Going beyond an old shockwave conjecture for improving upon Navier-Stokes.

Abstract

Nonequilibrium molecular dynamics (NEMD) computer simulations of steady shockwaves in dense fluids and rarefied gases produce detailed shockwave profiles of mechanical and thermal properties. The Boltzmann equation, under the assumption of local thermodynamic equilibrium (LTE), leads to the first-order (linear) continuum theory of hydrodynamic flow: Navier-Stokes-Fourier (NSF). (Expansion of the LTE Boltzmann equationin higher powers of gradients yields so-called Burnett second-order terms, etc.) NEMD simulations of strong shockwaves with high gradients are not well modeled by NSF theory. Many years ago, a conjecture for going "beyond Navier-Stokes" was proposed, applying the empirical observation of anisotropic thermal enhancement in the shock front to the temperature dependence of the NSF transport coefficients, whose dissipation determines the slope at the center of the shock profile: for weak shocks, the actual coefficients in NEMD simulations appear to be smaller than in NSF predictions, leading to steeper gradients being observed, while for strong shocks, the NEMD coefficients appear to be larger, leading to less steep shock rises than predicted by NSF calculations. In this paper, we show that adding significant Burnett nonlinearity into an LTE continuum theory reproduces the early shock rise and slope of NEMD profiles, for both weak and strong shocks in dense fluids, as well as strong shockwaves in the ideal gas. Moreover, we show that "Holian's conjecture" incorporates significant Burnett nonlinearity, but like all the other LTE continuum theories, it fails to describe the slow NEMD return to equilibrium beyond the shock front. We show that Maxwell relaxation has to be applied to the hydrodynamic variables themselves (rather than attempting indirect relaxation of their gradients) in order to more accurately model the entire shockwave profile. Non-LTE Maxwell relaxation is the only way to bring the entire profile into agreement with NEMD, most noticeably for strong shockwaves.