DYNAMICS OF A FREELY EVOLVING, 2-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 forcin g the total energy of the system is monotonically decreasing so that t his' idealized granular medium is ''cooling down.'' There are three no ndimensional control parameters: the coefficient of restitution r: the solid fraction 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, nu, N) parameter space. Previous studies have shown that granular cooling results in the formation of structures: both th e mass and the momentum spontaneously become nonuniform. Four differen t regimes (kinetic, shearing, clustered, and collapsed) have been iden tified. Starting with the almost elastic case, in which r is just less than 1, the kinetic regime resembles a classical nondissipative gas i n which there are no structures. When r is decreased (with fixed N and 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 col lision is binary, the space and time scales decrease geometrically wit h the cumulative number of collisions so that a multiparticle interact ion occurs. The regime boundaries (i.e., the critical values of r) in the (N, nu) plane have been delineated using event-driven numerical si mulations. Analytic considerations show that the results of the simula tions can be condensed by supposing that the critical values of r depe nd only on N and nu through the optical depth, lambda = root N pi nu/2 where d is the disk diameter.