Consider the flow in cylindrical coordinates with no dependence on \(\theta\). How is the streamfunction defined? What is the non-vanishing component of vorticity. Show that the \(E^{2}\) operator takes the form

\[\begin{equation*}E^{2}=\frac{\partial^{2}}{\partial r^{2}}-\frac{1}{r} \frac{\partial}{\partial r}+\frac{\partial^{2}}{\partial z^{2}} \tag{15.82}\end{equation*}\]

and \(E^{4} \psi=0\) for Stokes flow.