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

Abstract

Chapman-Enskog expansion is the orthodox approach to derive continuum flow models from Boltzmann's kinetic equation for dilute gases. Beyond the Navier-Stokes-Fourier order, these models known as Burnett hydrodynamic-regime equations violate a number of fundamental mechanical and thermodynamic principles in their original forms. This has generated a widely investigated problem in the kinetic theory of gases. In this short article, we derive a Burnett hydrodynamic-regime continuum model that is systematically consistent with all known mechanical and thermodynamic principles without using any series' expansion. Close comparison with the conventional Burnett hydrodynamic set of equations is considered and their linear stabilities around an equilibrium point under small perturbations are presented.

Proposal of a critical test of the Navier-Stokes-Fourier paradigm for compressible fluid continua

“…In contrast with our cautionary attitude displayed towards DSMC simulations of Boltzmann's original collision model, we see no objection to applying DSMC techniques to the proposed test at hand when using, say, Dadzie's stochastically modified Boltzmann model equation [9] in place of Boltzmann's original equation. Indeed, such an undertaking would presumably offer independent data bearing on the issue of the validity of the NSF equations for compressible gaseous continua.…”

“…However, in contrast with those of Burnett-and as with all other LIT-based constitutive equations [7]-our equations are regarded as describing the behavior only of continua, while not including noncontinua. This view is confirmed by the independent works of Dadzie [9] and Meng, et al [10], each grounded upon its own molecularly based modification of Boltzmann's gas-kinetic equation [11].…”

“…In this latter context, recent work by Dadzie, Reese, and their collaborators [9,10,[28][29][30][31][32][33][34][35], as well as that of other independent researchers [36,37], serving to molecularly mirror comparable macroscopic bivelocity developments based upon irreversible thermodynamic principles [8]-point to the fact that a fundamental problem exists with regard to the current statistical-mechanical modeling of compressible fluid continua. This owes to the failure of most existing molecular models to incorporate diffuse or dissipative (stochastic) volume transport phenomena within their scope.…”

“…(See also Refs. [9] and [10], which arrive at exactly these same bivelocity * hbrenner@mit.edu equations [8], starting, however, from a molecular rather than macroscopic basis-the former involving the addition of a stochastic contribution to the collisional term in Boltzmann's gas-kinetic equation [11].) According to its macroscopic LITbased derivation [8], bivelocity fluid mechanics is applicable to both liquids and gases.…”

Direct-Simulation Monte Carlo Investigation of a Berea Porous Structure

“…Darcy's law, which was concluded from experiments and derived from continuum methods [Navier-Stokes (NS) equations], is no longer valid without any slip modification (Song and Ehlig-Economides 2011;Swami et al 2012;Darabi et al 2012;Lv et al 2014). The inadequacy of NSF equations in describing gas in micro-and nanosize domains can be explained because these equations can, theoretically, describe only flows that are in the pure hydrodynamic regime (Mizzi et al 2007).…”