note the distance each piston travels Hydraulic Devices

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manometer barometer first instrument to accurately measure atmospheric pressure used mercury Pressure Indicators

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famous problem: Archimedes and the crown What happens when an object is placed in a fluid? Buoyancy

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for object in fluid: F w-o : gravitational force on object in fluid F b : buoyant force on object F b = ρ|g|V Buoyancy

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F b = ρ|g|V ρ is the density of the displaced fluid Buoyancy

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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

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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

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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

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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

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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

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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. Buoyancy

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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

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defined: the center of mass of the fluid that would occupy the submerged space that the object occupies Center of Buoyancy

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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. Center of Buoyancy

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instrument used to measure density has many uses Hydrometer

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Hydrodynamics: Fluids in Motion

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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

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assumptions: the density of the fluid is constant (incompressible) friction has no effect on fluid flow Ideal Fluids

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Streamlines not a physical reality laminar turbulent flow tube Ideal Fluids

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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. Ideal Fluids

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equation of flow continuity: Flow Continuity A 1 v 1 = A 2 v 2 requires tubes with smaller cross-sectional areas to have higher fluid velocities

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background equations: Bernoulli’s Principle ΔK = ½ρΔVv 2 2 – ½ρΔVv 1 2 ΔU = ρΔV|g|h 2 – ρΔV|g|h 1 Equation 17.12 Equation 17.13

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background equations: Bernoulli’s Principle W ncf = ΔK + ΔU W ncf = P 1 ΔV – P 2 ΔV Equation 17.14 Equation 17.15

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Bernoulli’s Equation: Bernoulli’s Principle P 1 + ½ρv 1 2 + ρ|g|h 1 = P 2 + ½ρv 2 2 + ρ|g|h 2

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if the velocity does not change: v 1 = v 2 Bernoulli’s Principle P 1 + ½ρv 1 2 + ρ|g|h 1 = P 2 + ½ρv 2 2 + ρ|g|h 2 P 1 + ρ|g|h 1 = P 2 + ρ|g|h 2

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if the elevation of the fluid does not change: h 1 = h 2 Bernoulli’s Principle P 1 + ½ρv 1 2 + ρ|g|h 1 = P 2 + ½ρv 2 2 + ρ|g|h 2 P 1 + ½ρv 1 2 = P 2 + ½ρv 2 2

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A faster-flowing fluid will have streamlines that are closer together. A lower-pressure fluid will have streamlines that are closer together. Bernoulli’s Principle

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airfoil: any device that generates lift as air flows along its surface hydrofoil: object that creates lift in liquid Lift

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Bernoulli principle Conadă effect Theories of Lift

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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

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lower coefficients of viscosity indicate that the fluids flow more easily viscosity is sometimes referred to as the “thickness” of a fluid Real Fluids

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particles closest to the walls move more slowly than those farther from the walls Real Fluids

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