# Influence of the first-order contributions to the partial temperatures on transport properties in polydisperse dense granular mixtures

## TL;DR

This paper revisits the Chapman--Enskog solution for polydisperse granular mixtures to explicitly calculate the first-order contributions to partial temperatures, revealing their significant impact on transport properties like bulk viscosity and cooling rate.

## Contribution

It introduces explicit formulas for the first-order temperature contributions in polydisperse mixtures, including inelastic effects, which were neglected in previous models.

## Key findings

- First-order temperature contributions significantly affect transport properties.
- Explicit formulas depend on mixture parameters like size, mass, and inelasticity.
- Effects are especially notable for mixtures with disparate masses and high inelasticity.

## Abstract

The Chapman--Enskog solution to the Enskog kinetic equation of polydisperse granular mixtures is revisited to determine the first-order contributions $\varpi_i$ to the partial temperatures. As expected, these quantities (which were neglected in previous attempts) are given in terms of the solution to a set of coupled integro-differential equations analogous to those for elastic collisions. The solubility condition for this set of equations is confirmed and the coefficients $\varpi_i$ are calculated by using the leading terms in a Sonine polynomial expansion. These coefficients are given as explicit functions of the sizes, masses, composition, density, and coefficients of restitution of the mixture. Within the context of small gradients, the results apply for arbitrary degree of inelasticity and are not restricted to specific values of the parameters of the mixture. In the case of elastic collisions, previous expressions of $\varpi_i$ for ordinary binary mixtures are recovered. Finally, the impact of the first-order coefficients $\varpi_i$ on the bulk viscosity $\eta_\text{b}$ and the first-order contribution $\zeta^{(1)}$ to the cooling rate is assessed. It is shown that the effect of $\varpi_i$ on $\eta_\text{b}$ and $\zeta^{(1)}$ is not negligible, specially for disparate mass ratios and strong inelasticity.