(i) Suppose that, with spherical polar coordinates, the Stokes streamfunction

$$\Psi_{\lambda}(r, \theta)=r^{\lambda} \sin ^{2} \theta \cos \theta$$

represents a Stokes flow and thus satisfies the equation $D^{2}\left(D^{2} \Psi_{\lambda}\right)=0$, where

$$D^{2}=\frac{\partial^{2}}{\partial r^{2}}+\frac{\sin \theta}{r^{2}} \frac{\partial}{\partial \theta} \frac{1}{\sin \theta} \frac{\partial}{\partial \theta} .$$

Show that the possible values of $\lambda$ are $5,3,0$ and $-2$. For which of these values is the corresponding flow irrotational? Sketch the streamlines of the flow for the case $\lambda=3$.

(ii) A spherical drop of liquid of viscosity $\mu_{1}$, radius $a$ and centre at $r=0$, is suspended in another liquid of viscosity $\mu_{2}$ which flows with streamfunction

$$\Psi \sim \Psi_{\infty}(r, \theta)=\alpha r^{3} \sin ^{2} \theta \cos \theta$$

far from the drop. The two liquids are of equal densities, surface tension is sufficiently strong to keep the drop spherical, and inertia is negligible. Show that

$$\Psi= \begin{cases}\left(A r^{5}+B r^{3}\right) \sin ^{2} \theta \cos \theta & (r<a), \\ \left(\alpha r^{3}+C+D / r^{2}\right) \sin ^{2} \theta \cos \theta & (r>a)\end{cases}$$

and obtain four equations determining the constants $A, B, C$ and $D$. (You need not solve these equations.)

[You may assume, with standard notation, that

$$\left.u_{r}=\frac{1}{r^{2} \sin \theta} \frac{\partial \Psi}{\partial \theta} \quad, \quad u_{\theta}=-\frac{1}{r \sin \theta} \frac{\partial \Psi}{\partial r} \quad, \quad e_{r \theta}=\frac{1}{2}\left\{r \frac{\partial}{\partial r}\left(\frac{u_{\theta}}{r}\right)+\frac{1}{r} \frac{\partial u_{r}}{\partial \theta}\right\} .\right]$$