The geometry of planar linear flows

We identify incompressible planar linear flows that are generalizations of the well known one-parameter family characterized by the ratio of in-plane extension to (out-of-plane) vorticity. The latter `canonical' family is classified into elliptic and hyperbolic linear flows with closed and open streamlines, respectively, corresponding to the extension-to-vorticity ratio being less or greater than unity; unity being the marginal case of simple shear flow. The novel flows possess an out-of-plane extension, but the streamlines may nevertheless be closed or open, allowing for an organization, in a three-dimensional parameter space, into regions of `eccentric' elliptic and hyperbolic flows, separated by a surface of degenerate linear flows with parabolic streamlines that are generalizations of simple shear. We discuss implications for various fluid mechanical scenarios.