Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 4 / 37B / i

The steady Stokes equations are $-\nabla p+\mu {\nabla }^2\mathbf{u}=0$, $\nabla \cdot \mathbf{u}=0$. Taking the curl gives ${\nabla }^2\mathbf{\omega }=0$. For an axisymmetric azimuthal component $\omega {\mathbf{e}}_{\varphi }$, the vector Laplacian's azimuthal component is $({\nabla }^2-1/(R^2{\sin }^2\theta ))\omega$. Expanding the derivatives showsWith $\omega =-D^2\Psi /(R\sin \theta )$, the vorticity equation consequently gives $D^4\Psi =0$. This derives the streamfunction equation from momentum rather than treating the scalar vorticity component as an ordinary scalar harmonic function.