Existence analysis of a single-phase flow mixture model with van der Waals pressure
Abstract
The transport of single-phase fluid mixtures in porous media is described by cross-diffusion equations for the mass densities. The equations are obtained in a thermodynamic consistent way from mass balance, Darcy's law, and the van der Waals equation of state for mixtures. The model consists of parabolic equations with cross diffusion with a hypocoercive diffusion operator. The global-in-time existence of weak solutions in a bounded domain with equilibrium boundary conditions is proved, extending the boundedness-by-entropy method. Based on the free energy inequality, the large-time convergence of the solution to the constant equilibrium mass density is shown. For the two-species model and specific diffusion matrices, an integral inequality is proved, which reveals a minimum principle for the mass fractions. Without mass diffusion, the two-dimensional pressure is shown to converge exponentially fast to a constant. Numerical examples in one space dimension illustrate this convergence.
Keywords
Cite
@article{arxiv.1612.04161,
title = {Existence analysis of a single-phase flow mixture model with van der Waals pressure},
author = {Ansgar Jüngel and Jiří Mikyška and Nicola Zamponi},
journal= {arXiv preprint arXiv:1612.04161},
year = {2016}
}