Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 38C (Fluid Dynamics)

Curling Navier-Stokes and using incompressibility gives ${\omega }_t+(u\cdot \nabla )\omega =(\omega \cdot \nabla )u+\nu {\nabla }^2\omega$. For pure swirl, $\omega =r^{-1}(rv{)}_r$, so regularity gives $v(r,t)=r^{-1}{\int }_0^{r}s\omega (s,t)ds$. Adding strain $(-\alpha r,v,2\alpha z)$ yieldsMultiplication by $2\pi r$ and integration proves $\dot{\Gamma }=0$. The decaying steady Burgers vortex is