Problem 2 (14 pts) Air with density p and viscosity flows through the gap between two...
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Problem 2 (14 pts) Air with density p and viscosity flows through the gap between two infinitely large parallel plates separated by a distance H. The bottom plate is fixed, and the top plate moves at a constant speed U along -x-direction. A manometer with two different liquids of densities p1 and p2 is connected to two static pressure taps at the bottom plate separated by a horizontal distance of L. The height differences of the liquid surfaces in the manometer are h and h2 as shown in the figure. Assume the air flow is viscous, steady, and fully developed, and assume p P1 and p < p2. The origin of the y coordinate is set to be at the bottom plate. Given: p, p, H, U, L, p1, p2 h1, h2, g. Questions: (1) (6 pts) Based on the reading from the manometer, determine the streamwise pressure gradient in the air flow, x 0 (2) (6 pts) The air flow is governed by the equation - +2 = 0. Solve this equation with proper boundary conditions to derive the solution for the velocity profile u(y). (3) (2 pts) Based on your solution of u(y) from (2), calculate the flow shear stress Tyr at the middle height y = H/2. U H L 8 Y X P1 h2 P2 h 9 Problem 2 (14 pts) Air with density p and viscosity flows through the gap between two infinitely large parallel plates separated by a distance H. The bottom plate is fixed, and the top plate moves at a constant speed U along -x-direction. A manometer with two different liquids of densities p1 and p2 is connected to two static pressure taps at the bottom plate separated by a horizontal distance of L. The height differences of the liquid surfaces in the manometer are h and h2 as shown in the figure. Assume the air flow is viscous, steady, and fully developed, and assume p P1 and p < p2. The origin of the y coordinate is set to be at the bottom plate. Given: p, p, H, U, L, p1, p2 h1, h2, g. Questions: (1) (6 pts) Based on the reading from the manometer, determine the streamwise pressure gradient in the air flow, x 0 (2) (6 pts) The air flow is governed by the equation - +2 = 0. Solve this equation with proper boundary conditions to derive the solution for the velocity profile u(y). (3) (2 pts) Based on your solution of u(y) from (2), calculate the flow shear stress Tyr at the middle height y = H/2. U H L 8 Y X P1 h2 P2 h 9
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