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.

Theory of thermostatted inhomogeneous granular fluids: A self-consistent density functional description

“…We consider a one dimensional model of granular fluid which is simple enough as to lend itself to analytic work, but is endowed with a sufficient complexity as to display inhomogeneous behavior 14,15,16,17,18,19,20,21,22,23,24 . One dimensional models may play a useful role since they can be employed to test approximations of more general applicability and allow us to link easily the structural properties to the dynamical behavior.…”

Untitled

“…In kinetic theory, granular fluids out of equilibrium were originally modeled by inelastic hard spheres (IHS) [2,3], which is the proto-typical model for dissipative short ranged hard core interactions. In general, similarity solutions are of interest, because they play an important role as asymptotic or limiting solutions of the Boltzmann equation at large times or at large velocities, and they frequently show overpopulated high energy tails when compared to the omni-present Gaussians.…”

Modeling and simulation of multilane traffic flow

“…These cause a smoothing of sudden density changes and prevent the formation of shock waves. This is the reason why density gradients and especially products of spatial derivatives are normally negligible which justifies the approximation made with equation (48). Apart from this, the diffusion terms are very helpful for efficient and stable numerical integration schemes.…”

“…In order to do this, we will apply a method that has been suggested by Sela and Goldhirsch [48]: First, we introduce the time averages…”