3. ( 25 pt) Flow between infinite parallel plates is shown below. The top plate has a velocity of VT​ to the right, and

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3 25 Pt Flow Between Infinite Parallel Plates Is Shown Below The Top Plate Has A Velocity Of Vt To The Right And 1 (55.16 KiB) Viewed 45 times

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3. ( 25 pt) Flow between infinite parallel plates is shown below. The top plate has a velocity of VT​ to the right, and the bottom plate a velocity of VB​ to the left. The flow is 2-d and incompressible. Assume that the flow is 1) fully developed, 2) steady, and 3) no pressure gradient in the x direction. The 2-D incompressible Navier-Stokes equations and conservation of mass are given by ∂x∂u​+∂y∂v​=0ρ(∂t∂u​+u∂x∂u​+v∂y∂u​)=−∂x∂P​+μ(∂x2∂2u​+∂y2∂2u​)+ρgx​ρ(∂t∂v​+u∂x∂v​+v∂y∂v​)=−∂y∂P​+μ(∂x2∂2v​+∂y2∂2v​)+ρgy​​ (a) (5pt) Show that v≡0 (b) (5 pt) Reduce the governing equations to the form specific to this problem. State the reasons for neglecting each term using the numbered assumptions above. (c) (5pt) State the boundary conditions (location and value). (d) (5pt ) Solve for the velocity profile in terms of the given variables. (e) (5 pt) Derive an expression for the value of y where the velocity is zero.