Consider fully developed laminar flow in the annular space formed by the two concentric cylinders shown in the diagram for Problem 8.36, but with pressure gradient, \(\partial p / \partial x\), and the inner cylinder stationary. Let \(r_{0}=R\) and \(r_{i}=k R\). Show that the velocity profile is given by

\[u=-\frac{R^{2}}{4 \mu} \frac{\partial p}{\partial x}\left[1-\left(\frac{r}{R}\right)^{2}+\left(\frac{1-k^{2}}{\ln (1 / k)}\right) \ln \frac{r}{R}\right]\]

Show that the volume flow rate is given by

\[Q=-\frac{\pi R^{4}}{8 \mu} \frac{\partial p}{\partial x}\left[\left(1-k^{4}\right)-\frac{\left(1-k^{2}\right)^{2}}{\ln (1 / k)}\right]\]

Compare the volume flow rate for the limiting case, \(k \rightarrow 0\), with the corresponding expression for flow in a circular pipe.