Extension of Haff's cooling law in granular flows

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) ~ exp[−2ετ ] (known as Haff's law), where “τ” is the average number of collisions suffered by a particle within time t, and ε = 1−α2 measures the degree of inelasticity, with α the coefficient of normal restitution. This decay law is extended for large times to E(t) ~ τ^(−d/2) in d dimensions, far into the nonlinear clustering regime. The theoretical predictions are quantitatively confirmed by computer simulations, and hold for small to moderate inelasticities with 0.6 < α < 1.