3.II.21C

Use separation of variables to determine the irrotational, incompressible flow

$\mathbf{u}=U \frac{a^{3}}{r^{3}}\left(\cos \theta \mathbf{e}_{r}+\frac{1}{2} \sin \theta \mathbf{e}_{\theta}\right)$

around a solid sphere of radius $a$ translating at velocity $U$ along the direction $\theta=0$ in spherical polar coordinates $r$ and $\theta$.

Show that the total kinetic energy of the fluid is

$K=\frac{1}{4} M_{f} U^{2},$

where $M_{f}$ is the mass of fluid displaced by the sphere.

A heavy sphere of mass $M$ is released from rest in an inviscid fluid. Determine its speed after it has fallen through a distance $h$ in terms of $M, M_{f}, g$ and $h$.

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