Grad’s 13 moments approximation for Enskog-Vlasov equation

Abstract

Hydrodynamic models of liquid-vapor flows have to face the difficulty of describing non-equilibrium regions next to interfaces. Depending on the flow regimes and the underlying theoretical models, different answers have been given. In particular, diffuse interface models (DIMs) provide, in principle, a unified description of the whole flow field by a set of PDE’s, not much more complex than Navier-Stokes-Fourier classical equations. Unfortunately, DIMs fail to provide a proper description of kinetic layers next to interfaces. In order to develop a model incorporating kinetic effects while keeping the relative simplicity of DIMs, macroscopic transport equations—moment equations — are derived from the Enskog-Vlasov equation. The Enskog-Vlasov equation extends the Enskog equation by accounting for the attractive forces between the gas molecules. Hence, it gives a van-der-Waals-like kinetic description of a non-ideal gas, including liquid-vapor phase change. Specifically, the equation describes the liquid phase, the vapor phase, and a diffusive transition region connecting both phases. While not the most accurate model, solutions of the Enskog-Vlasov equation exhibits all relevant phenomena occurring in the evaporation and condensation of rarefied or dense vapors. In this work, Grad’s moment method is used to derive a closed set of 13 moment equations. In the appropriate limits, these reduce to the Navier-Stokes-Fourier system for liquid and vapor. Our main interest is to study non-hydrodynamic effects, in particular transport in and across the transition region, and the interplay between the transition region and Knudsen layers. We present first results of this program, including the closed transport equations for 13 moments, discussion of the limits, and solutions in simple geometries.