The axisymmetric, irrotational flow generated by a solid sphere of radius $a$ translating at velocity $U$ in an inviscid, incompressible fluid is represented by a velocity potential $\phi(r, \theta)$. Assume the fluid is at rest far away from the sphere. Explain briefly why $\nabla^{2} \phi=0$.

By trying a solution of the form $\phi(r, \theta)=f(r) g(\theta)$, show that

$$\phi=-\frac{U a^{3} \cos \theta}{2 r^{2}}$$

and write down the fluid velocity.

Show that the total kinetic energy of the fluid is $k M U^{2} / 4$ where $M$ is the mass of the sphere and $k$ is the ratio of the density of the fluid to the density of the sphere.

A heavy sphere (i.e. $k<1$ ) is released from rest in an inviscid fluid. Determine its speed after it has fallen a distance $h$ in terms of $M, k, g$ and $h$.

Note, in spherical polars:

$$\begin{gathered} \boldsymbol{\nabla} \phi=\frac{\partial \phi}{\partial r} \mathbf{e}_{\mathbf{r}}+\frac{1}{r} \frac{\partial \phi}{\partial \theta} \mathbf{e}_{\theta} \\ \nabla^{2} \phi=\frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} \frac{\partial \phi}{\partial r}\right)+\frac{1}{r^{2} \sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial \phi}{\partial \theta}\right) \end{gathered}$$