Homogenization of the linearized ionic transport equations in random porous media

TL;DR

This paper extends homogenization techniques for ionic transport equations from periodic to random porous media, proving convergence, well-posedness, and Onsager properties of the effective tensor in stochastic settings.

Contribution

It introduces stochastic homogenization for ionic transport equations in random media, establishing convergence, symmetry, and positive definiteness of the effective tensor.

Findings

Proves convergence of stochastic homogenization for ionic transport.

Shows the effective tensor satisfies Onsager symmetry and positivity.

Establishes strong flux convergence in random porous media.

Abstract

In this paper we extend the homogenization results obtained in (G. Allaire, A. Mikeli\'c, A. Piatnitski, J. Math. Phys. 51 (2010), 123103) for a system of partial differential equations describing the transport of a N-component electrolyte in a dilute Newtonian solvent through a rigid periodic porous medium, to the case of random disperse porous media. We present a study of the nonlinear Poisson-Boltzmann equation in a random medium, establish convergence of the stochastic homogenization procedure and prove well-posedness of the two-scale homogenized equations. In addition, after separating scales, we prove that the effective tensor satisfies the so-called Onsager properties, that is the tensor is symmetric and positive definite. This result shows that the Onsager theory applies to random porous media. The strong convergence of the fluxes is also established.