consider an initially motionless incompressible Newtonian fluid between two infinite solid boundaries, one at y = 0 and the other at y = d. Beginning at t = 0, the lower boundary oscillates back and forth in its own plane with a velocity ux = U sin t

a Identify characteristic velocity, length, and time scales, and nondimensionalize the governing DE and boundary conditions. You should note that there are two combinations of dimensional parameters that represent characteristic time scales, 1 and d2/v. What is the physical significance of each? The nature of the solution for the velocity field depends on the magnitude of a single dimensionless parameter. What is it? What is its significance?

b) Assume that the boundary motion has been going on for a long period of time so that all initial transients have decayed away (t d2/v) and the velocity field is strictly periodic. Solve for the velocity field in this case. Also, calculate the shear stress at the moving wall. Does fluid inertia increase or decrease the average shear stress? (c) Determine the limiting form of the solution for d2/v 1 to determine the first effect of inertia. Explain the result in physical terms. (d) Solve the start-up flow problem, assuming that the fluid is at rest at t = 0. Do not evaluate the coefficients, but set up the integral for their calculation.