Linear planar strain and rotation streamlines ($\psi =\alpha xy-\frac{\beta }{2}(x^2+y^2)$)

The incompressible planar field $(\alpha x-\beta y,\beta x-\alpha y)$ has the displayed stream function under $(u,v)=({\psi }_y,-{\psi }_x)$ and vorticity $2\beta$. In rotated coordinates $s=(x+y)/\sqrt{2}$, $t=(x-y)/\sqrt{2}$, $\psi =((\alpha -\beta )s^2-(\alpha +\beta )t^2)/2$. For positive parameters its nonstationary streamlines are closed ellipses if $\alpha <\beta$, hyperbolas and straight separatrices if $\alpha >\beta$, and parallel straight lines with a stationary diagonal if $\alpha =\beta$.