HW_16_Soln - A wide flat belt moves vertically upward at...

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Page 1 of 2 A wide flat belt moves vertically upward at constant speed, U , through a large bath of viscous liquid as shown in the figure. The belt carries with it a layer of liquid of constant thickness, h . The motion is steady and fully-developed after a small distance above the liquid surface level. The external pressure is atmospheric (constant) everywhere. a. Simplify the governing equations to a form applicable for this particular problem. b. State the appropriate boundary conditions c. Determine the velocity profile in the liquid. d. Determine the volumetric flow rate per unit depth. SOLUTION: Make the following assumptions. 1. steady flow ( ) 0 t = " 2. planar flow ( ) 0 z u z = = " 3. fully-developed flow in the y -direction 0 y x u u y y = = 4. gravity acts only in the – y -direction 0, x z y g g g g = = = − Consider the continuity equation. ( ) N 0 #3 0 y x u u x y = + = 0 x u x = constant x u = (Note that u x does not vary with either y or z either.) Since there is no flow through the belt, 0 x u = (Call this condition #5.) (1) Consider the Navier-Stokes equation in the x -direction. ( ) N ( ) ( ) N ( ) N ( ) N ( ) N 2 2 2 2 0 #4 0 #1,#5 0 #5 0 #5 0 #3,#5 0 #3,#5 x x x x x x y x u u u u u p u u g t x y x x y ρ µ ρ = = = = = = + + = − + + + ±²³ 0 p x = (2) liquid atmosphere belt U gravity h x y
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