Diffusion of impurities in a granular gas

Abstract

Diffusion of impurities in a granular gas undergoing homogeneous cooling state is studied. The results are obtained by solving the Boltzmann--Lorentz equation by means of the Chapman--Enskog method. In the first order in the density gradient of impurities, the diffusion coefficient $D$ is determined as the solution of a linear integral equation which is approximately solved by making an expansion in Sonine polynomials. In this paper, we evaluate $D$ up to the second order in the Sonine expansion and get explicit expressions for $D$ in terms of the restitution coefficients for the impurity--gas and gas--gas collisions as well as the ratios of mass and particle sizes. To check the reliability of the Sonine polynomial solution, analytical results are compared with those obtained from numerical solutions of the Boltzmann equation by means of the direct simulation Monte Carlo (DSMC) method. In the simulations, the diffusion coefficient is measured via the mean square displacement of impurities. The comparison between theory and simulation shows in general an excellent agreement, except for the cases in which the gas particles are much heavier and/or much larger than impurities. In theses cases, the second Sonine approximation to $D$ improves significantly the qualitative predictions made from the first Sonine approximation. A discussion on the convergence of the Sonine polynomial expansion is also carried out.

9 figures. to appear in Phys. Rev. E

References in corpus (2)

• Studies of Mass and Size Effects in Three-Dimensional Vibrofluidized Granular Mixtures
• Shear viscosity for a heated granular binary mixture at low-density

Cited by in corpus (35)

• Enskog Theory for Polydisperse Granular Mixtures. I. Navier-Stokes order Transport
• Enskog Theory for Polydisperse Granular Mixtures II. Sonine Polynomial Approximation
• Modified Sonine approximation for the Navier-Stokes transport coefficients of a granular gas
• Granular Brownian motion
• Transport coefficients for an inelastic gas around uniform shear flow: Linear stability analysis
• Mass and heat fluxes for a binary granular mixture at low-density
• Navier-Stokes transport coefficients of $d$-dimensional granular binary mixtures at low density
• Mass transport of impurities in a moderately dense granular gas
• Steady state representation of the homogeneous cooling state of a granular gas
• On the Einstein relation in a heated granular gas
• Self-diffusion in granular gases: Green-Kubo versus Chapman-Enskog
• Enskog kinetic theory for multicomponent granular suspensions
• Transport coefficients for inelastic Maxwell mixtures
• First-order Chapman--Enskog velocity distribution function in a granular gas
• System of elastic hard spheres which mimics the transport properties of a granular gas
• Modified Sonine approximation for granular binary mixtures
• Segregation of an intruder in a heated granular dense gas
• Integral representation of the linear Boltzmann operator for granular gas dynamics with applications
• Granular mixtures modeled as elastic hard spheres subject to a drag force
• Diffusion transport coefficients for granular binary mixtures at low density. Thermal diffusion segregation
• Diffusion of intruders in granular suspensions: Enskog theory and random walk interpretation
• Mass transport of an impurity in a strongly sheared granular gas
• Stability of freely cooling granular mixtures at moderate densities
• Heat flux of driven granular mixtures at low density. Stability analysis of the homogeneous steady state
• Tracer diffusion coefficients in a moderately dense granular suspension. Stability analysis and thermal diffusion segregation
• Comment on "Kinetic theory models for granular mixtures with unequal granular temperature: Hydrodynamic velocity" [Phys. Fluids 33, 043321 (2021)]
• Diffusion of impurities in a moderately dense confined granular gas
• Impurity in a granular gas under nonlinear Couette flow
• Anomalous transport of impurities in inelastic Maxwell gases
• Assessment of kinetic theories for moderately dense granular binary mixtures: Shear viscosity coefficient
• Kinetic Theory for Binary Granular Mixtures at Low-Density
• Mean square displacement of intruders in freely cooling multicomponent granular mixtures
• A note on the violation of the Einstein relation in a driven moderately dense granular gas
• Kinetic model for transport in granular mixtures
• Kinetic theory of polydisperse granular mixtures: influence of the partial temperatures on transport properties. A review