Past exam of the fluids and waves course of the University of Oxford / 2015 / 1 / b / Solution

Substitute the given Stokes streamfunction representation into the divergence in spherical coordinates. The mixed derivatives cancel:Thus incompressibility is automatic for a smooth Stokes streamfunction. Since this is an axisymmetric flow with no azimuthal velocity, the only possibly nonzero component of its vorticity isDefine the differential operator in the Stokes streamfunction equation for irrotational flowThe stated equation is $\mathcal{L}\Psi =0$, and thereforeThe irrotational flow has a velocity potential locally, and globally on a regular simply connected domain. Regularity of the physical velocity field supplies the limiting interpretation on the polar axis, where the coordinate expressions contain $\sin \theta$ in the denominator.