HW_16_Soln

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 yy = = ∂∂ 4. gravity acts only in the – y -direction 0, xz y g ggg = == Consider the continuity equation. N 0#3 0 y x u u xy = += 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 22 0#4 0#1 ,#5 0#5 xxx x x x uuu u u p uu g txyx ρµ ρ = = = = = = ⎛⎞ ⎡⎤ ⎜⎟ ⎢⎥ ∂∂∂ ++ = + + + ∂∂∂∂ ⎣⎦ ⎝⎠ ±²³ 0 p x ∴= (2) liquid atmosphere belt U gravity h x y
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Page 2 of 2 Note that along the free surface of the liquid film the pressure remains constant (= p atm ). Hence, from Eqn. (2) the pressure everywhere in the film will be the same, i.e. p ( x , y ) = p atm (call this condition #6.). Now consider the Navier-Stokes equation in the y -direction. () N N N N N N 22 0#5 0#1 0#3 0#6 yy y y y xy y g uu u u u p g tx y y ρµ ρ = =− = == = ⎛⎞ ⎡⎤ ⎜⎟ ⎢⎥ ∂∂ ++= + + + ⎣⎦ ⎝⎠ Note that since u y is neither a function of y or z
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This note was uploaded on 03/09/2010 for the course ME 309 taught by Professor Merkle during the Spring '08 term at Purdue.

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

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