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HW_21_Soln

# HW_21_Soln - ME 309 Fall 2008 Section 2(Merkle Homework 21...

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ME 309 Fall 2008 Section 2 (Merkle) Homework # 21 Due Mon 20 Oct 2008 Problem 21-1 . The text develops the velocity profile for the flow between parallel plates from an analysis of a control volume. Repeat this analysis by starting from the Navier- Stokes equations. Specifically, given that the flow is fully developed in the x direction, is planar with no variations and zero velocity in z , is incompressible without body forces and is at steady state, find the velocity profile between the plates. The continuity equation is given in Eq. 5.20 and the Navier-Stokes equations in Eq. 5.27. You may use only the x and y components of the Navier-Stokes and ignore the z component. Solution: Given: Navier-Stokes equations plus the continuity equation; Continuity (conservation of mass): 0 = + + z w y v x u Navier-Stokes x- direction + + + = + + + + 2 2 2 2 2 2 z u y u x u g x p z u w y u v x u u t u x μ ρ ρ ρ ρ ρ y -direction + + + = + + + + 2 2 2 2 2 2 z v y v x v g y p z v w y v v x v u t v y μ ρ ρ ρ ρ ρ Assumptions: incompressible, no body forces, two-dimensional, fully developed Analysis: Starting from the continuity equation, the first term is dropped because of the fully developed assumption; the third term is dropped because of the two-dimensional assumption. This leaves the result: 0 = y v , or, upon integrating, v = const. Evaluating this ‘velocity profile’ for the v component by evaluating the constant of integration at the wall where v = 0, we obtain: v = 0 Apply to the y -component of the momentum equation: + + + = + + + + 2 2 2 2 2 2 z v y v x v g y p z v w y v v x v u t v y μ ρ ρ ρ ρ ρ The first term is zero because of steady state condition The second term is zero because of fully developed flow and because v = 0 The third term is zero because v = 0 The fourth term is zero because v = 0 and because there is no variation in z The first term on the RHS is zero because there is no gravity. The viscous terms (in

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HW_21_Soln - ME 309 Fall 2008 Section 2(Merkle Homework 21...

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