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

TL;DR

The paper develops Navier–Stokes hydrodynamics for a confined, multicomponent granular gas modeled by the Δ-model at moderate densities using the inelastic Enskog equation. It applies the Chapman–Enskog expansion to first order in spatial gradients, solving for the diffusion coefficients $D_{ij}$ and $D_i^T$ and the viscosities $\eta$ and $\eta_b$ via a leading Sonine polynomial approximation; the results are expressed in terms of restitution $\alpha_{ij}$, concentrations, masses, diameters, and density. An explicit application to binary mixtures yields a thermal diffusion factor $\Lambda$ that characterizes segregation driven by temperature gradients and gravity, with phase diagrams showing when larger particles migrate toward the cold or hot boundary. The work demonstrates that Δ-model predictions for transport at moderate densities are generally milder in their inelasticity effects than the conventional inelastic hard-spheres model and offers a tractable framework for comparing theory with simulations and experiments on confined granular mixtures.

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.

Figures (11)

• Figure 1: Plot of the (scaled) shear viscosity coefficient $\eta^*(\alpha)/\eta^*(1)$ as a function of the coefficient of restitution $\alpha$ for $d=2$ and two different values of the solid volume fraction $\phi$: $\phi=0$ (a) and $\phi=0.314$ (b). The solid lines correspond to the results obtained from Eq. \ref{['5.25']}. Symbols refer to MD simulations carried out in Ref. SRB14 for $\phi=0.314$.
• Figure 2: Plot of the (dimensionless) diffusion coefficients $D_{11}^*(\alpha)/D_{11}^*(1)$, $D_{12}^*(\alpha)/D_{12}^*(1)$, and $D_1^{*T}(\alpha)/D_1^{*T}(1)$ vs the (common) coefficient of restitution $\alpha_{ij}=\alpha$ for $d=2$, $x_1=0.5$, $\phi_i=0$, and three different binary mixtures: $\sigma_1/\sigma_2=2$, $m_1/m_2=2$ (a); $\sigma_1/\sigma_2=2$, $m_1/m_2=0.5$ (b); and $\sigma_1/\sigma_2=3$, $m_1/m_2=9$ (c). Here, $D_{11}^*(1)$, $D_{12}^*(1)$, and $D_1^{*T}(1)$ refer to the values of the diffusion coefficients for elastic collisions ($\alpha=1$).
• Figure 3: Plot of the (dimensionless) diffusion coefficient $D_{11}^*(\alpha)/D_{11}^*(1)$ vs the (common) coefficient of restitution $\alpha_{ij}=\alpha$ for $d=2$, $x_1=0.5$, and two different mixtures: $\sigma_1/\sigma_2=0.5$, $m_1/m_2=0.4$, and $\sigma_1/\sigma_2=2$, $m_1/m_2=2$. Three different values of the solid volume fraction $\phi$ have been considered: $\phi=0$ (a), $\phi=0.1$ (b), and $\phi=0.2$ (c).
• Figure 4: Plot of the (dimensionless) diffusion coefficient $D_{12}^*(\alpha)/D_{12}^*(1)$ vs the (common) coefficient of restitution $\alpha_{ij}=\alpha$ for $d=2$, $x_1=0.5$, and two different mixtures: $\sigma_1/\sigma_2=0.5$, $m_1/m_2=0.4$, and $\sigma_1/\sigma_2=2$, $m_1/m_2=2$. Three different values of the solid volume fraction $\phi$ have been considered: $\phi=0$ (a), $\phi=0.1$ (b), and $\phi=0.2$ (c).
• Figure 5: Plot of the (dimensionless) diffusion coefficient $D_{1}^{*T}(\alpha)/D_{1}^{*T}(1)$ vs the (common) coefficient of restitution $\alpha_{ij}=\alpha$ for $d=2$, $x_1=0.5$, and two different mixtures: $\sigma_1/\sigma_2=0.5$, $m_1/m_2=0.4$, and $\sigma_1/\sigma_2=2$, $m_1/m_2=2$. Three different values of the solid volume fraction $\phi$ have been considered: $\phi=0$ (solid line), $\phi=0.1$ (dashed line), and $\phi=0.2$ (dash-dotted line).