[Paper Review] Kinetic Theory of Polydisperse Granular Mixtures: Influence of the Partial Temperatures on Transport Properties—A Review

Moisés García Chamorro|arXiv (Cornell University)|Jun 14, 2022

Granular flow and fluidized beds139 references21 citations

TL;DR

This paper reviews the influence of partial temperatures—deviations from energy equipartition—on transport properties in polydisperse granular mixtures using kinetic theory. It shows that inelastic collisions and flow gradients independently drive partial temperature differences, significantly affecting diffusion and bulk viscosity, with analytical predictions validated by simulations.

ABSTRACT

It is well-recognized that granular media under rapid flow conditions can be modeled as a gas of hard spheres with inelastic collisions. At moderate densities, a fundamental basis for the determination of the granular hydrodynamics is provided by the Enskog kinetic equation conveniently adapted to account for inelastic collisions. A surprising result (compared to its molecular gas counterpart) for granular mixtures is the failure of the energy equipartition, even in homogeneous states. This means that the partial temperatures Ti (measuring the mean kinetic energy of each species) are different to the (total) granular temperature T. The goal of this paper is to provide an overview on the effect of different partial temperatures on the transport properties of the mixture. Our analysis addresses first the impact of energy nonequipartition on transport which is only due to the inelastic character of collisions. This effect (which is absent for elastic collisions) is shown to be significant in important problems in granular mixtures such as thermal diffusion segregation. Then, an independent source of energy nonequipartition due to the existence of a divergence of the flow velocity is studied. This effect (which was already analyzed in several pioneering works on dense hard-sphere molecular mixtures) affects to the bulk viscosity coefficient. Analytical (approximate) results are compared against Monte Carlo and molecular dynamics simulations, showing the reliability of kinetic theory for describing granular flows.

Motivation & Objective

• To analyze the impact of partial temperatures on transport coefficients in polydisperse granular mixtures.
• To distinguish between inelastic collision-induced and flow-gradient-induced energy nonequipartition.
• To assess the reliability of kinetic theory in predicting transport properties under nonequipartition conditions.
• To compare analytical results with Monte Carlo and molecular dynamics simulations.
• To clarify the role of partial temperatures in phenomena like thermal diffusion segregation.

Proposed method

• Uses the Enskog kinetic equation adapted for inelastic collisions in granular mixtures.
• Applies the Chapman–Enskog method to expand distribution functions in hydrodynamic gradients.
• Treats partial temperatures as scalars dependent on velocity gradients and collisional inelasticity.
• Derives Navier–Stokes transport coefficients (diffusion, bulk viscosity) including first-order corrections from partial temperature gradients.
• Compares analytical results with DSMC and molecular dynamics simulations.
• Considers both dilute and moderately dense systems with size- and mass-disparate species.

Experimental results

Research questions

• RQ1 How does energy nonequipartition due to inelastic collisions affect transport coefficients in granular mixtures?
• RQ2 What is the contribution of flow velocity divergence to partial temperature differences and bulk viscosity?
• RQ3 To what extent do analytical predictions of transport coefficients match simulation data under nonequipartition?
• RQ4 How do partial temperatures influence thermal diffusion segregation in granular mixtures?
• RQ5 What is the validity of the Maxwellian assumption in the presence of partial temperature differences?

Key findings

• Inelastic collisions cause significant energy nonequipartition, even in homogeneous cooling states, with partial temperatures differing from the total granular temperature.
• The inelasticity-induced partial temperature difference strongly affects diffusion and thermal diffusion segregation, particularly in systems with disparate masses or sizes.
• Flow velocity divergence contributes to partial temperature gradients, influencing the bulk viscosity coefficient, a mechanism already known in elastic mixtures but now extended to granular systems.
• Analytical results for transport coefficients, including partial temperature corrections, show good agreement with DSMC and molecular dynamics simulations.
• The failure of energy equipartition invalidates the standard assumption of equal partial and granular temperatures, especially in strongly inelastic or polydisperse mixtures.
• The study confirms that partial temperature gradients are essential for accurate modeling of granular hydrodynamics beyond the Navier–Stokes level.