Microscopic molecular dynamics characterization of the second-order non-Navier-Fourier constitutive laws in the Poiseuille gas flow

Abstract

The second-order non-Navier-Fourier constitutive laws, expressed in a compact algebraic mathematical form, were validated for the force-driven Poiseuille gas flow by the deterministic atomic-level microscopic molecular dynamics (MD). Emphasis is placed on how completely different methods (a second-order continuum macroscopic theory based on the kinetic Boltzmann equation, the probabilistic mesoscopic direct simulation Monte Carlo, and, in particular, the deterministic microscopic MD) describe the non-classical physics, and whether the second-order non-Navier-Fourier constitutive laws derived from the continuum theory can be validated using MD solutions for the viscous stress and heat flux calculated directly from the molecular data using the statistical method. Peculiar behaviors (non-uniform tangent pressure profile and exotic instantaneous heat conduction from cold to hot [R. S. Myong, “A full analytical solution for the force-driven compressible Poiseuille gas flow based on a nonlinear coupled constitutive relation,” Phys. Fluids 23(1), 012002 (2011)]) were re-examined using atomic-level MD results. It was shown that all three results were in strong qualitative agreement with each other, implying that the second-order non-Navier-Fourier laws are indeed physically legitimate in the transition regime. Furthermore, it was shown that the non-Navier-Fourier constitutive laws are essential for describing non-zero normal stress and tangential heat flux, while the classical and non-classical laws remain similar for shear stress and normal heat flux.

Numerical Simulation of Hypersonic Rarefied Flows Using the Second-Order Constitutive Model of the Boltzmann Equation

“…Interestingly, the existence of the hyperbolic sine form in the dissipation (or production) term of second-order constitutive equation can be explained in heuristic way [18,19] by recognizing that the net change in the number of gas molecules due to the Boltzmann collision integral may be described by gain minus loss, that is, exp (nonequilibrium) À exp (Ànonequilibrium) ,so that the leading term of dissipation in the cumulant expansion becomes sinh.…”

Coupled constitutive relations: a second law based higher-order closure for hydrodynamics

“…The classical Navier–Stokes–Fourier (NSF) equations are known to fail in describing small-scale flows, for which the Knudsen number—defined as the ratio of the molecular mean free path to a characteristic hydrodynamic length scale—is sufficiently large [1,2]. It is well established that the traditional NSF equations cannot describe strong non-equilibrium effects, which occur at high Knudsen numbers; for instance, the classical NSF equations are not able to describe the heat flux parallel to flow direction which is not forced by temperature gradient [3,4], non-uniform pressure profile and characteristic temperature dip in Poiseuille flow [5–7], non-Fourier heat flux in a lid-driven cavity where heat flows from low temperature to high temperature [8,9], etc.…”

Numerical simulation of three-dimensional high-speed flows using a second-order nonlinear model

“…The model, named as the nonlinear coupled constitutive relations (NCCR), takes a form of nonlinear algebraic system and can be implemented more easily in the hyperbolic conservation laws. The second-order molecular dynamic characterization of NCCR model was validated in Poiseuille flow research through the deterministic microscopic MD method [16]. Myong also constructed an uncoupled computational algorithm for this model, which was applied successfully in one-dimensional shock wave structure and two-dimensional simple flow [14,17].…”

“…a 9-dimensional space9 into the real line . A more compact vector form of the nonlinear equation systems(16) can be rewritten as: , are attempted to be used for solving aforementioned systems of nonlinear algebraic equations(17). In Fixed-point iteration, the first challenge is to construct an available convergent iterative expression for NCCR model.…”