Dynamical approach to weakly dissipative granular collisions

TL;DR

This paper introduces a dynamical model for weakly dissipative granular collisions using viscous terms, providing analytical solutions for collision outcomes that improve upon traditional phenomenological models.

Contribution

It develops a velocity-dependent dissipation model for granular collisions and derives analytical expressions for collision time and restitution coefficient.

Findings

Analytical expressions for collision time and final velocities.

Velocity-dependent coefficient of restitution.

Good agreement with numerical simulations.

Abstract

Granular systems present surprisingly complicated dynamics. In particular, nonlinear interactions and energy dissipation play important roles in these dynamics. Usually, constant coefficients of restitution are introduced phenomenologically to account for energy dissipation when grains collide. The collisions are assumed to be instantaneous and to conserve momentum. Here, we improve on this phenomenology by introducing the dissipation through a viscous (velocity dependent) term in the equations of motion for two colliding grains. Using a first order approximation, we solve the equations of motion in the low viscosity regime. This approach allows us to calculate the collision time, the final velocity of each grain, and a coefficient of restitution that depends on the relative velocity of the grains. We compare our analytic results with those obtained by numerical integration of the equations of motion.