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Hydrostatics: Fluids at RestHydrostatics: Fluids at Rest
• applying Newtonian principles to fluids
• hydrostatics—the study of stationary fluids in which all forces are in equilibrium
Fluid MechanicsFluid Mechanics
• hydrodynamics—the study of fluids in motion
Fluid MechanicsFluid Mechanics
• abbreviation: ρ• mass per unit volume• g/cm³ is commonly used• SI unit: kg/m³
DensityDensity
• specific gravity: density relative to water
• dimensionless number• numerically equal to the
density of the substance in g/cm³
DensityDensity
• 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
• 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
• 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
• gauge pressure (Pg) often used with piping systems
• absolute pressure (P)
Units of PressureUnits of Pressure
• pressure changes with depth
• density is usually assumed to be constant throughout depth
• y = d2 = d1 + Δd• ΣF = 0 N
Incompressible FluidsIncompressible Fluids
• ΣFy = Fd1 + Fd2 + Fw = 0 N• to calculate the pressure at
any depth d:
Incompressible FluidsIncompressible Fluids
Pd = Pref + ρgd
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
• usually referring to gases, since their density is not constant with height/depth
Compressible FluidsCompressible Fluids
P = Pref ePref
ρref- |g|h
• must remember that temperature also affects the pressure of a gas
Compressible FluidsCompressible Fluids
• Pascal’s principle: the external pressure applied to a completely enclosed incompressible fluid is distributed in all directions throughout the fluid
Hydraulic DevicesHydraulic Devices
• machines that transmit forces via enclosed liquids
• small input forces can generate large output forces
Hydraulic DevicesHydraulic Devices
• note the cross-sectional areas of each
• Fout = nFin
Hydraulic DevicesHydraulic Devices
• note the distance each piston travels
Hydraulic DevicesHydraulic Devices
• manometer• barometer
• first instrument to accurately measure atmospheric pressure
• used mercury
Pressure IndicatorsPressure Indicators
• famous problem: Archimedes and the crown
• What happens when an object is placed in a fluid?
BuoyancyBuoyancy
• for object in fluid:• Fw-o: gravitational force
on object in fluid• Fb: buoyant force on
object• Fb = ρ|g|V
BuoyancyBuoyancy
• Fb = ρ|g|V• ρ is the density of the
displaced fluid
BuoyancyBuoyancy
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• defined: the center of mass of the fluid that would occupy the submerged space that the object occupies
Center of BuoyancyCenter 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.
Center of BuoyancyCenter of Buoyancy
• instrument used to measure density
• has many uses
HydrometerHydrometer
Hydrodynamics: Fluids in MotionHydrodynamics: Fluids in Motion
• 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
• assumptions:• the density of the fluid is
constant (incompressible)• friction has no effect on
fluid flow
Ideal FluidsIdeal Fluids
• Streamlines• not a physical reality• laminar• turbulent• flow tube
Ideal FluidsIdeal 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.
Ideal FluidsIdeal Fluids
• equation of flow continuity:
Flow ContinuityFlow Continuity
A1v1 = A2v2
• requires tubes with smaller cross-sectional areas to have higher fluid velocities
• background equations:
Bernoulli’s PrincipleBernoulli’s Principle
ΔK = ½ρΔVv22 – ½ρΔVv1
2
ΔU = ρΔV|g|h2 – ρΔV|g|h1
Equation 17.12
Equation 17.13
• background equations:
Bernoulli’s PrincipleBernoulli’s Principle
Wncf = ΔK + ΔU
Wncf = P1ΔV – P2ΔV
Equation 17.14
Equation 17.15
• Bernoulli’s Equation:
Bernoulli’s PrincipleBernoulli’s Principle
P1 + ½ρv12 + ρ|g|h1 =
P2 + ½ρv22 + ρ|g|h2
• 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
• 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
• 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
• airfoil: any device that generates lift as air flows along its surface
• hydrofoil: object that creates lift in liquid
LiftLift
• Bernoulli principle• Conadă effect
Theories of LiftTheories of Lift
• 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
• 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
• particles closest to the walls move more slowly than those farther from the walls
Real FluidsReal Fluids