Enskog Theory for Polydisperse Granular Mixtures. I. Navier-Stokes order Transport

TL;DR

This paper develops a hydrodynamic theory for polydisperse granular mixtures using the revised Enskog approach, deriving macroscopic equations and constitutive relations at Navier-Stokes order applicable to various densities and restitution coefficients.

Contribution

It provides the first derivation of macroscopic balance equations and constitutive relations for polydisperse granular mixtures based on the revised Enskog theory at Navier-Stokes order.

Findings

Derived macroscopic balance equations for mass, momentum, and energy.

Obtained exact integral-differential equations for distribution functions.

Established a framework for approximate solutions in the companion paper.

Abstract

A hydrodynamic description for an $s$-component mixture of inelastic, smooth hard disks (two dimensions) or spheres (three dimensions) is derived based on the revised Enskog theory for the single-particle velocity distribution functions. In this first portion of the two-part series, the macroscopic balance equations for mass, momentum, and energy are derived. Constitutive equations are calculated from exact expressions for the fluxes by a Chapman-Enskog expansion carried out to first order in spatial gradients, thereby resulting in a Navier-Stokes order theory. Within this context of small gradients, the theory is applicable to a wide range of restitution coefficients and densities. The resulting integral-differential equations for the zeroth- and first-order approximations of the distribution functions are given in exact form. An approximate solution to these equations is required for practical purposes in order to cast the constitutive quantities as algebraic functions of the macroscopic variables; this task is described in the companion paper.