Potential of a translating sphere ($\Phi =-U{a}^3\cos \theta /(2r^2)$)

For a sphere translating through an unbounded inviscid incompressible fluid at rest far away, the laboratory fluid velocity written in coordinates centred on the moving sphere is the gradient of the displayed dipole potential. Its radial velocity at the surface equals $U\cos \theta$. The sphere-rest-frame velocity instead needs the additional uniform potential $-Ur\cos \theta$.