= Solution
Use <cylindrical coordinates> $x=\rho\cos\phi$, $y=\rho\sin\phi$. The <velocity field> becomes
$$
\boxed{\mathbf v=\alpha z\rho\,\widehat{\boldsymbol\phi},\qquad
v_\rho=v_z=0.}
$$
For 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.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2015/paper-1-height-dependent-rotation.png]
{title=Circular flow at fixed heights reverses direction across the stationary middle plane}
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