Navier--Stokes transport coefficients for a model of a confined quasi-two-dimensional granular binary mixture

TL;DR

This paper derives Navier--Stokes transport coefficients for a confined quasi-two-dimensional granular binary mixture using the Boltzmann equation and Chapman--Enskog method, accounting for inelastic collisions and nonzero partial temperature contributions.

Contribution

It provides explicit forms of transport coefficients for inelastic granular mixtures considering arbitrary inelasticity, mass, size ratios, and confinement effects, extending previous low-density results.

Findings

Transport coefficients expressed in terms of restitution, concentration, mass, and size ratios.

Explicit forms for diffusion, shear viscosity, and temperature-related quantities.

Quantification of Onsager relation violations in granular mixtures.

Abstract

The Navier--Stokes transport coefficients for a model of a confined quasi-two-dimensional granular binary mixture of inelastic hard spheres are determined from the Boltzmann kinetic equation. A normal or hydrodynamic solution to the Boltzmann equation is obtained via the Chapman--Enskog method for states near the local version of the homogeneous time-dependent state. The mass, momentum, and heat fluxes are determined to first order in the spatial gradients of the hydrodynamic fields, and the associated transport coefficients are identified. As expected, they are given in terms of the solutions of a set of coupled linear integral equations. In addition, in contrast to previous results obtained for low-density granular mixtures, there are also nonzero contributions to the first-order approximations to the partial temperatures $T_i^{(1)}$ and the cooling rate $\zeta^{(1)}$. Explicit forms for the diffusion transport coefficients, the shear viscosity coefficient, and the quantities $T_i^{(1)}$ and $\zeta^{(1)}$ are obtained by assuming the steady-state conditions and by considering the leading terms in a Sonine polynomial expansion. The above transport coefficients are given in terms of the coefficients of restitution, concentration, and the masses and diameters of the components of the mixture. The results apply in principle for arbitrary degree of inelasticity and are not limited to specific values of concentration, mass and/or size ratios. As a simple application of these results, the violation of the Onsager reciprocal relations for a confined granular mixture is quantified in terms of the parameter space of the problem.