Past exam of the mathematics course of the University of Cambridge / 2009 / ib / Paper 1 / 5D / Solution

The divergence of this axisymmetric velocity field is $r^{-1}{\partial }_r(-\alpha r^2)+{\partial }_z(2\alpha z)=-2\alpha +2\alpha =0$. Thus it satisfies incompressibility. The apparent axis singularity in the azimuthal component is removable, since $u_{\theta }\sim \gamma \beta r$ as $r\to 0$.

Only the axial vorticity component is nonzero:The cross product is $\mathbf{u}\times \mathbf{\omega }=(u_{\theta }w,\alpha rw,0)$. Its components are independent of $z$ and $\theta$, so its curl also has only an axial component:Meanwhile $w^{\prime }=-2\beta rw$ and $w^{\prime \prime }=(-2\beta +4{\beta }^2{r}^2)w$, givingThus the required steady vorticity equation holds with $\nu =\alpha /(2\beta )$. This is the balance between vortex stretching and viscous diffusion in a Burgers vortex.