Show that, in cylindrical polar co-ordinates, the streamfunction $\psi(r, \phi)$ for the velocity $\mathbf{u}=\left(u_{r}(r, \phi), u_{\phi}(r, \phi), 0\right)$ and vorticity $(0,0, \omega(r, \phi))$ of two-dimensional Stokes flow of incompressible fluid satisfies the equations

$$\mathbf{u}=\left(\frac{1}{r} \frac{\partial \psi}{\partial \phi},-\frac{\partial \psi}{\partial r}, 0\right), \quad \nabla^{2} \omega=-\nabla^{4} \psi=0$$

Show also that the pressure $p(r, \phi)$ satisfies $\nabla^{2} p=0$.

A stationary rigid circular cylinder of radius $a$ occupies the region $r \leqslant a$. The flow around the cylinder tends at large distances to a simple shear flow, with velocity given in cartesian coordinates $(x, y, z)$ by $\mathbf{u}=(\Gamma y, 0,0)$. Inertial forces may be neglected.

By solving the equation for $\psi$ in cylindrical polars, determine the flow field everywhere. Determine the torque on the cylinder per unit length in $z$.

[Hint: in cylindrical polars

$$\nabla^{2} V=\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial V}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} V}{\partial \phi^{2}}$$

The off-diagonal component of the rate-of-strain tensor is given by

$$\left.e_{r \phi}=\frac{1}{2}\left(\frac{1}{r} \frac{\partial u_{r}}{\partial \phi}+r \frac{\partial}{\partial r}\left(\frac{u_{\phi}}{r}\right)\right) .\right]$$