Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2015 / Paper 1 / 20 / a / Solution

Use cylindrical coordinates $x=\rho \cos \varphi$, $y=\rho \sin \varphi$. The velocity field becomesFor each fixed height, particles follow concentric circular streamlines around the vertical axis. At $z>0$ the motion is anticlockwise viewed from above, while at $z<0$ it is clockwise. Within a horizontal plane the angular velocity is $\alpha z$, independent of radius, and the speed is $\alpha |z|\rho$. Both the plane $z=0$ and the vertical axis are stationary. The rotation rate varies with height, so the entire liquid is not one rigidly rotating body.