Hybrid Burnett Equations: A New Method of Stabilizing

Abstract

In the original work by Burnett the pressure tensor and the heat current contain two time derivates. Those are commonly replaced by spatial derivatives using the equations to zero order in the Knudsen number. The resulting conventional Burnett equations were shown by Bobylev to be linearly unstable. In this paper it is shown that the original equations of Burnett have a singularity. A hybrid of the original and conventional equations is constructed which is shown to be linearly stable. It 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 2 Ma 2 can be neglected, the only difference from the conventional Burnett equations is the change of coefficients ̟2 → ̟3, ̟3 → ̟3.

Beyond the Navier–Stokes equations: Burnett hydrodynamics

“…An analysis of some papers related to Bobylev's instability and a discussion of the so-called regularized 13-moment equations for the shock wave structure can be found in [185]. Other possibilities and/or extensions are also available [186][187][188][189]. We hope that in the near future the situation will be clarified at least for the normal shock wave problem.…”

A thermo-mechanically consistent Burnett regime continuum flow equation without Chapman–Enskog expansion

“…As such, these Burnett shear stress and heat flux terms are, in principle, mechanically inconsistent. As the above Burnett continuum flow equations are also known to be linearly unstable (Karlin & Gorban 2002), the methods used to correct all of these defects vary from modifications of the material derivative or its alternative forms, to ad hoc additions of selected terms from higherorder Chapman-Enskog contributions (Jin & Slemrod 2001;Woods 1983;Söderholm 2007;Bobylev & Windfall 2012). Furthermore, not all terms appearing in equations (2.2) and (2.3) have an established physical significance.…”

Diffuse volume transport in fluids

“…Their possible interrelation stems from Soderholm's [33] observation that Bobylev's instability disappears if Burnett's inequality 2 = 3 were, on purely empirical grounds, to be replaced by the equality 2 = 3 , such as is true of our three-fold bi-velocity equality (6.40). Since Onsager reciprocity further requires that (6.43) be satisfied independently of any other considerations, Sonderholm's suggestion implicitly requires that all three Burnett coefficients be equal, a conclusion in accord with our triple equality (6.40).…”