Inelastic collapse and clumping in a one-dimensional granular medium

Abstract

The dynamics of a one-dimensional gas of inelastic point particles is investigated. To model inelastic collisions, it is supposed that the relative velocity of two colliding particles is reduced by a factor r, where 0<r<1. The constant r is the coefficient of restitution. Because the collisions are inelastic, particles can collide infinitely often in finite time so that the relative separations and velocities of adjacent particles on the line become zero. The minimal example of this ‘‘inelastic collapse’’ requires r<7−4 7/8 ≊0.0718. With this restriction, three particles condense into a single lump in a finite time: The particle in the middle is sandwiched between the monotonically converging outer particles. When r is greater than 7−4 7/8, more than three particles are needed to trigger inelastic collapse and it is shown that r is close to 1 the minimum number scales as −ln(1−r)/(1−r). The simplest statistical problem is the ‘‘cooling law’’ of a uniformly excited gas confined between inelastic boundaries. A scaling argument suggests that the mean square velocity (the ‘‘granular temperature’’) of the particles decreases like t−2. Numerical simulations show that this scaling is correct only if the total number of particles in the domain is less than the number required to trigger collapse (e.g., roughly 88 if r=0.95). When the number of particles is much greater than this minimum, and before the first collapse, clusters form throughout the medium. Thus a state with uniform particle density is unstable to the formation of aggregates and inelastic collapse is the finite-amplitude expression of this instability.