Relaxation time for the temperature in a dilute binary mixture from classical kinetic theory

TL;DR

This paper investigates the relaxation time for temperature equilibration in a dilute binary mixture using the full Boltzmann equation and Chapman-Enskog method, concluding that temperature differences equilibrate quickly relative to hydrodynamical timescales.

Contribution

It provides a detailed kinetic theory analysis of temperature relaxation in binary mixtures without simplifying assumptions on collision terms.

Findings

Temperature relaxation time is much smaller than hydrodynamical times.

No diffusion force or mass diffusion appears in the system.

Different initial temperatures equilibrate rapidly.

Abstract

The system of our interest is a dilute binary mixture, in which we consider that the species have different temperatures as an initial condition. To study their time evolution, we use the full version of the Boltzmann equation, under the hypothesis of partial local equilibrium for both species. Neither a diffusion force nor mass diffusion appears in the system. We also estimate the time in which the temperatures of the components reach the full local equilibrium. In solving the Boltzmann equation, we imposed no assumptions on the collision term. We work out its solution by using the well known Chapman-Enskog method to first order in the gradients. The time in which the temperatures relax is obtained following Landau's original idea. The result is that the relaxation time for the temperatures is much smaller than the characteristic hydrodynamical times but greater than a collisional time. The main conclusion is that there is no need to study binary mixtures with different temperatures when hydrodynamical properties are sought.