Fluid Dynamics | Part IB, 2004

Use separation of variables to determine the irrotational, incompressible flow

u=Ua3r3(cosθer+12sinθeθ)\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 aa translating at velocity UU along the direction θ=0\theta=0 in spherical polar coordinates rr and θ\theta.

Show that the total kinetic energy of the fluid is

K=14MfU2,K=\frac{1}{4} M_{f} U^{2},

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

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

Typos? Please submit corrections to this page on GitHub.