Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of...

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Hydrostatics: Fluids at Rest

Transcript of Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of...

Page 1: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

Hydrostatics: Fluids at RestHydrostatics: Fluids at Rest

Page 2: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• applying Newtonian principles to fluids

• hydrostatics—the study of stationary fluids in which all forces are in equilibrium

Fluid MechanicsFluid Mechanics

Page 3: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• hydrodynamics—the study of fluids in motion

Fluid MechanicsFluid Mechanics

Page 4: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

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

DensityDensity

Page 5: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• specific gravity: density relative to water

• dimensionless number• numerically equal to the

density of the substance in g/cm³

DensityDensity

Page 6: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• 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 PressureUnits of Pressure

Page 7: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• 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 PressureUnits of Pressure

Page 8: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• 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 PressureUnits of Pressure

Page 9: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• gauge pressure (Pg) often used with piping systems

• absolute pressure (P)

Units of PressureUnits of Pressure

Page 10: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• pressure changes with depth

• density is usually assumed to be constant throughout depth

• y = d2 = d1 + Δd• ΣF = 0 N

Incompressible FluidsIncompressible Fluids

Page 11: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• ΣFy = Fd1 + Fd2 + Fw = 0 N• to calculate the pressure at

any depth d:

Incompressible FluidsIncompressible Fluids

Pd = Pref + ρgd

Page 12: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

Incompressible FluidsIncompressible Fluids

• 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

Pd = Pref + ρgd

Page 13: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• usually referring to gases, since their density is not constant with height/depth

Compressible FluidsCompressible Fluids

P = Pref ePref

ρref- |g|h

Page 14: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• must remember that temperature also affects the pressure of a gas

Compressible FluidsCompressible Fluids

Page 15: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

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

Hydraulic DevicesHydraulic Devices

Page 16: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• machines that transmit forces via enclosed liquids

• small input forces can generate large output forces

Hydraulic DevicesHydraulic Devices

Page 17: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• note the cross-sectional areas of each

• Fout = nFin

Hydraulic DevicesHydraulic Devices

Page 18: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• note the distance each piston travels

Hydraulic DevicesHydraulic Devices

Page 19: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• manometer• barometer

• first instrument to accurately measure atmospheric pressure

• used mercury

Pressure IndicatorsPressure Indicators

Page 20: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• famous problem: Archimedes and the crown

• What happens when an object is placed in a fluid?

BuoyancyBuoyancy

Page 21: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• for object in fluid:• Fw-o: gravitational force

on object in fluid• Fb: buoyant force on

object• Fb = ρ|g|V

BuoyancyBuoyancy

Page 22: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• Fb = ρ|g|V• ρ is the density of the

displaced fluid

BuoyancyBuoyancy

Page 23: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• 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

BuoyancyBuoyancy

Page 24: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• 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

BuoyancyBuoyancy

Page 25: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• 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.

BuoyancyBuoyancy

Page 26: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• 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.

BuoyancyBuoyancy

Page 27: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• 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.

BuoyancyBuoyancy

Page 28: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• 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.

BuoyancyBuoyancy

Page 29: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• 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 BuoyancyCenter of Buoyancy

Page 30: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

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

Center of BuoyancyCenter of Buoyancy

Page 31: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• 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 BuoyancyCenter of Buoyancy

Page 32: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• instrument used to measure density

• has many uses

HydrometerHydrometer

Page 33: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

Hydrodynamics: Fluids in MotionHydrodynamics: Fluids in Motion

Page 34: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• assumptions:• the fluid flows smoothly• the velocity of the fluid

does not change with time at a fixed location in the fluid path

Ideal FluidsIdeal Fluids

Page 35: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• assumptions:• the density of the fluid is

constant (incompressible)• friction has no effect on

fluid flow

Ideal FluidsIdeal Fluids

Page 36: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• Streamlines• not a physical reality• laminar• turbulent• flow tube

Ideal FluidsIdeal Fluids

Page 37: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• 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 FluidsIdeal Fluids

Page 38: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• equation of flow continuity:

Flow ContinuityFlow Continuity

A1v1 = A2v2

• requires tubes with smaller cross-sectional areas to have higher fluid velocities

Page 39: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• background equations:

Bernoulli’s PrincipleBernoulli’s Principle

ΔK = ½ρΔVv22 – ½ρΔVv1

2

ΔU = ρΔV|g|h2 – ρΔV|g|h1

Equation 17.12

Equation 17.13

Page 40: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• background equations:

Bernoulli’s PrincipleBernoulli’s Principle

Wncf = ΔK + ΔU

Wncf = P1ΔV – P2ΔV

Equation 17.14

Equation 17.15

Page 41: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• Bernoulli’s Equation:

Bernoulli’s PrincipleBernoulli’s Principle

P1 + ½ρv12 + ρ|g|h1 =

P2 + ½ρv22 + ρ|g|h2

Page 42: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• if the velocity does not change: v1 = v2

Bernoulli’s PrincipleBernoulli’s Principle

P1 + ½ρv12 + ρ|g|h1 =

P2 + ½ρv22 + ρ|g|h2

P1 + ρ|g|h1 = P2 + ρ|g|h2

Page 43: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• if the elevation of the fluid does not change: h1 = h2

Bernoulli’s PrincipleBernoulli’s Principle

P1 + ½ρv12 + ρ|g|h1 =

P2 + ½ρv22 + ρ|g|h2

P1 + ½ρv12 = P2 + ½ρv2

2

Page 44: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• A faster-flowing fluid will have streamlines that are closer together.

• A lower-pressure fluid will have streamlines that are closer together.

Bernoulli’s PrincipleBernoulli’s Principle

Page 45: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• airfoil: any device that generates lift as air flows along its surface

• hydrofoil: object that creates lift in liquid

LiftLift

Page 46: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• Bernoulli principle• Conadă effect

Theories of LiftTheories of Lift

Page 47: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• 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 FluidsReal Fluids

Page 48: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

• lower coefficients of viscosity indicate that the fluids flow more easily

• viscosity is sometimes referred to as the “thickness” of a fluid

Real FluidsReal Fluids

Page 49: Hydrostatics: Fluids at Rest. applying Newtonian principles to fluids hydrostatics—the study of stationary fluids in which all forces are in equilibrium.

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

Real FluidsReal Fluids