Kinematics of flows on curved, deformable media

TL;DR

This paper studies the kinematics of geodesic flows on curved, deformable media, analyzing singularities and oscillatory behaviors on surfaces with constant and varying curvature, including a torus.

Contribution

It provides explicit solutions to Raychaudhuri equations for flows on curved media and explores the effects of curvature on singularity formation and flow behavior.

Findings

Existence of singular and non-singular solutions depending on initial conditions.

Curvature sign influences the flow dynamics and singularity development.

Oscillatory flow features depend on geometric parameters like the radii ratio of a torus.

Abstract

In this article, we first investigate the kinematics of specific geodesic flows on two dimensional media with constant curvature, by explicitly solving the evolution (Raychaudhuri) equations for the expansion, shear and rotation along the flows. We point out the existence of singular (within a finite value of the time parameter) and non-singular solutions and illustrate our results through a `phase' diagram. This diagram demonstrates under which initial conditions (or combinations thereof) we end up with a singularity in the congruence and when, if at all, we encounter non--singular solutions for the kinematic variables. Our analysis illustrates the differences which arise due to a positive or negative value of the curvature. Subsequently, we move on to geodesic flows on two dimensional spaces with varying curvature. As an example, we discuss flows on a torus, where interesting oscillatory features of the expansion, shear and rotation emerge, which are found to depend on the ratio of the radii of the torus. The singular (within a finite time)/non--singular nature of the solutions are also discussed. Finally, we arrive at some general statements and point out similarities or dissimilarities that arise in comparison to our earlier work on media in flat space.