= Linear planar strain and rotation streamlines
{title2=$\psi=\alpha xy-\frac\beta2(x^2+y^2)$}
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 $s=(x+y)/\sqrt2$, $t=(x-y)/\sqrt2$, $\psi=((\alpha-\beta)s^2-(\alpha+\beta)t^2)/2$. 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 $\alpha=\beta$.
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