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EML 5709 Homework 5 Spring 1997

1. To establish how Couette flow forms, solve the following unsteady problem. The fluid in the gap between two parallel infinite plates is initially at rest. The plates are normal to the y-axis. The gap width is h. From time t=0 onward, the top plate is given a constant velocity U in the positive x-direction. Find the unsteady flow using the separation of variables technique. Note, to obtain a Sturm-Liouville problem in the y-direction, the smart thing to do is to substract out the steady Couette flow solution. Solution.
2. For the flow of the previous question, plot a typical velocity profile between the plates at an early time, an intermediate time, a long time and infinite time. Solution.
3. For the profiles of the previous question, state what mathematical function(s) approximately describes each profile (linear, error function, sinusoidal, ...) Solution.
4. For the flow of the previous question, find a mathematical approximation for the velocity profile near the top plate for very small times. Solution.
5. Find the mass flow through a pipe of circular cross section in terms of the pressure gradient, density, viscosity, and radius. Solution.
6. For what Reynolds numbers is the expression for the mass flow derived in the previous question valid? Is it valid for pipes leading to faucets in your house? Solution.
7. In the gap between two concentric pipes is a viscous fluid. Assume that the inner pipe is at rest, but the outer rotates with a constant angular velocity . Calculate the torque needed to keep the outer cylinder rotating and the one to keep the inner cylinder at rest. Solution.