Compressible multicomponent flow in porous media with Maxwell‐Stefan diffusion

Abstract

We introduce a Darcy‐scale model to describe compressible multicomponent flow in a fully saturated porous medium. In order to capture cross‐diffusive effects between the different species correctly, we make use of the Maxwell–Stefan theory in a thermodynamically consistent way. For inviscid flow, the model turns out to be a nonlinear system of hyperbolic balance laws. We show that the dissipative structure of the Maxwell‐Stefan operator permits to guarantee the existence of global classical solutions for initial data close to equilibria. Furthermore, it is proven by relative entropy techniques that solutions of the Darcy‐scale model tend in a certain long‐time regime to solutions of a parabolic limit system.

Asymptotic derivation of multicomponent compressible flows with heat conduction and mass diffusion

“…By contrast, situations that involve mean flow lead to questions in the realm of hyperbolic or hyperbolic-parabolic systems. There are available analyses for isothermal viscous flows of multicomponent systems [3] and even for multicomponent compressible Euler flows [25][26][27]30]. For analyses of non-isothermal multicomponent systems that include effects of heat-conduction we refer to [5,21,22,31].…”

“…There has been recent interest in the convergence from compressible multicomponent Euler equations in the high-friction limit to Maxwell-Stefan systems, a problem pertaining to the subject of relaxation approximations. A number of studies have appeared regarding isothermal flows [4,14,16,25] achieving in the limit the classical Maxwell-Stefan system [4], or (for more general chemical potentials) porous media variants or even fourth order diffusions [16,25].…”