An axisymmetric incompressible Stokes flow has the Stokes stream function $\Psi(R, \theta)$ in spherical polar coordinates $(R, \theta, \phi)$. Give expressions for the components $u_{R}$ and $u_{\theta}$ of the flow field in terms of $\Psi$, and show that

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

where

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

Write down the equation satisfied by $\Psi$.

Verify that the Stokes stream function

$$\Psi(R, \theta)=\frac{1}{2} U \sin ^{2} \theta\left(R^{2}-\frac{3}{2} a R+\frac{1}{2} \frac{a^{3}}{R}\right)$$

represents the Stokes flow past a stationary sphere of radius $a$, when the fluid at large distance from the sphere moves at speed $U$ along the axis of symmetry.

A sphere of radius a moves vertically upwards in the $z$ direction at speed $U$ through fluid of density $\rho$ and dynamic viscosity $\mu$, towards a free surface at $z=0$. Its distance $d$ from the surface is much greater than $a$. Assuming that the surface remains flat, show that the conditions of zero vertical velocity and zero tangential stress at $z=0$ can be approximately met for large $d / a$ by combining the Stokes flow for the sphere with that of an image sphere of the same radius located symmetrically above the free surface. Hence determine the leading-order behaviour of the horizontal flow on the free surface as a function of $r$, the horizontal distance from the sphere's centre line.

What is the size of the next correction to your answer as a power of $a / d ?$ [Detailed calculation is not required.]

[Hint: For an axisymmetric vector field $\mathbf{u}$,

$$\nabla \times \mathbf{u}=\left(\frac{1}{R \sin \theta} \frac{\partial}{\partial \theta}\left(u_{\phi} \sin \theta\right),-\frac{1}{R} \frac{\partial}{\partial R}\left(R u_{\phi}\right), \frac{1}{R} \frac{\partial}{\partial R}\left(R u_{\theta}\right)-\frac{1}{R} \frac{\partial u_{R}}{\partial \theta}\right)$$