Hydrostatics: Fluids at Rest

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 Fluid Mechanics

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 Density

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 = × 10 5 Pa torr bars and millibars (mb) 1 atm = bar = 1013 mb Units of Pressure

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

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

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

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

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

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

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

note the distance each piston travels Hydraulic Devices

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

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

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

F b = 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. 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. Center of Buoyancy

instrument used to measure density has many uses Hydrometer

Hydrodynamics: 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 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. Ideal Fluids

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

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

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

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

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

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

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

Bernoulli principle Conadă effect Theories 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 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 Real Fluids