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Hydrostatics: Fluids at Rest

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

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Hydrostatics: Fluids at Rest

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  1. Hydrostatics: Fluids at Rest

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

  3. Fluid Mechanics • hydrodynamics—the study of fluids in motion

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

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

  6. 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.

  7. 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.

  8. 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

  9. Units of Pressure • gauge pressure (Pg) often used with piping systems • absolute pressure (P)

  10. Incompressible Fluids • pressure changes with depth • density is usually assumed to be constant throughout depth • y = d2 = d1 + Δd • ΣF = 0 N

  11. Incompressible Fluids • ΣFy = Fd1 + Fd2 + Fw = 0 N • to calculate the pressure at any depth d: Pd = Pref + ρgd

  12. 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

  13. ρref - |g|h Pref P = Prefe Compressible Fluids • usually referring to gases, since their density is not constant with height/depth

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

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

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

  17. Hydraulic Devices • note the cross-sectional areas of each • Fout = nFin

  18. Hydraulic Devices • note the distance each piston travels

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

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

  21. Buoyancy • for object in fluid: • Fw-o: gravitational force on object in fluid • Fb: buoyant force on object • Fb = ρ|g|V

  22. Buoyancy • Fb = ρ|g|V • ρ is the density of the displaced fluid

  23. 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

  24. 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

  25. 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.

  26. 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.

  27. 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.

  28. 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.

  29. 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.

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

  31. 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.

  32. Hydrometer • instrument used to measure density • has many uses

  33. Hydrodynamics: Fluids in Motion

  34. 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

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

  36. Ideal Fluids • Streamlines • not a physical reality • laminar • turbulent • flow tube

  37. 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.

  38. Flow Continuity • equation of flow continuity: A1v1 = A2v2 • requires tubes with smaller cross-sectional areas to have higher fluid velocities

  39. Bernoulli’s Principle • background equations: ΔK = ½ρΔVv22 – ½ρΔVv12 Equation 17.12 ΔU = ρΔV|g|h2 – ρΔV|g|h1 Equation 17.13

  40. Bernoulli’s Principle • background equations: Wncf = ΔK + ΔU Equation 17.14 Wncf = P1ΔV – P2ΔV Equation 17.15

  41. P1 + ½ρv12 + ρ|g|h1 = P2 + ½ρv22 + ρ|g|h2 Bernoulli’s Principle • Bernoulli’s Equation:

  42. P1 + ½ρv12 + ρ|g|h1 = P2 + ½ρv22 + ρ|g|h2 Bernoulli’s Principle • if the velocity does not change: v1 = v2 P1 + ρ|g|h1 = P2 + ρ|g|h2

  43. P1 + ½ρv12 + ρ|g|h1 = P2 + ½ρv22 + ρ|g|h2 Bernoulli’s Principle • if the elevation of the fluid does not change: h1 = h2 P1 + ½ρv12 = P2 + ½ρv22

  44. 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.

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

  46. Theories of Lift • Bernoulli principle • Conadă effect

  47. 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)

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

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

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