In a pipe of length L and of inner radius R, a pressure difference of p is driving the flow of a liquid of density

In a pipe of length L and of inner radius R, a pressure difference of Δp is driving the flow of a liquid of density ρ and viscosity μ. With the help of some magical technology, the walls of the pipe are made to move with velocity V in the direction opposite to the liquid motion. Assuming laminar flow and ignoring gravity, derive the expressions for:

(a) Axial velocity u as a function of radius r.

(b) Total flow rate Q.