Hydrodynamics of a one-dimensional granular medium

Abstract

The question whether one-dimensional granular systems can be described by hydrodynamic equations is the main theme of the present work. Numerical simulations are used to create a database with which theory is compared. The system investigated in the numerical work is that of a one-dimensional collection of point particles colliding inelastically. The dependence of the dynamical properties on both the degree of inelasticity and the number of particles is investigated. A hydrodynamic theory which describes the large-scale motion of such systems has been developed. It is shown that the standard set of hydrodynamic fields (density, velocity, and granular temperature) is insufficient for this purpose and that an additional hydrodynamic field corresponding to the third moment of the fluctuating velocity field must be added to that set. The results of a linear stability analysis of the derived hydrodynamic equations are in a close agreement with those of the numerical simulations. The question of the effects of velocity correlations on the hydrodynamics is addressed as well. It is shown that these correlations, though not negligible, do not affect the hydrodynamic equations. The form of the single particle initial distribution function is shown to slightly affect the form of the hydrodynamic equations for transient times. Except for this minor effect the hydrodynamic equations possess a universal form. Possible implications for higher dimensional systems are mentioned.