Development of a Generalized Version of the Poisson-Nernst-Planck Equations Using the Hybrid Mixture Theory: Presentation of 2D Numerical Examples

A numerical scheme for the transient solution of generalized version of the Poisson-Nernst-Planck equations is presented. The finite element method is used to establish the coupled non-linear matrix system of equations capable of solving the present problem iteratively. The Poisson-Nernst-Planck equations represent a set of diffusion equations for charged species, i.e. dissolved ions. These equations are coupled to the ‘internally’ induced electrical field and to the velocity field of the fluid. The Nernst-Planck equations describing the diffusion of the ionic species and the Gauss’ law in used are, however, coupled in both directions. The governed set of equations is derived from a simplified version of the so-called hybrid mixture theory (HMT). This theory is a special version of the more ‘classical’ continuum mixture theories in the sense that it works with averaged equations at macro-scale and that it includes the volume fractions of phases in its structure. The background to the Poisson-Nernst-Planck equations can by the HMT approach be described by using the postulates of mass conservation of constituents together with the Gauss’ law used together with consistent constitutive laws. The HMT theory includes the constituent forms of the quasi-static version of Maxwell’s equations making it suitable for analyzes of the kind addressed in this work. Within the framework of HTM constitutive equations has been derived using the postulate of entropy inequality together with the technique of identifying properties by Lagrange multipliers. These results will be used in obtaining a closed set of equations for the present problem.