The extended gas-kinetic theory from Pullin equation: the relaxation rates, transport coefficients and model equation

Abstract

The Borgnakke-Larsen model, widely used in rarefied flow predictions, serves as the mainstream energy-exchange kernel for polyatomic gases. However, it lacks integrability and does not guarantee detailed balance, limiting theoretical foundations for near-continuum relaxation mechanisms, transport coefficients, and relaxation model equations. In this work, we adopt the Pullin equation, which possesses an integrable collision kernel and satisfies detailed balance, to analyze near-continuum relaxation. Considering only translational and rotational degrees, we obtain explicit analytical expressions for the relaxation rates of macroscopic variables including stress, temperatures, and heat fluxes by approximating the distribution function in mixed Hermite and Laguerre spaces. Based on the same elementary moments, we derive transport coefficients via Chapman-Enskog expansion, rigorously confirming a long-standing speculation that thermal conductivity depends on the degree of thermal non-equilibrium; under equilibrium, the results reduce to those of Mason and Monchick. Using the correct relaxation rates, we propose a novel Rykov-type relaxation model that captures the coupled relaxation of translational and rotational heat fluxes, a mechanism ignored in the widely used Rykov equation. The model is validated against benchmark test cases.