Extensions to the Navier-Stokes-Fourier Equations for Rarefied Transport: Entropy Variable Formulation, Heat Transfer and Dispersion of a Non-Linear Entropy Stable Closure

Abstract

We assess the non-linear version of the Entropy Stable Extension to the Navier-Stokes-Fourier equations which were originally derived in arXiv:2407.17334. Using an identity for the derivative of the inverse of linearized collision operator, we show that the Entropy Stable Extension takes the form of a fourth-order quasi-cubic system of equations when written in entropy variables. Furthermore, we find that the quasi-linear contributions to the extension retain the highest order derivative contributions and the entropy stability of the full extension. This quasi-linear approximation is applied to the stationary heat transfer between parallel plates where we find that a specification of the boundary conditions that interpolates between the continuum and collisions limits can yield temperature profiles that better agree with DSMC predictions than Navier-Stokes-Fourier equations with Smoluchowski boundary conditions, while also giving a heat flux prediction that better agrees with the Sherman-Lees heat flux than the Navier-Stokes-Fourier prediction. We conclude with a brief study of dispersion of plane waves according to the linearized equations in one dimension where we show that all frequency modes are stable and relate this to entropy stability.