Extension of Haff's cooling law in granular flows

TL;DR

This paper extends Haff's law for energy decay in granular flows to large times and dimensions, supported by simulations, applicable for moderate inelasticities.

Contribution

It generalizes the known energy decay law to later times and higher dimensions, including nonlinear clustering effects.

Findings

Energy decays as t^{-2} initially, then as τ^{-d/2} at large times.

The extended law is confirmed by computer simulations.

Applicable for inelasticities with 0.6<α<1.

Abstract

The total energy E(t) in a fluid of inelastic particles is dissipated through inelastic collisions. When such systems are prepared in a homogeneous initial state and evolve undriven, E(t) decays initially as t^{-2} \aprox exp[ - 2\epsilon \tau] (known as Haff's law), where \tau is the average number of collisions suffered by a particle within time t, and \epsilon=1-\alpha^2 measures the degree of inelasticity, with \alpha the coefficient of normal restitution. This decay law is extended for large times to E(t) \aprox \tau^{-d/2} in d-dimensions, far into the nonlinear clustering regime. The theoretical predictions are quantitatively confirmed by computer simulations, and holds for small to moderate inelasticities with 0.6< \alpha< 1.