ECH 141—Fluid Mechanics Winter 2017 Exam 2 2/1/2017 (One hour, Closed book, no notes) 1. In a coating process, a sheet of paper is pulled through a narrow slot filled with a coating fluid. the entrance to the slot (x=0) the dynamic pressure is P1, at the exit (x=L) the coating is exposed to air at and is at dynamic pressure P0, and the distance between the paper surfaces and the coating surfaces is H, as shown in the diagram below. The paper moves horizontally in the x-direction with velocity U, and the pressures at the entrance and exit are not equal. It may be assumed that L >> H. a) (20 pts.) Derive an expression for the velocity profile vX(y) in the region 0 ≤y ≤H. Assume that the pressures P1and P0are known. b) (10 pts.) Derive an expression for the flow rate Q of coating fluid through the slot, per unit width in the z-direction. c) (10 pts.) Derive an expression for the force FX, per unit width in the z-direction, needed to pull the paper sheet through the slot at velocity U. At Solutiona) Assume unidirectional, fully-developed, steady flow, then vx=vxy( )and vy=vz=0. The y-component of the Navier-Stokes equations shows that ∂P∂y=0 or P=P x( )onlyso that the x-component simplifies to μd2vxdy2=dPdx=−P1−P0()L=−ΔPL. Integrating twice yields vxy( )=−12μΔPLy2+C1y+C2.