For a two-dimensional flow in plane polar coordinates $(r, \theta)$, state the relationship between the streamfunction $\psi(r, \theta)$ and the flow components $u_{r}$ and $u_{\theta}$. Show that the vorticity $\omega$ is given by $\omega=-\nabla^{2} \psi$, and deduce that the streamfunction for a steady two-dimensional Stokes flow satisfies the biharmonic equation

$$\nabla^{4} \psi=0$$

A rigid stationary circular disk of radius $a$ occupies the region $r \leqslant a$. The flow far from the disk tends to a steady straining flow $\mathbf{u}_{\infty}=(-E x, E y)$, where $E$ is a constant. Inertial forces may be neglected. Calculate the streamfunction, $\psi_{\infty}(r, \theta)$, for the far-field flow.

By making an appropriate assumption about its dependence on $\theta$, find the streamfunction $\psi$ for the flow around the disk, and deduce the flow components, $u_{r}(r, \theta)$ and $u_{\theta}(r, \theta)$.

Calculate the tangential surface stress, $\sigma_{r \theta}$, acting on the boundary of the disk.

$[$ Hints: In plane polar coordinates $(r, \theta)$,

$$\begin{gathered} \boldsymbol{\nabla} \cdot \mathbf{u}=\frac{1}{r} \frac{\partial\left(r u_{r}\right)}{\partial r}+\frac{1}{r} \frac{\partial u_{\theta}}{\partial \theta}, \quad \omega=\frac{1}{r} \frac{\partial\left(r u_{\theta}\right)}{\partial r}-\frac{1}{r} \frac{\partial u_{r}}{\partial \theta} \\ \nabla^{2} V=\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial V}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} V}{\partial \theta^{2}}, \quad e_{r \theta}=\frac{1}{2}\left(r \frac{\partial}{\partial r}\left(\frac{u_{\theta}}{r}\right)+\frac{1}{r} \frac{\partial u_{r}}{\partial \theta}\right) \end{gathered}$$