3D linearized stability analysis of various forms of Burnett equations

Abstract

Burnett equations were originally derived in 1935 by Burnett by employing the Chapman-Enskog expansion to Classical Boltzmann equation to second order in Knudsen number Kn. Since then several variants of these equations have been proposed in the literature; these variants have differing physical and numerical properties. In this paper, we consider three such variants which are known in the literature as ‘the Original Burnett (OB) equations’, the Conventional Burnett (CB) equations’ and the recently formulated by the authors ‘the Simplified Conventional (SCB) equations.’ One of the most important issues in obtaining numerical solutions of the Burnett equations is their stability under small perturbations. In this paper, we perform the linearized stability (known as the Bobylev Stability) analysis of three-dimensional Burnett equations for all the three variants (OB, CB, and SCB) for the first time in the literature on this subject. By introducing small perturbations in the steady state flow field, the trajectory curve and the variation in attenuation coefficient with wave frequency of the characteristic equation are obtained for all the three variants of Burnett equations to determine their stability. The results show that the Simplified Conventional Burnett (SCB) equations are unconditionally stable under small wavelength perturbations. However, the Original Burnett (OB) and the Conventional Burnett (CB) equations are unstable when the Knudsen number becomes greater than a critical value and the stability condition worsens in 3D when compared to the stability condition for 1-D and 2-D equations. The critical Knudsen number for 3-D OB and CB equations is 0.061 and 0.287 respectively.