(i) Assuming that axisymmetric incompressible flow $\mathbf{u}=\left(u_{R}, u_{\theta}, 0\right)$, with vorticity $(0,0, \omega)$ in spherical polar coordinates $(R, \theta, \phi)$ satisfies the equations

$$\mathbf{u}=\nabla \times\left(0,0, \frac{\Psi}{R \sin \theta}\right), \quad \omega=-\frac{1}{R \sin \theta} D^{2} \Psi$$

where

$$D^{2} \equiv \frac{\partial^{2}}{\partial R^{2}}+\frac{\sin \theta}{R^{2}} \frac{\partial}{\partial \theta}\left(\frac{1}{\sin \theta} \frac{\partial}{\partial \theta}\right)$$

show that for Stokes flow $\Psi$ satisfies the equation

$$D^{4} \Psi=0$$

(ii) A rigid sphere of radius $a$ moves at velocity $U \hat{\mathbf{z}}$ through viscous fluid of density $\rho$ and dynamic viscosity $\mu$ which is at rest at infinity. Assuming Stokes flow and by applying the boundary conditions at $R=a$ and as $R \rightarrow \infty$, verify that $\Psi=(A R+B / R) \sin ^{2} \theta$ is the appropriate solution to $(*)$ for this flow, where $A$ and $B$ are to be determined.

(iii) Hence find the velocity field outside the sphere. Without direct calculation, explain why the drag is in the $z$ direction and has magnitude proportional to $U$.

(iv) A second identical sphere is introduced into the flow, at a distance $b \gg a$ from the first, and moving at the same velocity. Justify the assertion that, when the two spheres are at the same height, or when one is vertically above the other, the drag on each sphere is the same. Calculate the leading correction to the drag in each case, to leading order in $a / b$.

[You may quote without proof the fact that, for an axisymmetric function $F(R, \theta)$,

$$\nabla \times(0,0, F)=\left(\frac{1}{R \sin \theta} \frac{\partial}{\partial \theta}(\sin \theta F),-\frac{1}{R} \frac{\partial}{\partial R}(R F), 0\right)$$

in spherical polar coordinates $(R, \theta, \phi)$.]