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

Abstract

A critical, albeit simple experimental and/or molecular-dynamic (MD) simulation test is proposed whose outcome would, in principle, establish the viability of the Navier-Stokes-Fourier (NSF) equations for compressible fluid continua. The latter equation set, despite its longevity as constituting the fundamental paradigm of continuum fluid mechanics, has recently been criticized on the basis of its failure to properly incorporate volume transport phenomena-as embodied in the proposed bivelocity paradigm [H. Brenner, Int. J. Eng. Sci. 54, 67 (2012)]-into its formulation. Were the experimental or simulation results found to accord, even only qualitatively, with bivelocity predictions, the temperature distribution in a gas-filled, thermodynamically and mechanically isolated circular cylinder undergoing steady rigid-body rotation in an inertial reference frame would not be uniform; rather, the temperature would be higher at the cylinder wall than along the axis of rotation. This radial temperature nonuniformity contrasts with the uniformity of the temperature predicted by the NSF paradigm for these same circumstances. Easily attainable rates of rotation in centrifuges and readily available tools for measuring the expected temperature differences render experimental execution of the proposed scheme straightforward in principle. As such, measurement-via experiment or MD simulation-of, say, the temperature difference ΔT between the gas at the wall and along the axis of rotation would provide quantitative tests of both the NSF and bivelocity hydrodynamic models, whose respective solutions for the stated set of circumstances are derived in this paper. Independently of the correctness of the bivelocity model, any temperature difference observed during the proposed experiment or simulation, irrespective of magnitude, would preclude the possibility of the NSF paradigm being correct for fluid continua, except for incompressible flows.

Molecular-level study of compressible gaseous continua

“…For Poiseuille flow [14,15] and the gas flow within a circular cylinder [16][17][18], viscous effects dominate the system, and the gas does not compress much; therefore, we consider them to test the applicability of NSF.…”

Weak solutions and convergent numerical schemes of modified compressible Navier–Stokes equations

“…The standard Navier-Stokes equations take the form (3) and are obtained by adding a diffusive flux to the Euler equations.…”

“…The regularization (G(u)u x ) is conservative and we will refer to (3) as being conservative. Using the entropy variables, (3) can be stated as u w w t + g(w) x = (G(w)w x ) x , x ∈ . (4) We require that G is symmetric and positive semi-definite.…”

Bivelocity Picture in the Nonrelativistic Limit of Relativistic Hydrodynamics

“…(3.12) into Eq. (2.24), the fundamental relation between the two fluid velocities in the nonrelativistic limit is derived as 13) where the diffusion current ν i can be obtained from Eq. (3.1).…”