Dynamics of inelastically colliding spheres with Coulomb friction: Relaxation of translational and rotational energy

Abstract

We investigate the free cooling of inelastic rough spheres in the presence of Coulomb friction. Depending on the coefficients of normal restitution $ε$ and Coulomb friction $μ$, we find qualitatively different asymptotic states. For nearly complete normal restitution ($ε$ close to 1) and large $μ$, friction does not change the cooling properties qualitatively compared to a constant coefficient of tangential restitution. In particular, the asymptotic state is characterized by a constant ratio of rotational and translational energies, both decaying according to Haff's law. However, for small $ε$ and small $μ$, the dissipation of rotational energy is suppressed, so that the asymptotic state is characterized by constant rotational energy while the translational energy continues to decay as predicted by Haff's law. Introducing either surface roughness for grazing collisions or cohesion forces for collisions with vanishing normal load, causes the rotational energy to decay according to Haff's law again in the asymptotic long-time limit with, however, an intermediate regime of approximately constant rotational energy.