The half-space problem of evaporation and condensation for polyatomic gases and entropy inequalities

TL;DR

This paper analyzes the steady Boltzmann equation for polyatomic gases in a half-space, deriving conditions for evaporation or condensation based on entropy inequalities and boundary parameters, with numerical exploration of these relations.

Contribution

It introduces new relations between boundary and far-field parameters for polyatomic gases, considering evaporation and condensation in a half-space setting, using entropy inequalities.

Findings

Relations depend on the Mach number sign at infinity.

Quantitative differences exist for various internal degrees of freedom.

Numerical analysis confirms qualitative similarities across different gases.

Abstract

This study investigates the steady Boltzmann equation in one spatial variable for a polyatomic single-component gas in a half-space. Inflow boundary conditions are assumed at the half-space boundary, where particles entering the half-space are distributed as a Maxwellian, an equilibrium distribution characterized by macroscopic parameters of the boundary. At the far end, the gas tends to an equilibrium distribution, which is also Maxwellian. Using conservation laws and an entropy inequality, we derive relations between the macroscopic parameters of the boundary and at infinity required for the existence of solutions. The relations vary depending on the sign of the Mach number at infinity, which dictates whether evaporation or condensation takes place at the interface between the gas and the condensed phase. We explore the obtained relations numerically. This investigation reveals that, although the relations are qualitatively similar for various internal degrees of freedom of the gas, clear quantitative differences exist.