OBurnett Equations: Thermodynamically Consistent Continuum Theory Beyond the Navier–Stokes Regime

Abstract

Rarefied gas flows are highly nonequilibrium flows whose flow physics cannot be discerned accurately within the framework of the Navier–Stokes equations. The Burnett equations and the Grad moment equations, which form a super-set of the Navier–Stokes equations, have been proposed in the literature to model such flows but not much success has been achieved because of some inherent limitations of these equations. In this review article, we mainly focus on the recently proposed Onsager-Burnett equations (Singh et al., 2017, “Derivation of stable Burnett equations for rarefied gas flows,” Phys. Rev. E 96, p. 013106) for rarefied gas flows, and the progress achieved so far by solving these equations for some benchmark flow problems. Like Burnett and Grad equations, the OBurnett equations form a super-set of the Navier–Stokes equations and belong to the class of higher order continuum transport equations. However, there are two fundamental aspects where the significance of the OBurnett equations is clearly visible. First, the OBurnett equations are unconditionally stable as well as thermodynamically consistent unlike the conventional Burnett and Grad moment equations. Second, the OBurnett constitutive relations for the stress tensor and the heat flux vector do not have any higher order derivatives of velocity, pressure, or temperature. This is quite significant since now the equations need the same number of boundary conditions as that of the Navier–Stokes equations. As such, the OBurnett equations form a complete theory, which cannot be said for the conventional Burnett equations. These two important aspects help to set the OBurnett equations apart from the rest of the higher order continuum theories. The results of the OBurnett equations are compiled for two benchmark rarefied flow problems: force-driven compressible Poiseuille flow and the normal shock wave flow problem. For force-driven compressible Poiseuille flow, the OBurnett equations successfully capture the nonequilibrium effects such as nonuniform pressure profile and presence of normal stresses and tangential heat flux in the flow. The accurate description of highly nonequilibrium internal structure of normal shocks has always been the stringent test for the higher order continuum theories. The results of the OBurnett equations for normal shocks show that there is no theoretical upper Mach number limit for the equations. Further, the equations predict smooth shock structures at all Mach numbers, existence of heteroclinic trajectory, positive entropy generation throughout the shock, and significant improvement over the results of the Navier–Stokes equations. Finally, the recently proposed Grad's second problem, which has the potential to become a benchmark problem, is discussed. The solution of Grad's second problem for different interaction potentials (Maxwell and hard-sphere molecules) within the Burnett hydrodynamics is also presented at length and some important remarks are made in this context.