Transport properties in a model of confined granular mixtures at moderate densities

TL;DR

This paper derives hydrodynamic equations and transport coefficients for a confined granular mixture at moderate densities, accounting for inelastic collisions and analyzing segregation phenomena due to temperature gradients.

Contribution

It provides a theoretical framework for transport properties in inelastic granular mixtures using Enskog theory and Chapman--Enskog expansion, applicable to various particle parameters.

Findings

Transport coefficients depend on restitution, concentration, mass, diameter, and density.

Derived criteria for particle segregation based on temperature gradients and gravity.

Transport coefficients are expressed through coupled integral equations and approximate solutions.

Abstract

This work derives the Navier--Stokes hydrodynamic equations for a model of a confined, quasi-two-dimensional, $s$-component mixture of inelastic, smooth, hard spheres. Using the inelastic version of the revised Enskog theory, macroscopic balance equations for mass, momentum, and energy are obtained, and constitutive equations for the fluxes are determined through a first-order Chapman--Enskog expansion. As for elastic collisions, the transport coefficients are given in terms of the solutions of a set of coupled linear integral equations. Approximate solutions to these equations for diffusion transport coefficients and shear viscosity are achieved by assuming steady-state conditions and considering leading terms in a Sonine polynomial expansion. These transport coefficients are expressed in terms of the coefficients of restitution, concentration, the masses and diameters of the mixture's components, and the system's density. The results apply to moderate densities and are not limited to particular values of the coefficients of restitution, concentration, mass, and/or diameter ratios. As an application, the thermal diffusion factor is evaluated to analyze segregation driven by temperature gradients and gravity, providing criteria that distinguish whether larger particles accumulate near the hotter or colder boundaries.