Homogenization of the linearized ionic transport equations in rigid periodic porous media
Grégoire Allaire, Andro Mikelić, Andrey Lvovich Piatnitski · Journal of Mathematical Physics · 2010
In this paper we undertake the rigorous homogenization of a system of partial differential equations describing the transport of a N-component electrolyte in a dilute Newtonian solvent through a rigid porous medium. The motion is governed by a small static electric field and a small hydrodynamic force, which allows us to use O’Brien's linearized equations as the starting model. We establish convergence of the homogenization procedure and discuss the homogenized equations. Even if the symmetry of the effective tensor is known from the literature [J. R. Looker and S. L. Carnie, Transp. Porous Media, 65, 107 (2006)], its positive definiteness does not seem to be known. Based on the rigorous study of the underlying equations, we prove that the effective tensor satisfies Onsager properties, namely is symmetric positive definite. This result justifies the approach of many authors who use Onsager theory as starting point.