The geometry of planar linear flows

TL;DR

This paper classifies and analyzes a broad family of incompressible planar linear flows, extending known models to include out-of-plane extension and exploring their streamline geometries and classifications.

Contribution

It introduces a generalized classification of planar linear flows with out-of-plane extension, expanding the understanding of flow geometries beyond classical models.

Findings

Flows are classified into elliptic and hyperbolic types based on extension-to-vorticity ratio.

Streamline geometries can be closed or open, with a parameter space separating different flow regimes.

A surface of degenerate flows with parabolic streamlines generalizes simple shear flow.

Abstract

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.