7334346 - u=0=V xh u y 1 Consider incompressible viscous...

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u=V y h u=0 x 1. Consider incompressible viscous flow between two very long plates, separated by a distance h. The bottom plate is stationary, while the top plate moves at a velocity of V. The flow is subjected to a constant pressure gradient dp/dx. Show that reversed flow (u < 0) will occur near the lower flat plate, whenever the parameter {h 2 / (2μV)} dp/dx is greater than unity. Hint: Solve for u as a function of y, as done in the class for Couette flow. Determine when the velocity slope u/ y becomes negative at the lower plate. Solution: We have, Flow is incompressible, one-dimensional, viscous flow. Flow in x-direction, For a Couette flow: = , = v 0 w 0 i.e. no flow in y & z-dir Conservation of mass: ∂ ∂ +∂ ∂ +∂ ∂ = u x v y w z 0 ∂ ∂ = u x 0 ……….. [1] Conservation of momentum : (Navier-Stoke’s equation) For x-dir n : ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ =- ∂ ∂ + +∂ +∂ u t u u x v u y w u z 1ρ P x μρ 2u x2 2u y2 2u z2 Assuming, steady state flow i.e. ∂∂ = , t 0 & ∂ ∂ = , = , = , ∂∂ = u x 0 v 0 w
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