Diffusion of intruders in a granular gas thermostatted by a bath of elastic hard spheres

TL;DR

This work develops a Boltzmann-kinetic framework for tracer diffusion of intruders in a dilute granular gas immersed in a molecular gas bath at fixed temperature $T_g$. By applying the Chapman–Enskog method to first order in spatial gradients and using a leading Sonine approximation, it derives four diffusion coefficients $D_T$, $D$, $D_0$, and $D_0^U$ that depend on mass/diameter ratios, restitution coefficients, and bath parameters; in steady-state, these reduce to explicit forms and agree with Brownian-limit Langevin models. The study also analyzes homogeneous steady states, determining the temperature ratios $\chi=T/T_g$ and $\chi_0=T_0/T_g$, with small kurtoses $c$ and $c_0$ supporting Maxwellian approximations. As a key application, it derives a segregation criterion via the thermal diffusion factor $\Lambda$, revealing how gravity, temperature gradients, and finite mass/size ratios drive Brazil-nut vs reverse Brazil-nut effects, including novel mass-ratio dependent behavior not captured by coarse-grained Langevin models. The results provide a general framework for predicting diffusion and segregation in granular suspensions with interstitial gases, with potential experimental realizations using low-Reynolds-number fluids.

Abstract

The Boltzmann kinetic equation is considered to compute the transport coefficients associated with the mass flux of intruders in a granular gas. Intruders and granular gas are immersed in a gas of elastic hard spheres (molecular gas). We assume that the granular particles are sufficiently rarefied so that the state of the molecular gas is not affected by the presence of the granular gas. Thus, the gas of elastic hard spheres can be considered as a thermostat (or bath) at a fixed temperature $T_g$. In the absence of spatial gradients, the system achieves a steady state where the temperature of the granular gas $T$ differs from that of the intruders $T_0$ (energy nonequipartition). Approximate theoretical predictions for the temperature ratio $T_0/T_g$ and the kurtosis $c_0$ associated with the intruders compare very well with Monte Carlo simulations for conditions of practical interest. For states close to the steady homogeneous state, the Boltzmann equation for the intruders is solved by means of the Chapman--Enskog method to first order in the spatial gradients. As expected, the diffusion transport coefficients are given in terms of the solutions of a set of coupled linear integral equations which are approximately solved by considering the first-Sonine approximation. In dimensionless form, the transport coefficients are nonlinear functions of the mass and diameter ratios, the coefficients of restitution, and the (reduced) bath temperature. Interestingly, previous results derived from a suspension model based on an effective fluid-solid interaction force are recovered when $m/m_g\to \infty$ and $m_0/m_g\to \infty$, where $m$, $m_0$, and $m_g$ are the masses of the granular, intruders, and molecular gas particle, respectively. Finally, as an application of our results, thermal diffusion segregation is exhaustively analysed.

Figures (11)

• Figure 1: Plot of the kurtosis $c_0$ associated with the distribution function of the intruders as a function of the coefficient of normal restitution $\alpha$ for $d=3$, $\phi=0.0052$, $T_g^*=1000$, and four different values of the mass ratio $m_0/m_g$ [from top to bottom, $m_0/m_g=20, 50, 100$, and 1000]. Moreover, in all the curves $m_0/m=10$, $\sigma_0/\sigma=5$, and $\sigma_0/\sigma_g=(m_0/m_g)^{1/3}$. The solid lines are the theoretical results while the symbols are the DSMC simulation results. The dashed line is the result obtained from the Fokker--Planck approach \ref{['2.9']} to the operator $J_{0g}[f_0,f_g]$. Diamonds refer to DSMC simulations implemented using the time-driven approach GG22.
• Figure 2: Temperature ratio $\chi_0\equiv T_0/T_g$ versus the (common) coefficient of normal restitution $\alpha_0=\alpha$ for $d=3$, $\phi=0.0052$, $T_g^*=1000$, and four different values of the mass ratio $m_0/m_g$ [from top to bottom, $m_0/m_g=20, 50, 100$, and 1000]. Moreover, in all the curves $m_0/m=10$, $\sigma_0/\sigma=5$, and $\sigma_0/\sigma_g=(m_0/m_g)^{1/3}$. The solid lines are the theoretical results while the symbols are the Monte Carlo simulation results. The dashed line is the result obtained by using the Fokker--Planck approach \ref{['2.9']} to the operator $J_{0g}[f_0,f_g]$. Diamonds refer to DSMC simulations implemented using the time-driven approach GG22.
• Figure 3: Plot of the (scaled) thermal diffusion coefficient $D_T(\alpha)/D_T(1)$ versus the (common) coefficient of restitution $\alpha=\alpha_0$ for $d=3$, $\phi=0.0052$, $T_g^*=10$, and four different values of the mass ratio $m_0/m_g$ [$m_0/m_g=20, 50, 100$, and 1000]. In all the curves $m_0/m=8$, $\sigma_0/\sigma=2$, and $\sigma_0/\sigma_g=(m_0/m_g)^{1/3}$. The dashed line refers to the expression \ref{['6.8']} derived in the Brownian limiting case for the ratio $D_T(\alpha)/D_T(1)$.
• Figure 4: Plot of the (scaled) mutual diffusion coefficient $D(\alpha)/D(1)$ versus the (common) coefficient of restitution $\alpha=\alpha_0$ for $d=3$, $\phi=0.0052$, $T_g^*=10$, and four different values of the mass ratio $m_0/m_g$ [$m_0/m_g=20, 50, 100$, and 1000]. In all the curves $m_0/m=8$, $\sigma_0/\sigma=2$, and $\sigma_0/\sigma_g=(m_0/m_g)^{1/3}$. The dashed line refers to the expression \ref{['6.12']} derived in the Brownian limiting case for the ratio $D(\alpha)/D(1)$.
• Figure 5: Plot of the (scaled) tracer diffusion coefficient $D_0(\alpha)/D_0(1)$ versus the (common) coefficient of restitution $\alpha=\alpha_0$ for $d=3$, $\phi=0.0052$, $T_g^*=10$, and four different values of the mass ratio $m_0/m_g$ [$m_0/m_g=20, 50, 100$, and 1000]. In all the curves $m_0/m=8$, $\sigma_0/\sigma=2$, and $\sigma_0/\sigma_g=(m_0/m_g)^{1/3}$. The dashed line refers to the expression \ref{['5.15.3']} derived in the Brownian limiting case for the ratio $D_0(\alpha)/D_0(1)$.