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ME 362 Navier-Stokes Equation Page 1 of 2 Differential Equations of Mass and Momentum Conservations: For incompressible flow: Continuity: u v 0 x y NS equation: DV g P 2V Dt where D (V ) (V ) u (V ) v (V ) Dt t x y 2 2 2 (V ) 2 (V ) 2 (V ) x y Solution Procedure: Step 1: Assumptions Used to simplify NS equation (reducing PDE to ODE) Common assumptions are: o Incompressible o Steady state o Fully developed and are constant V 0 t V 0 x (time independent) (space independent) Other assumptions may include: o Neglect gravity o No pressure gradient g0 P 0 Step 2: Boundary Conditions Needed for solving integration constants in Step 5 Sketching the problem will help to define boundary conditions ME 362 Navier-Stokes Equation Page 2 of 2 Step 3: Continuity Equation u v 0 x y Step 4: Momentum Equation x – momentum: y – momentum: D P (u ) g x 2 (u ) Dt x D P (v) g y 2 (v) Dt y Step 5: Find the Velocity Profile Integrate the resulting momentum equation(s) from Step 4 Apply boundary conditions from Step 2 to solve for integration constants Step 6: Solve for Flow Rate or Wall Shear Stress flow rate: Q u ( y )dy wall shear stress: yx ( u v ) y x Example: A conveyor belt, which moves at constant velocity Uc, transports oil from an oil bath to a conduit above. The vacuum pump assists in the process by imposing a negative pressure gradient in the flow direction (i.e., dp/dx < 0). Assume the flow is fully developed in the x-direction. Find the velocity profile and flow rate. Vacuum pump Conveyor belt L Uc x y H g Oil bath (, )