Flows in the disk-and-tube system (Fig. 4D.3)⁹ 

(a) A fluid in a circular tube is caused to move tangentially by a tightly fitting rotating disk at the liquid surface at z = 0; the bottom of the tube is located at z = L. Find the steady-state velocity distribution v_θ(r, z), when the angular velocity of the disk is fl. Assume that creeping flow prevails throughout, so that there is no secondary flow. Find the limit of the solution as L → ∞. 

(b) Repeat the problem for the unsteady flow. The fluid is at rest before t = 0, and the disk suddenly begins to rotate with an angular velocity Ω at t = 0. Find the velocity distribution v_θ(r, z, t) for a column of fluid of height L. Then find the solution for the limit as L → ∞. 

(c) If the disk is oscillating sinusoidally in the tangential direction with amplitude Ω₀, obtain the velocity distribution in the tube when the "oscillatory steady state" has been attained. Repeat the problem for a tube of infinite length.