Ion transport in porous media: derivation of the macroscopic equations using up-scaling and properties of the effective coefficients

In this work we undertake the upscaling of a system of partial differential equations describing transport of a dilute N -component electrolyte in a Newtonian solvent through a rigid porous medium. The motion is governed by a small static electric field and a small hydrodynamic force, which allows us to calculate the linear response regime in a way initially proposed by O’Brien. The O’Brien partial linearization requires a fast and accurate solution of the underlying nonlinear Poisson-Boltzmann equation. We present an analysis of it, with the discussion of the boundary layer appearing as the Debye-Hückel parameter becomes large. Next we present briefly the corresponding two-scale asymptotic expansion and reduce the obtained two-scale equations to a coarse scale model. Our previous rigorous study assures that the ∗Ecole Polytechnique, CMAP, UMR CNRS 7641, 91128 Palaiseau Cedex, France ([email protected]) †Ecole Polytechnique, CMAP, UMR CNRS 7641, 91128 Palaiseau Cedex, France ([email protected]) ‡Université de Montpellier 2, Laboratoire Modélisation Mésoscopique et Chimie Théorique (LMCT), Institut de Chimie Séparative de Marcoule ICSM UMR 5257, CEA / CNRS / Université de Montpellier 2 / ENSCM Centre de Marcoule , Bât. 426, BP 17171, 30207 Bagnols sur Cèze Cedex, France ([email protected]) §Université de Lyon, Lyon, F-69003, France; Université Lyon 1, Département de mathématiques, Institut Camille Jordan, UMR 5208, 43, Bd du 11 novembre 1918, 69622 Villeurbanne Cedex, France ([email protected]) ¶Narvik University College, Postbox 385, 8505 Narvik, Norway; Lebedev Physical Institute RAS, Leninski ave., 53, 119991 Moscow, Russia ([email protected])