We study the inhomogeneous clustered regime of a freely cooling granular gas of rough particles in two dimensions using large-scale event driven simulations and scaling arguments. During collisions, rough particles dissipate energy in both the normal
and tangential directions of collision. In the inhomogeneous regime, translational kinetic energy and the rotational energy decay with time $t$ as power-laws $t^{-theta_T}$ and $t^{-theta_R}$. We numerically determine $theta_T approx 1$ and $theta_R approx 1.6$, independent of the coefficients of restitution. The inhomogeneous regime of the granular gas has been argued to be describable by the ballistic aggregation problem, where particles coalesce on contact. Using scaling arguments, we predict $theta_T=1$ and $theta_R=1$ for ballistic aggregation, $theta_R$ being different from that obtained for the rough granular gas. Simulations of ballistic aggregation with rotational degrees of freedom are consistent with these exponents.
The kinetic energy of a freely cooling granular gas decreases as a power law $t^{-theta}$ at large times $t$. Two theoretical conjectures exist for the exponent $theta$. One based on ballistic aggregation of compact spherical aggregates predicts $the
ta= 2d/(d+2)$ in $d$ dimensions. The other based on Burgers equation describing anisotropic, extended clusters predicts $theta=d/2$ when $2le d le 4$. We do extensive simulations in three dimensions to find that while $theta$ is as predicted by ballistic aggregation, the cluster statistics and velocity distribution differ from it. Thus, the freely cooling granular gas fits to neither the ballistic aggregation or a Burgers equation description.
We analyze a recent experiment [Phys. Rev. Lett., {bf103}, 224501 (2009)] in which the shock, created by the impact of a steel ball on a flowing monolayer of glass beads, is quantitatively studied. We argue that radial momentum is conserved in the pr
ocess, and hence show that in two dimensions the shock radius increases in time $t$ as a power law $t^{1/3}$. This is confirmed in event driven simulations of an inelastic hard sphere system. The experimental data are compared with the theoretical prediction, and is shown to compare well at intermediate times. At late times, the experimental data exhibit a crossover to a different scaling behavior. We attribute this to the problem becoming effectively three dimensional due to accumulation of particles at the shock front, and propose a simple hard sphere model which incorporates this effect. Simulations of this model capture the crossover seen in the experimental data.
