Fluid flows down a long cylindrical pipe of length b much larger than radius a, from a reservoir maintained at pressure P₀ (which connects to the pipe at x = 0) to a free end at large x, where the pressure is negligible. In this problem, we try to understand the velocity field v_x(ω̅ , x) as a function of radius ω̅ and distance x down the pipe, for a given discharge (i.e., mass flow per unit time) Ṁ . Assume that the Reynolds number is small enough for the flow to be treated as laminar all the way down the pipe.

(a) Close to the entrance of the pipe (small x), the boundary layer will be thin, and the velocity will be nearly independent of radius. What is the fluid velocity outside the boundary layer in terms of its density and Ṁ?

(b) How far must the fluid travel along the pipe before the vorticity diffuses into the center of the flow and the boundary layer becomes as thick as the radius? An order-of-magnitude calculation is adequate, and you may assume that the pipe is much longer than your estimate.

(c) At a sufficiently great distance down the pipe, the profile will cease evolving with x and settle down into the Poiseuille form derived in Sec. 13.7.6, with the discharge Ṁ given by the Poiseuille formula. Sketch how the velocity profile changes along the pipe, from the entrance to this final Poiseuille region.

(d) Outline a procedure for computing the discharge in a long pipe of arbitrary cross section.