Existence and two-scale convergence of the generalised Poisson–Nernst–Planck problem with non-linear interface conditions

Abstract

The paper is devoted to the existence and rigorous homogenisation of the generalised Poisson–Nernst–Planck problem describing the transport of charged species in a two-phase domain. By this, inhomogeneous conditions are supposed at the interface between the pore and solid phases. The solution of the doubly non-linear cross-diffusion model is discontinuous and allows a jump across the phase interface. To prove an averaged problem, the two-scale convergence method over periodic cells is applied and formulated simultaneously in the two phases and at the interface. In the limit, we obtain a non-linear system of equations with averaged matrices of the coefficients, which are based on cell problems due to diffusivity, permittivity and interface electric flux. The first-order corrector due to the inhomogeneous interface condition is derived as the solution to a non-local problem.

Validating the Nernst–Planck transport model under reaction-driven flow conditions using RetroPy v1.0

“…Fickian diffusion utilizes the diagonal diffusivity matrix assumption, which is valid for many electrolytes (Dreyer et al, 2013). A diagonal diffusivity matrix means that short-range interactions between ions are negligible, which restricts our model to dilute solutions (Kontturi et al, 2008).…”

Stress-controlled hysteresis and long-time dynamics of implicit differential equations arising in hypoplasticity

“…For other representatives of incrementally nonlinear constitutive equations, see the models by Armstrong-Frederick [2], endochronic [30], octolinear [10], and CLoE [9]. For mathematical modelling granular and multiphase media we cite [1,11,12,13,22,23,24], while for well-posedness analysis we refer to [8,16,25].…”