Tracer Diffusion in Granular Suspensions: Testing the Enskog Kinetic Theory with DSMC and Molecular Dynamics

TL;DR

The paper investigates tracer diffusion of an intruder in a granular suspension by combining Enskog kinetic theory with drag and Langevin-type solvent forcing, and validates the theory against molecular dynamics and direct simulation Monte Carlo results. The authors derive the tracer diffusion coefficient via Chapman–Enskog expansion up to the second Sonine approximation and corroborate it with Green–Kubo-based VACF analysis, comparing predictions for $D$, intruder temperature $T_0$, and VACF against simulations across varying restitution $\alpha$, mass ratios $m_0/m$, and friction $\gamma$. They demonstrate that the Maxwellian approximation for particle distributions and the second Sonine correction yield the best agreement, identifying the friction regime where Enskog theory remains reliable for granular suspensions. The findings support the applicability of Enskog kinetic theory to particle-laden flows with solvent effects and provide quantitative benchmarks for future modeling of granular suspensions in fluids.

Abstract

We investigate the diffusion of an intruder in a granular gas, with both components modeled as smooth hard spheres, both immersed in a low viscosity carrier fluid to form a particle-laden suspension. In this system, dissipative particle collisions coexist with the action of a solvent. The latter is modeled via a viscous drag force and a stochastic Langevin-like force proportional to the background fluid temperature. Building on previous kinetic theory and random-walk results of the tracer diffusion coefficient [R. Gómez González, E. Abad, S. Bravo Yuste, and V. Garzó, Phys. Rev. E \textbf{108}, 024903 (2023)], where random-walk predictions were compared with Chapman--Enskog results up to the second Sonine approximation, we assess the robustness of the Enskog framework by incorporating molecular dynamics (MD) simulations, using direct simulation Monte Carlo (DSMC) results as an intermediate reference. In particular, we focus on the intruder velocity autocorrelation function, considering intruders different masses (from 0.01 to 100 times the mass of the granular particles), and analyse the behavior of the intruder temperature and diffusion coefficient. Our results clarify the influence of the friction parameter and the conditions under which Enskog kinetic theory reliably describes intruder diffusion in granular suspensions.

Figures (10)

• Figure 1: Evolution of the average kinetic energy, $K$, per particle following Eq. \ref{['Langevin']} (black line). The volume fraction is $\phi=0.10$ and restitution coefficient $\alpha=0.8$. The intruder has the same properties as the bath particles.
• Figure 2: Tracer diffusion coefficient as a function of the friction coefficient $\gamma$ for different volume fractions, as labeled. Simulation data are shown as open circles, lines represent the theory and closed symbols indicate the DSMC calculations.
• Figure 3: Pair distribution function of the system with volume fraction $\phi=0.10$ and friction coefficient $\gamma=1 (m T_b)^{1/2}\sigma$, for different restitution coefficients, as labeled.
• Figure 4: Velocity autocorrelation function. The upper panel shows it for different values of the restitution coefficient (volume fraction $\phi=0.10$ in all cases). The lower panel depicts it for different volume fractions (restitution coefficient, $\alpha=0.8$).
• Figure 5: Temperature as a function of $\alpha$ for different volume fractions $\phi$, as labeled. Open symbols, lines, and closed symbols represent the simulation data, theory, and DSMC calculations, respectively.