2 Overview Continuity Equation Navier-Stokes Equation (a bit of vector notation...)Examples (all laminar flow)Flow between stationary parallel horizontal platesFlow between inclined parallel platesPipe flow (Hagen Poiseuille)
3 Conservation of Mass in Differential Equation Form Mass flux out of differential volumeRate of change of mass in differential volumeMass flux into differential volume
4 Continuity Equation Mass flux out of differential volume Higher order termoutinRate of mass decrease1-d continuity equation
5 Continuity Equation u, v, w are velocities in x, y, and z directions 3-d continuity equationu, v, w are velocities in x, y, and z directionsVector notationIf density is constant...or in vector notationTrue everywhere! (contrast with CV equations!)
7 Navier-Stokes Equations Derived by Claude-Louis-Marie Navier in 1827General Equation of Fluid MotionBased on conservation of ___________ with forces…_______________________________U.S. National Academy of Sciences has made the full solution of the Navier-Stokes Equations a top prioritymomentumGravityPressureShear
8 Navier-Stokes Equations h is vertical (positive up)Inertial forces [N/m3], a is Lagrangian accelerationIs acceleration zero when dV/dt = 0?Pressure gradient (not due to change in elevation)Shear stress gradientWhat is in a static fluid? _____Zero!
9 DxOver what time did this change of velocity occur (for a particle of fluid)?Why no term?
10 Notation: Total Derivative Eulerian Perspective Total derivative (chain rule)Material or substantial derivativeLagrangian accelerationN-S
11 Application of Navier-Stokes Equations The equations are nonlinear partial differential equationsNo full analytical solution existsThe equations can be solved for several simple flow conditionsNumerical solutions to Navier-Stokes equations are increasingly being used to describe complex flows.
12 Navier-Stokes Equations: A Simple Case No acceleration and no velocity gradientsxyz could have any orientation1Let y be vertical upwardFor constant g
13 Infinite Horizontal Plates: Laminar Flow Derive the equation for the laminar, steady, uniform flow between infinite horizontal parallel plates.yxx1yHydrostatic in yz
14 Infinite Horizontal Plates: Laminar Flow Pressure gradient in x balanced by shear gradient in yNo a so forces must balance!Now we must find A and B… Boundary Conditions
15 Infinite Horizontal Plates: Boundary Conditions No slip conditionatuu = 0 at y = 0 and y = aletbe___________negativeWhat can we learn about t?
16 Laminar Flow Between Parallel Plates UqauyxNo fluid particles are acceleratingWrite the x-component
17 Flow between Parallel Plates u is only a function of yh and p are only functions of xGeneral equation describing laminar flow between parallel plates
18 Flow Between Parallel Plates: Integration Uqauyx
19 Boundary Conditions u = 0 at y = 0 u = U at y = a Boundary conditions
21 Example: Oil Skimmer g = 8430 N/m3 m = 1x10-2 Ns/m2 An oil skimmer uses a 5 m wide x 6 m long moving belt above a fixed platform (q=30º) to skim oil off of rivers (T=10 ºC). The belt travels at 3 m/s. The distance between the belt and the fixed platform is 2 mm. The belt discharges into an open container on the ship. The fluid is actually a mixture of oil and water. To simplify the analysis, assume crude oil dominates. Find the discharge and the power required to move the belt.hg = 8430 N/m3lm = 1x10-2 Ns/m230º
22 Example: Oil Skimmer (per unit width) Q = 0.0027 m2/s dominatesQ = m2/s(per unit width)In direction of beltQ = m2/s (5 m) = m3/s
23 Example: Oil Skimmer Power Requirements How do we get the power requirement?___________________________What is the force acting on the belt?Remember the equation for shear?_____________ Evaluate at y = a.Power = Force x Velocity [N·m/s]Shear force (t · L · W)t = m(du/dy)
24 Example: Oil Skimmer Power Requirements (shear by belt on fluid)FV= 3.46 kW
25 Example: Oil Skimmer Where did the Power Go? Where did the energy input from the belt go?Lifting the oil (potential energy)Heating the oil (thermal energy)Dh = 3 m
26 Example : Oil Skimmer Was it Really Laminar Flow? We assumed that the flow was laminar (based on the small flow dimension of 2 mm)We need to check our assumption!!!!m3/s= 1.36 m/s0.002 m* 5 m1.36 m/sLaminar
27 Example: No flowFind the velocity of a vertical belt that is 5 mm from a stationary surface that will result in no flow of glycerin at 20°C (m = 0.62 Ns/m2 and g =12300 N/m3)
28 Laminar Flow through Circular Tubes Different geometry, same equation development (see Munson, et al. p 367)Apply equation of motion to cylindrical sleeve (use cylindrical coordinates)
29 Laminar Flow through Circular Tubes: Equations R is radius of the tubeMax velocity when r = 0Velocity distribution is paraboloid of revolution therefore _____________ _____________average velocity (V) is 1/2 vmaxQ = VA =VpR2
30 Laminar Flow through Circular Tubes: Diagram ShearVelocityLaminar flowShear at the wallTrue for Laminar or Turbulent flow
31 The Hagen-Poiseuille Equation cv pipe flowConstant cross sectionh or zLaminar pipe flow equationsFrom Navier-StokesCV equations!
32 Example: Laminar Flow (Team work) Calculate the discharge of 20ºC water through a long vertical section of 0.5 mm ID hypodermic tube. The inlet and outlet pressures are both atmospheric. You may neglect minor losses. What is the total shear force?What assumption did you make? (Check your assumption!)
33 SummaryNavier-Stokes Equations and the Continuity Equation describe complex flow including turbulenceThe Navier-Stokes Equations can be solved analytically for several simple flowsNumerical solutions are required to describe turbulent flows
36 Euler’s Equation Along a Streamline Inviscid flow (frictionless)x along a streamlineVelocity normal to streamline is zerov = u = velocity in x direction
37 We’ve assumed: frictionless and along a streamline Euler’s EquationWe’ve assumed: frictionless and along a streamlineSteadyx is the only independent variable(Multiplying by dx converts from a force balance equation to an energy equation)Euler’s equation along a streamline
38 Bernoulli Equation Integrate for constant density Bernoulli Equation Euler’s equationIntegrate for constant densityBernoulli EquationThe Bernoulli Equation is a statement of the conservation of ____________________p.e.k.e.Mechanical Energy
39 Hydrostatic Normal to Streamlines? x, u along streamliney, v perpendicular to streamline (v = 0)
40 Laminar Flow between Parallel Plates hadyUluyqdlq
41 Equation of Motion: Force Balance +pressure--shear+gravity+q=accelerationl
42 Equation of MotionhButqlLaminar flow assumption!
43 Limiting cases y U x u q Motion of plate Pressure gradient Hydrostatic pressureLinear velocity distributionBoth plates stationaryParabolic velocity distribution