Past exam of the mathematics course of the University of Cambridge / 2004 / ib / Paper 3 / 21C (Fluid Dynamics)

Let the fluid be at rest at infinity, and write its velocity potential as $\varphi =f(r)\cos \theta$, the angular dependence selected by translation along the polar axis. The separated Laplace equation iswith radial solutions $r$ and $r^{-2}$. Decay at infinity eliminates the former. The no-penetration condition on the moving sphere is $u_r(a)=U\cos \theta$, giving $f(r)=-U{a}^3/(2r^2)$. Thus the potential flow around a translating sphere isFor density $\rho$, integrate its kinetic energy over the exterior. The angular factor satisfies $\int ({\cos }^2\theta +{\sin }^2\theta /4)d\Omega =2\pi$, and ${\int }_a^{\infty }{r}^{-4}dr=1/(3a^3)$. HenceThe added mass of a sphere is consequently $M_f/2$. For a heavy sphere, $M>M_f$, falling through distance $h$, the net gravitational work after buoyancy is $(M-M_f)gh$. Its own kinetic energy plus that of the fluid is $(M+M_f/2)U^2/2$. Conservation of energy from rest therefore givesThis is the unbounded inviscid-fluid model; walls or viscous dissipation would alter the flow and its energy.