Problem 5.95 - simplified as JB dy u d 2 2(4 General...

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Problem 5.95 [Difficulty: 3] Given: N-S equations and simplification assumptions Find: Fluid Velocity Solution: Governing Equations: 0 u (Continuity Equation) B J u u u 2 p - ρ (Momentum Equation) Assumptions: (1) Incompressible fluid (2) Two dimensional, fully developed flow driven by Lorentz force (2) Zero pressure gradient Write the 2D continuity and momentum equations in Cartesian coordinates: 0 y u x u (1) JB ) y u x u ( x p ) y u v x u ρ (u 2 2 2 2 (2)
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) y v x v ( y p ) y v v x v ρ (u 2 2 2 2 (3) Simplify the above equations: ) ( 0 0 y u u x u v Using the assumption of zero pressure gradient, equation (3) vanishes, and equation (2) can be
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Unformatted text preview: simplified as JB dy u d 2 2 (4) General solution for equation (4) is 2 1 2 2 1 C y C y JB u (5) Apply the no slip boundary conditions into equation (5), we get 2 1 2 2 1 2 2 4 2 1 ) 2 ( 2 4 2 1 ) 2 ( C h C h JB h u C h C h JB h u Therefore, C 1 =0 and 2 2 8 h JB C The fluid velocity is given as ) 4 ( 8 ) ( 2 2 y h JB y u...
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Problem 5.95 - simplified as JB dy u d 2 2(4 General...

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