Kinematics of a Fluid Ellipse in a Linear Flow

TL;DR

This paper introduces a four-parameter kinematic model for fluid parcels in a time-varying ellipse within linear flows, establishing a unique relationship between ellipse parameters and flow properties, and deriving physical properties of elliptical fluid rings.

Contribution

It presents the ellipse/flow equivalence, a novel result linking ellipse evolution to the velocity gradient, and extends Stokes' theorem to matrix form involving the velocity gradient tensor.

Findings

Derived explicit formulas for ellipse parameter evolution.

Established the ellipse/flow equivalence principle.

Analyzed the kinetic energy components of elliptical fluid rings.

Abstract

A four-parameter kinematic model for the position of a fluid parcel in a time-varying ellipse is introduced. For any ellipse advected by an arbitrary linear two-dimensional flow, the rates of change of the ellipse parameters are uniquely determined by the four parameters of the velocity gradient matrix, and vice versa. This result, termed ellipse/flow equivalence, provides a stronger version of the well-known result that a linear velocity field maps an ellipse into another ellipse. Moreover, ellipse/flow equivalence is shown to be a manifestation of Stokes' theorem. This is done by deriving a matrix-valued extension of the classical Stokes' theorem that involves a spatial integral over the velocity gradient tensor, thus accounting for the two strain terms in addition to the divergence and vorticity. General expressions for various physical properties of an elliptical ring of fluid are also derived. The ellipse kinetic energy is found to be composed of three portions, associated respectively with the circulation, the rate of change of the moment of inertia, and the variance of parcel angular velocity around the ellipse. A particular innovation is the use of four matrices, termed the IJKL basis, that greatly facilitate the required calculations.