Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 4 / 37A / c / Solution

Let the sphere have radius $a$; set $a=1$ for the unit sphere implicit in the question. Sinceone findsOn $r=a$, the no-slip boundary condition requires $\mathbf{u}=\mathbf{U}$ for every surface normal. The coefficient of
$(\mathbf{U}\cdot \mathbf{n})\mathbf{n}$ must vanish, while the coefficient of $\mathbf{U}$ must equal one:Solving,For a unit sphere,Substitution reproduces the standard translating sphere in Stokes flow.