Introduction To Fluids

Density ρ = m/V ρ = m/V  ρ: density (kg/m 3 )  m: mass (kg)  V: volume (m 3 )

Pressure p = F/A p = F/A  p : pressure (Pa)  F: force (N)  A: area (m 2 )

Pressure The pressure of a fluid is exerted in all directions. The force on a surface caused by pressure is always normal to the surface.

The Pressure of a Liquid p = ρgh p = ρgh  p: pressure (Pa)  ρ: density (kg/m3)  g: acceleration constant (9.8 m/s 2 )  h: height of liquid column (m)

Absolute Pressure p = p o + ρgh p = p o + ρgh  p: pressure (Pa)  p o : atmospheric pressure (Pa)  ρgh: liquid pressure (Pa)

Piston Density of Hg 13,400 kg/m 2 Problem 25 cm A Area of piston: 8 cm 2 Weight of piston: 200 N What is total pressure at point A?

Floating is a type of equilibrium

Archimedes’ Principle: a body immersed in a fluid is buoyed up by a force that is equal to the weight of the fluid displaced. Buoyant Force: the upward force exerted on a submerged or partially submerged body.

Calculating Buoyant Force F buoy = ρVg F buoy : the buoyant force exerted on a submerged or partially submerged object. V: the volume of displaced liquid. ρ: the density of the displaced liquid.

Buoyant force on submerged object mg F buoy = ρVg Note: if F buoy < mg, the object will sink deeper!

Buoyant force on submerged object mg F buoy = ρVg SCUBA divers use a buoyancy control system to maintain neutral buoyancy (equilibrium!)

Buoyant force on floating object mg F buoy = ρVg If the object floats, we know for a fact F buoy = mg!

Fluid Flow Continuity Conservation of Mass results in continuity of fluid flow. Conservation of Mass results in continuity of fluid flow. The volume per unit time of water flowing in a pipe is constant throughout the pipe. The volume per unit time of water flowing in a pipe is constant throughout the pipe.

Fluid Flow Continuity A 1 v 1 = A 2 v 2 A 1 v 1 = A 2 v 2 –A 1, A 2 : cross sectional areas at points 1 and 2 –v 1, v 2 : speed of fluid flow at points 1 and 2

Fluid Flow Continuity V = Avt V = Avt –V: volume of fluid (m 3 ) –A: cross sectional areas at a point in the pipe (m 2 ) –v : speed of fluid flow at a point in the pipe (m/s) –t: time (s)

Bernoulli’s Theorem The sum of the pressure, the potential energy per unit volume, and the kinetic energy per unit volume at any one location in the fluid is equal to the sum of the pressure, the potential energy per unit volume, and the kinetic energy per unit volume at any other location in the fluid for a non-viscous incompressible fluid in streamline flow.

Bernoulli’s Theorem All other considerations being equal, when fluid moves faster, the pressure drops.

Bernoulli’s Theorem p + ρg h + ½ ρv 2 = Constant p + ρg h + ½ ρv 2 = Constant –p : pressure (Pa) –ρ : density of fluid (kg/m 3 ) –g: gravitational acceleration constant (9.8 m/s 2 ) –h: height above lowest point (m) –v: speed of fluid flow at a point in the pipe (m/s)

Bernoulli’s Theorem p 1 + ρg h 1 + ½ ρv 1 2 = p 2 + ρg h 2 + ½ ρv 2 2