Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 38A / b / Solution

The only vorticity component is $\omega =(1/r)(rv)\prime$. Its steady equation is $-\alpha r\omega \prime =2\alpha \omega +\nu (\omega \prime \prime +r^{-1}\omega \prime )$, solved by $\omega ={\omega }_0{e}^{-\alpha r^2/(2\nu )}$. Regularity at zero givesIts maximum satisfies $e^s=1+2s$, $s=\alpha r^{\ast 2}/(2\nu )$, so $r^{\ast }=O(\sqrt{\nu /\alpha })$.