Two-scale model of quasi-steady flow of electrolyte in weakly piezoelectric porous media

Abstract

This paper presents a new homogenized model of two-component electrolyte transport through a weakly piezoelectric porous medium. The model relevant to the microscopic scale describes quasi-stationary states of the medium while reflecting essential physical phenomena, such as electrochemical interactions in a dilute Newtonian solvent under assumptions of slow flow. The dimensional analysis of the mathematical model introduces scaling of the viscosity, electric permittivity, piezoelectric coupling, and dielectric tensor. The micromodel is linearized around the reference state represented by the thermodynamic equilibrium and further upscaled through the asymptotic homogenization method. Due to the scaling of parameters, the derived limit model retains the characteristic length associated with the pore size and the electric double-layer thickness. The upscaling procedure gives a fully coupled macroscopic model describing the electrolyte flow in terms of a global pressure and streaming potentials of the two ionic species in the weakly piezoelectric matrix. By virtue of the characteristic responses, quantities of interest are reconstructed at the microscopic scale using the resolved macroscopic fields. The coupling between electrochemical and mechanical phenomena influenced by the skeleton piezoelectricity is illustrated using numerical examples. The model, which was motivated by the cortical bone porous tissue, is widely applicable in designing new biomaterials involving piezoelectric stimulation.