Streamline classification of a planar linear saddle or centre

For $(u,v)=(y,ax)$, the stream function $\psi =(y^2-a{x}^2)/2$ gives hyperbolic streamlines and a saddle when $a>0$, while $a<0$ gives elliptical streamlines and a centre.

The incompressible linear flow $(u,v)=(ϵx-\gamma y,\gamma x-ϵy)$ has streamfunction $\psi =ϵxy-\gamma (x^2+y^2)/2$. Its velocity matrix squares to $({ϵ}^2-{\gamma }^2)I$. Thus strain-dominated flow has saddle trajectories, rotation-dominated flow has closed elliptical trajectories, and equal nonzero strain and rotation gives simple shear flow along parallel lines with a stationary line. Pure rotation is counterclockwise for positive $\gamma$.