Hybrid Burnett Equations. A New Method of Stabilizing

TL;DR

This paper introduces a hybrid approach to the Burnett equations that maintains linear stability by combining the original and zero-order approximations, with potential applications in fluid dynamics modeling.

Contribution

A new hybrid method for Burnett equations that ensures linear stability by retaining time derivatives and adjusting coefficients, improving upon previous unstable formulations.

Findings

Hybrid equations are linearly stable.

For stationary flow, equations differ mainly in coefficients.

The method avoids third derivatives of temperature.

Abstract

In the Chapman & Enskog version of the Burnett equations the two time derivatives in the pressure tensor and heat current are replaced by spatial derivatives using the equations to zero order in the Knudsen number. Bobylev showed that the resulting conventional Burnett equations are linearly unstable. In this paper it is shown that if the time derivatives are instead kept, the equations. A hybrid of the two possibilities is proposed which gives equations which are shown to be linearly stable. The system contains two parameters. For the simplest choice of parameters the hybrid equations have no third derivative of the temperature but the inertia term contains second spatial derivatives. For stationary flow, when terms $Kn^2Ma^2$ can be neglected, the only difference from the conventional Burnett equations is the change of coefficients $\varpi_{2}\to \varpi_{3},\varpi_{3}\to \varpi_{3}.$