Model of coarsening and vortex formation in vibrated granular rods

TL;DR

This paper presents a phenomenological model explaining the spontaneous formation of vortices and long-range order in vibrated granular rods, combining Cahn-Hilliard and Ginzburg-Landau equations to describe density and tilt dynamics.

Contribution

It introduces a coupled theoretical framework for coarsening and vortex formation in vibrated granular rods, linking experimental observations with mathematical modeling.

Findings

Spontaneous formation and growth of ordered islands due to coarsening.

Vortex solutions correspond to rods rotating around cores near islands.

The model aligns with experimental phenomena of vortex and order emergence.

Abstract

Neicu and Kudrolli observed experimentally spontaneous formation of the long-range orientational order and large-scale vortices in a system of vibrated macroscopic rods. We propose a phenomenological theory of this phenomenon, based on a coupled system of equations for local rods density and tilt. The density evolution is described by modified Cahn-Hilliard equation, while the tilt is described by the Ginzburg-Landau type equation. Our analysis shows that, in accordance to the Cahn-Hilliard dynamics, the islands of the ordered phase appear spontaneously and grow due to coarsening. The generic vortex solutions of the Ginzburg-Landau equation for the tilt correspond to the vortical motion of the rods around the cores which are located near the centers of the islands.