Preprint Article Version 1 This version is not peer-reviewed

# Kinematics of a Fluid Ellipse in a Linear Flow

Version 1 : Received: 5 February 2018 / Approved: 6 February 2018 / Online: 6 February 2018 (06:42:11 CET)

A peer-reviewed article of this Preprint also exists.

Lilly, J.M. Kinematics of a Fluid Ellipse in a Linear Flow. Fluids 2018, 3, 16. Lilly, J.M. Kinematics of a Fluid Ellipse in a Linear Flow. Fluids 2018, 3, 16.

Journal reference: Fluids 2018, 3, 16
DOI: 10.3390/fluids3010016

## 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 relationship, called the geometric 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.

## Subject Areas

elliptical vortex; linear flow; Kida vortex; Stokes' theorem; Ball's theorem; moment of inertia; matrix basis