Homogenization of electromagnetohydrodynamic (EMHD) two-phase fluid flow through porous media.*

This work provides the derivation of a new model for immiscible flow of a wetting phase inside a non-wetting phase through the pores of a homogeneous porous medium under the effect of external electric and magnetic field. The model assumes both fluid phases to be incompressible and Newtonian, and that the solid matrix is rigid and impermeable. The porous medium, characterized by a length L, is divided into periodically structured regions of smaller length l. For mathematical and physical understanding of two-phase flow, diffuse interface model is used. The interface between the two fluids is governed by the Cahn-Hilliard equation and the influence of electro-magnetic fields is incorporated in the Navier-Stokes equation by the Lorentz force term. The research employs a two-scale asymptotic homogenization approach to upscale Navier-Stokes–Cahn–Hilliard (NSCH) equation system and derive the EM-permeability tensor by extending Darcy's law to account for multiple phases and incorporating the influence of external electromagnetic fields. The finite element method is utilized to solve the derived equations. The results indicate that wall zeta potential, capillary number (Ca), and the intensities of external magnetic and electric fields represented by the Hartmann number (Ha) and a non-dimensional parameter (S),respectively, affect the EM-permeability.