Abstract
By the two-scale homogenization approach we justify a two-scale model of ion transport through a layered membrane, with flows being driven by a pressure gradient and an external electrical field. By up-scaling, the electroosmotic flow equations in horizontal thin slits separated by thin solid layers are approximated by a homogenized system of macroscale equations in the form of the Poisson equation for induced vertical electrical field and Onsager's reciprocity relations between global fluxes (hydrodynamic and electric) and forces (horizontal pressure gradient and external electrical field). In addition, the two-scale approach provides macroscopic mobility coefficients in the Onsager relations. On this way, the cross-coupling kinetic coefficient is obtained in a form which does involves the ζ -potential among the data provided the surface current is negligible.
Highlights
In numerous studies on the electrolyte flows in rocks, the pore pressure p and the streaming potential interplay through the equation j = L0 p r, where j is the current density, r is the saturated rock conductivity, and L0 is the electrokinetic crosscoupling term
By the two-scale homogenization approach we justify a two-scale model of ion transport through a layered membrane, with flows being driven by a pressure gradient and an external electrical field
Restricting ourselves to one-dimensional flows, we derive a representation formula for L0 by the two-scale homogenization technique [11,12], starting from the equations of the ions transport through a layered membrane with a periodical structure
Summary
Restricting ourselves to one-dimensional flows, we derive a representation formula for L0 by the two-scale homogenization technique [11,12], starting from the equations of the ions transport through a layered membrane with a periodical structure. On this way we arrive at electro-osmotic macro-equations, whereas electrokinetic coupling coefficients can be determined from micro-equations defined on the periodicity cell. The two-scale homogenization is a well established method in the theory of partial differential equations with rapidly oscillating periodic coefficients This method has a lot of important applications in various branches of physics, mechanics and modern technology: porous media, composite and perforated materials, thermal conduction, acoustics, electromagnetism.
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