Long Time Dynamics of Nonequilibrium Electroconvection

TL;DR

This paper analyzes the long-term behavior of the Nernst-Planck-Stokes system modeling electroconvection, establishing the existence of a finite-dimensional attractor and electroneutrality in the zero Debye length limit.

Contribution

It provides quantitative bounds on solutions, proves the existence of a global attractor with finite fractal dimension, and demonstrates electroneutrality as Debye length approaches zero.

Findings

Existence of a compact global attractor with finite fractal dimension.

Space-time averaged electroneutrality in the zero Debye length limit.

Quantitative bounds on solutions of the NPS system over long times.

Abstract

The Nernst-Planck-Stokes (NPS) system models electroconvection of ions in a fluid. We consider the system, for two oppositely charged ionic species, on three dimensional bounded domains with Dirichlet boundary conditions for the ionic concentrations (modelling ion selectivity), Dirichlet boundary conditions for the electrical potential (modelling an applied potential), and no-slip boundary conditions for the fluid velocity. In this paper, we obtain quantitative bounds on solutions of the NPS system in the long time limit, which we use to prove 1) the existence of a compact global attractor with finite fractal (box-counting) dimension and 2) space-time averaged electroneutrality $\rho\approx 0$ in the singular limit of Debye length going to zero, $\epsilon\to 0$.