Ion transport in porous media: derivation of the macroscopic equations using upscaling and properties of the effective coefficients |
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Authors: | Grégoire Allaire Robert Brizzi Jean-François Dufrêche Andro Mikelić Andrey Piatnitski |
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Institution: | 1. CMAP, UMR CNRS 7641, Ecole Polytechnique, 91128, Palaiseau Cedex, France 2. Laboratoire Modélisation Mésoscopique et Chimie Théorique (LMCT), Université de Montpellier 2, Montpellier, France 3. Institut de Chimie Séparative de Marcoule ICSM UMR 5257, CEA / CNRS / Université de Montpellier 2 / ENSCM Centre de Marcoule, Bat. 426, BP 17171, 30207, Bagnols sur Cèze Cedex, France 4. Université de Lyon, CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan, 43, blvd. du 11 novembre 1918, 69622, Villeurbanne Cedex, France 5. Narvik University College, P.O. Box 385, 8505, Narvik, Norway 6. Lebedev Physical Institute RAS, Leninski ave., 53, 119991, Moscow, Russia
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Abstract: | In this work, we undertake a numerical study of the effective coefficients arising in 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, around a nonlinear Poisson–Boltzmann equilibrium with given surface charges of arbitrary size. This approach allows us to calculate the linear response regime in a way initially proposed by O’Brien. The O’Brien linearization requires a fast and accurate solution of the underlying 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 briefly discuss the corresponding two-scale asymptotic expansion and reduce the obtained two-scale equations to a coarse scale model. Our previous rigorous study proves that the homogenized coefficients satisfy Onsager properties, namely they are symmetric positive definite tensors. We illustrate with numerical simulations several characteristic situations and discuss the behavior of the effective coefficients when the Debye–Hückel parameter is large. Simulated qualitative behavior differs significantly from the situation when the surface potential is given (instead of the surface charges). In particular, we observe the Donnan effect (exclusion of co-ions for small pores). |
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