Interior Electroneutrality in Nernst-Planck-Navier-Stokes Systems

TL;DR

This paper analyzes the behavior of ionic charge density in fluid systems described by Nernst-Planck-Navier-Stokes equations as the Debye length approaches zero, establishing conditions for electroneutrality and convergence in various boundary regimes.

Contribution

It provides rigorous proofs of charge neutrality convergence in the vanishing Debye length limit for different boundary conditions in ionic fluid models.

Findings

Charge density converges to zero in stable boundary regimes as Debye length vanishes.

Electroneutrality is achieved exponentially fast in time under electroneutral boundary conditions.

Bounds are established for unstable boundary regimes that are uniform in time and Debye length.

Abstract

We consider the limit of vanishing Debye length for ionic diffusion in fluids, described by the Nernst-Planck-Navier-Stokes system. In the asymptotically stable cases of blocking (vanishing normal flux) and uniform selective (special Dirichlet) boundary conditions for the ionic concentrations, we prove that the ionic charge density $\rho$ converges in time to zero in the interior of the domain, in the limit of vanishing Debye length ($\epsilon\to 0$). For the unstable regime of Dirichlet boundary conditions for the ionic concentrations, we prove bounds that are uniform in time and $\epsilon$. We also consider electroneutral boundary conditions, for which we prove that electroneutrality $\rho\to 0$ is achieved at any fixed $\epsilon> 0$, exponentially fast in time in $L^p$, for all $1\le p<\infty$. The results hold for two oppositely charged ionic species with arbitrary ionic diffusivities, in bounded domains with smooth boundaries.