Past exam of the advanced fluid dynamics course of the University of Oxford / 2023 / 2 / a / Solution

Expand the given translating-sphere Stokes flow into its Stokeslet and potential dipole parts:The second parenthesis is $\nabla [(\mathbf{U}\cdot \mathbf{x})/r^3]$ and has zero curl outside the sphere. For the first,Thus the vorticity of translating-sphere Stokes flow isThe result is independent of the irrotational finite-size dipole correction. It holds in the exterior $r>a$; the singular solid harmonics are used only outside their source.