Solution of the Multicomponent Boltzmann Equation Based on an Extended Set of Observables

TL;DR

This paper develops a perturbative solution to the multicomponent Boltzmann equation using an extended set of observables, deriving Euler and Navier-Stokes equations with detailed transport terms for each component.

Contribution

It introduces a modified density scaling scheme allowing independent component densities and derives detailed multicomponent momentum and energy equations including new source terms.

Findings

Derivation of species-specific momentum and energy balance equations.

First order Navier-Stokes equations with partial viscosities and conductivities.

Calculation of first order corrections to Maxwell-Stefan diffusion and transport coefficients.

Abstract

We present the perturbative solution of the multicomponent Boltzmann kinetic equation based on the set of observables including the hydrodynamic velocity and temperature for each component. The solution is obtained by modifying the formal density scaling scheme by Enskog, such that the density of each component is scaled independently. As a result we obtain the species momentum and energy balance equations with the source terms describing the transfer of corresponding quantities between different components. In the zero order approximation those are the Euler equations with the momentum and heat diffusion included in the form of the classical Maxwell-Stefan diffusion terms. The first order approximation results in equations of a Navier-Stokes type with the partial viscosity and heat conductivity including only the correlations of the particles of the same component. The first order corrections to the Maxwell-Stefan terms as well as the contributions bilinear in gradients and differences of observables are calculated. The first order momentum source term is shown to include thermal diffusion. The nondiagonal (in component indexes) components of viscosity and heat conductivity appear as second order contributions.