swirling-flow.md

Swirling flow

Here we apply a time dependent velocity field {cite}Crouseilles15a The swirling flow distorts the initial distribution and reaching a maximum distortion at $t=T/2$. At that point the flow reverses and the distribution returns to its initial value.

$$ \partial_t f + \partial_x \left(\sin^2(\pi x) \sin(2\pi y)g(t) f\right) + \partial_y \left(-\sin^2(\pi y) \sin (2\pi x) g(t)f\right) = 0. $$

where $g(t) = cos(\pi t / T ) $.