Nonlinear transport of tracer particles immersed in a strongly sheared dilute gas with inelastic collisions

TL;DR

This paper addresses nonlinear tracer diffusion in a dilute granular gas under uniform shear flow by solving the inelastic Boltzmann equation via two complementary routes: inelastic Maxwell models (IMM) for exact collisional moments and the Vega Reyes et al. (VGS) kinetic model for inelastic hard spheres (IHS). A generalized Chapman–Enskog–like expansion about the shear-flow reference state yields tensor diffusion transport, revealing strong anisotropy and shear-induced cross effects in the diffusion tensors D_{kℓ}, D_{p,kℓ}, and D_{T,kℓ}. The results show that IMM provides highly accurate rheology and diffusion tensors in steady USF, with VGS giving qualitatively correct trends and explicit velocity distributions, both converging toward IHS benchmarks in many regimes. The study further applies the framework to thermal diffusion segregation, producing phase diagrams that elucidate how size, mass, and inelasticity govern segregation in sheared granular media, with practical implications for mixing and processing of granular mixtures.

Abstract

The diffusion of tracer particles immersed in a granular gas under uniform shear flow (USF) is analyzed within the framework of the inelastic Boltzmann equation. Two different but complementary approaches are followed to achieve exact results. First, we maintain the structure of the Boltzmann collision operator but consider inelastic Maxwell models (IMM). Using IMM allows us to compute the collisional moments of the Boltzmann operator without knowing the velocity distribution functions of the granular binary mixture explicitly. Second, we consider a kinetic model of the Boltzmann equation for inelastic hard spheres (IHS). This kinetic model is based on the equivalence between a gas of elastic hard spheres subjected to a drag force proportional to the particle velocity and a gas of IHS. We solve the Boltzmann--Lorentz kinetic equation for tracer particles using a generalized Chapman--Enskog--like expansion around the shear flow distribution. This reference distribution retains all hydrodynamic orders in the shear rate. The mass flux is obtained to first order in the deviations of the concentration, pressure, and temperature from their values in the reference state. Due to the velocity space anisotropy induced by the shear flow, the mass flux is expressed in terms of tensorial quantities rather than the conventional scalar diffusion coefficients. The exact results derived here are compared with those previously obtained for IHS by using different approximations [JSTAT P02012 (2007)]. The comparison generally shows reasonable quantitative agreement, especially for IMM results. Finally, we study segregation by thermal diffusion as an application of the theory. The phase diagrams illustrating segregation are shown and compared with IHS results, demonstrating qualitative agreement.

Figures (16)

• Figure 1: Plot of the ratio $R_{2,c_x}(c_x)=\varphi_{2,x}(c_x)/\varphi_{2,x}^{\text{el}}(c_x)$ obtained by means of the VGS kinetic model vs the (scaled) velocity $c_x$ for three different values of the coefficient of restitution $\alpha_{22}$: $\alpha_{22} = 0.9$, $0.7$, and $0.5$.
• Figure 2: Plot of the ratio $R_{1,c_x}(c_x)=\varphi_{1,x}(c_x)/\varphi_{1,x}^{\text{el}}(c_x)$ obtained by means of the VGS kinetic model vs the (scaled) velocity $c_x$ for $m_1/m_2 = \sigma_1/\sigma_2 = 0.5$, and three different values of the (common) coefficient of restitution $\alpha_{22} =\alpha_{12} = \alpha$: $\alpha = 0.9$, $0.7$, and $0.5$.
• Figure 3: Plot of the (reduced) elements of the pressure tensor $P_{2,k\ell}^*$ as functions of the coefficient of restitution $\alpha_{22}$ for a three-dimensional single gas. The solid lines are the approximate results derived for IHS from the leading Sonine approximation, the dashed lines correspond to the results obtained for IMM, and the dotted lines refer to the results of the VGS kinetic model. Symbols are the Monte Carlo simulations for IHS obtained in the paper MG02a.
• Figure 4: Plot of the (reduced) elements of the pressure tensor $P_{1,k\ell}^*$ as functions of the (common) coefficient of restitution $\alpha_{22}=\alpha_{12}\equiv \alpha$ for a three-dimensional single system ($d=3$) in the case $\sigma_1/\sigma_2=m_1/m_2=0.5$. The solid lines are the approximate results derived for IHS from the leading Sonine approximation, the dashed lines correspond to the results obtained for IMM, and the dotted lines refer to the results of the VGS kinetic model.
• Figure 5: Plot of the (reduced) elements of the self-diffusion tensor $D_{xx}^{\text{self}*}$, $D_{yy}^{\text{self}*}$, $D_{zz}^{\text{self}*}$, $D_{xy}^{\text{self}*}$, and $D_{yx}^{\text{self}*}$ as functions of the (common) coefficient of restitution $\alpha_{22}=\alpha_{12}=\alpha$ for a three-dimensional system in the case $\sigma_1/\sigma_2=m_1/m_2=1$. The solid lines are the approximate results derived for IHS from the leading Sonine approximation, the dashed lines correspond to the results obtained for IMM, and the dotted lines refer to the results of the VGS kinetic model. Note that $D_{p,yy}^{\text{self}*}=D_{p,zz}^{\text{self}*}$ in the results obtained from IMM and the kinetic model.