Electro-osmotic flows through topographically complicated porous media: role of electropermeability tensor

Abstract

In the present work, we consider a framework for characterizing electro-osmotic flows in topographically complicated porous media and derive an effective up-scaled transport parameter to quantify this. We term this parameter the electro-permeability, which characterizes the electro-osmotic flow through composite porous media in analogy with Darcy's law. The electro-permeability tensor, thus introduced, serves a simple means of relating the volume flow rate with the applied electric field without going into the intricacies of the microstructure of the porous domain. First, we consider cases where the solid fractions have a fractal dimension generated by the Mandelbrot set, purely for the sake of demonstration. Based on such considerations, we employ the method of homogenization to obtain the effective electro-permeability parameter from the numerical simulations executed over a representative volume element. Our derived electro-permeability tensor components exhibit functional relationships with the solid or liquid fraction as well as the topography of the porous medium. Having established these functional relationships, we evaluate the tensor components for a binary composite porous medium in which one constituent has markedly high ζ potential than the other constituent, for illustration with potential relevance in microfluidics. We establish the sensitivity of the electro-permeability tensor on the domain morphology, solid fraction, ratio of solid fractions of the two phases having the two different ζ potential values, and the ζ potential contrast and compare it with equivalent Darcy permeability for the same. We thus provide a simple mathematical framework that may be immensely helpful for devising a computationally efficient way of characterizing electro-osmosis through topographically complicated porous media.