2004-17.md

course	course_year	question_number	tags	title	year
Fluid Dynamics	IB	17	IB 2004 Fluid Dynamics	3.II.21C	2004

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 $h$.