Energy nonequipartition in a collisional model of a confined quasi-two-dimensional granular mixture

TL;DR

This paper develops a collisional kinetic model for a confined quasi-two-dimensional granular mixture, analyzing energy nonequipartition and validating predictions with simulations.

Contribution

It introduces an exact scaling solution for the kinetic equations and derives explicit temperature ratios considering energy transfer mechanisms.

Findings

Kinetic partial temperatures differ, indicating energy nonequipartition.

Theoretical predictions agree well with simulations for temperature ratios.

Deviations are due to non-Gaussian velocity distributions and microsegregation.

Abstract

A collisional model of a confined quasi-two-dimensional granular mixture is considered to analyze homogeneous steady states. The model includes an effective mechanism to transfer the kinetic energy injected by vibration in the vertical direction to the horizontal degrees of freedom of grains. The set of Enskog kinetic equations for the velocity distribution functions of each component is derived first to analyze the homogeneous state. As in the one-component case, an exact scaling solution is found where the time dependence of the distribution functions occurs entirely through the granular temperature $T$. As expected, the kinetic partial temperatures $T_i$ of each component are different and hence, energy equipartition is broken down. In the steady state, explicit expressions for the temperature $T$ and the ratio of partial kinetic temperatures $T_i/T_j$ are obtained by considering Maxwellian distributions defined at the partial temperatures $T_i$. The (scaled) granular temperature and the temperature ratios are given in terms of the coefficients of restitution, the solid volume fraction, the (scaled) parameters of the collisional model, and the ratios of mass, concentration, and diameters. In the case of a binary mixture, the theoretical predictions are exhaustively compared with both direct simulation Monte Carlo and molecular dynamics simulations with a good agreement. The deviations are identified to be originated in the non-Gaussianity of the velocity distributions and on microsegregation patterns, which induce spatial correlations not captured in the Enskog theory.