Given that the circulation round every closed material curve in an inviscid, incompressible fluid remains constant in time, show that the velocity field of such a fluid started from rest can be written as the gradient of a potential, $\phi$, that satisfies Laplace's equation.

A rigid sphere of radius a moves in a straight line at speed $U$ in a fluid that is at rest at infinity. Using axisymmetric spherical polar coordinates $(r, \theta)$, with $\theta=0$ in the direction of motion, write down the boundary conditions on $\phi$ and, by looking for a solution of the form $\phi=f(r) \cos \theta$, show that the velocity potential is given by

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

Calculate the kinetic energy of the fluid.

A rigid sphere of radius $a$ and uniform density $\rho_{b}$ is submerged in an infinite fluid of density $\rho$, under the action of gravity. Show that, when the sphere is released from rest, its initial upwards acceleration is

$$\frac{2\left(\rho-\rho_{b}\right) g}{\rho+2 \rho_{b}}$$

[Laplace's equation for an axisymmetric scalar field in spherical polars is:

$$\left.\frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} \frac{\partial \phi}{\partial r}\right)+\frac{1}{r^{2} \sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial \phi}{\partial \theta}\right)=0 .\right]$$