Dynamics of a freely evolving, two-dimensional granular medium

We consider the dynamics of an ensemble of identical, inelastic, hard disks in a doubly periodic domain. Because there is no external forcing the total energy of the system is monotonically decreasing so that this idealized granular medium is ``cooling down.'' There are three nondimensional control parameters: the coefficient of restitution r, the solid fraction \ensuremath{\nu}, and the total number of disks in the domain N. Our goal is a comprehensive description of the phenomenology of granular cooling in the (r,\ensuremath{\nu},N) parameter space. Previous studies have shown that granular cooling results in the formation of structures: both the mass and the momentum spontaneously become nonuniform. Four different regimes (kinetic, shearing, clustered, and collapsed) have been identified. Starting with the almost elastic case, in which r is just less than 1, the kinetic regime resembles a classical nondissipative gas in which there are no structures. When r is decreased (with fixed N and \ensuremath{\nu}) the system evolves into the shearing regime in which most of the energy and momentum resides in the gravest hydrodynamic shear mode. At still smaller values of r the clustered regime appears as an extended transient. Large clusters of disks form, collide, breakup, and reform. From the clustered state the gas eventually either evolves into the shearing regime or, alternatively, collapses. The collapsed regime is characterized by a dynamical singularity in which a group of particles collides infinitely often in a finite time. While each individual collision is binary, the space and time scales decrease geometrically with the cumulative number of collisions so that a multiparticle interaction occurs. The regime boundaries (i.e., the critical values of r) in the (N,\ensuremath{\nu}) plane have been delineated using event-driven numerical simulations. Analytic considerations show that the results of the simulations can be condensed by supposing that the critical values of r depend only on N and \ensuremath{\nu} through the optical depth, \ensuremath{\lambda}\ensuremath{\equiv}d\ensuremath{\surd}N\ensuremath{\pi}\ensuremath{\nu}/2 where d is the disk diameter. \textcopyright{} 1996 The American Physical Society.