Ion transport through deformable porous media: derivation of the macroscopic equations using upscaling

We study the homogenization (or upscaling) of the transport of a multicomponent electrolyte in a dilute Newtonian solvent through a deformable porous medium. The pore scale interaction between the flow and the structure deformation (modeled by linearized elasticity 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 the velocity 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 small hydrodynamic 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 of time scale. It turns out that the deformation of the porous medium is weakly coupled to the electrokinetics system in the sense that it does not influence electrokinetics although the latter one yields an osmotic pressure term in the mechanical equations. As a consequence, the effective tensor satisfies Onsager properties, namely is symmetric positive definite.