Ion transport through deformable porous media: derivation of the macroscopic equations using upscaling
Résumé
We study the upscaling or homogenization of the transport of a multicomponentelectrolyte in a dilute Newtonian solvent through a deformable porous medium.The pore scale interaction between the flow and the structure deformation 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 andthe 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 homogenizationof these linearized equations for a suitable choice of time scale. It turns out thatthe deformation of the porous medium is weakly coupled to the electrokinetics systemin the sense that it does not influence electrokinetics although the latter one yieldsan osmotic pressure term in the mechanical equations. As a byproduct we find that theeffective tensor satisfies Onsager properties, namely is symmetric positive definite.
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