Swirling flow
In this problem we use a time-dependent velocity field
\[\begin{aligned} u_x(x,y,t) &= \sin^2(\pi x) \sin(2 \pi y) g(t) \\ u_y(x,y,t) &= -\sin^2(\pi y) \sin(2 \pi x) g(t) \end{aligned}\]
This represents a swirling flow that distorts the vorticity field, reaching a maximum distortion at :math:t=T/2. At that point the flow reverses and the vorticity profile returns to its initial value.