Ion transport through deformable porous media:derivation of the macroscopic equations using upscaling
Résumé
We study the homogenization (or upscaling) of the transport of a multicomponentelectrolyte in a dilute Newtonian solvent through a deformable porous medium. The porescale interaction between the flow and the structure deformation (modeled by linearizedelasticity equations) is taken into account. After a careful adimensionalization process, we first consider so-called equilibrium solutions, in the absence of external forces, for which thevelocity and diffusive fluxes vanish and the electrostatic potential is the solution of a Poisson–Boltzmann equation. When the motion is governed by a small static electric field and smallhydrodynamic and elastic forces, we use O’Brien’s argument to deduce a linearized model.Then we perform the homogenization of these linearized equations for a suitable choice oftime scale. It turns out that the deformation of the porous medium is weakly coupled tothe electrokinetics system in the sense that it does not influence electrokinetics although thelatter one yields an osmotic pressure term in the mechanical equations. As a consequence,the effective tensor satisfies Onsager properties, namely is symmetric positive definite.