Kinematics of deformable media

TL;DR

This paper analytically studies the deformation kinematics in 2D and 3D media by solving evolution equations, revealing how initial conditions influence singularity formation and deformation behavior.

Contribution

It provides explicit analytical solutions to the Raychaudhuri equations for deformation kinematics, enabling detailed analysis of initial condition effects and singularity development.

Findings

Identification of initial conditions leading to finite-time singularities

Analysis of deformation behavior in different media dimensions

Discussion of potential applications to fluid and spacetime flows

Abstract

We investigate the kinematics of deformations in two and three dimensional media by explicitly solving (analytically) the evolution equations (Raychaudhuri equations) for the expansion, shear and rotation associated with the deformations. The analytical solutions allow us to study the dependence of the kinematical quantities on initial conditions. In particular, we are able to identify regions of the space of initial conditions that lead to a singularity in finite time. Some generic features of the deformations are also discussed in detail. We conclude by indicating the feasibility and utility of a similar exercise for fluid and geodesic flows in flat and curved spacetimes.