3 A. Continuity Equation Mass flow is conserved Flow Rate 3A. Continuity EquationMass flow is conservedFlow RateContinuity Equation
4 1. Definition of Fluid Flow 4(a) Volume Flux: Measured in units of:Imperial: gallons per minSI: cubic meters per second(b) Relate to “flow speed” vA=cross section area of pipev=speed of flow
5 2. Mass Flux 5 (a) Mass Flux: Measured in units of: SI: kg/sec (b) Relate to “volume flux”=density of fluidA=cross section area of pipev=speed of flow
6 3. Continuity Equation6Based upon conservation of mass, when a fluid (liquid or gas) is forced through a smaller pipe, the speed must increase.
7 3b. Incompressible Fluids 7Unlike gasses, liquids are difficult to compress. Hence the density is constant. The continuity equation reduces to the velocity inversely proportional to the cross section area:
8 B. Bernoulli Equation Torricelli's law (1643) Bernoulli effect 8B. Bernoulli EquationTorricelli's law (1643)Bernoulli effectBernoulli equation (1738)
9 1. Torricelli's law (1643)9In brief, the velocity of a fluid exiting at the bottom of a tank of depth “h” is independent of the fluid’s density (i.e. the fluid analogy of Galileo’s law that all bodies fall at same rate independent of mass)
10 102. Bernoulli Effect1738, Daniel Bernoulli notes that pressure decreases when a fluid’s velocity increases.
11 Eulerian Flow (1757) 11 Note on your Lab: If no friction (no viscosity), the volumetric flow rate out of a pipe of radius “r” with pressure difference P will hence be proportional to the square of the radius:
12 123. Bernoulli Equation1738, at any point in the fluid, the sum of the pressure, kinetic energy density and potential energy density is a constantFrom this can derive Pascal’s law of depth, Torricelli’s law and Bernoulli effect
13 3b. Venturi Tube13Can be used to measure flow rate v1 of a liquid (or gas) from the observed pressure difference (inferred from “h”) when cross section area is decreased.
14 144. Rayleigh’s drag forceFor high velocity, drag force (on object with surface area “A”) increases quadratically with velocity due to inelastic collisions of object with molecules of fluid (density ).The coefficient of drag “Cd” depends upon geometry. Hence “power” goes like the cube of velocity (i.e. windmills most efficient at high velocities).
15 15C. ViscosityDefinitionDrag ForceReynolds number
16 161. Fluid ShearInviscid fluid, no viscosity fluid all flows at same rate“Eulerian flow”
17 1. Fluid Shear 17 With viscosity, there is a flow gradient. Near the wall the velocity is zeroVelocity increases linearly as move away from wall.Velocity gradient: ratio of velocity to distance “h” from wall
18 1c. Definition of Viscosity 181c. Definition of ViscosityViscosity defined by ratio of force per wall area to the velocity gradient.
19 Viscosity Units SI Unit: Poiseulle=Pl=Pasec 19Viscosity UnitsSI Unit: Poiseulle=Pl=PasecCgs: poise “P”=gm/(cms)=0.1 PlMost common: centipoise: 100 cP=PImperial: reyn=lbsec/inch2=6.894 Pl
21 212. Drag ForceStoke's Law (1851) for sphere (radius “r”) moving at slow speed “v” in a fluid of viscosity “”:If falling, terminal velocity reached (density of object )
22 222. ViscometerMeasure viscosity of fluid from terminal velocity (“settling velocity”) of object. When drag plus buoyant force equal gravity we have:
23 233. Poiseuille’s Law (1838)Pressure required to make fluid flow with (average) velocity “v” in pipe of length “L”, radius “r”Expressed in terms of volumetric flow “Q”, we find an r4 dependence!
24 244. Reynold’s NumberDimensionless ratio (invented by Stokes 1851, popularized by Reynolds 1883) of the kinetic effects to the frictional effects.R< viscosity dominates, Stoke’s law validR=1000 Rayleigh’s drag force dominatesR>2000 unstableR>3000 turbulence
25 25NotesDemo PHET Bernoulli:Demo PHET for Gas Law: