Global well-posedness and stability of electro-kinetic flows

TL;DR

This paper proves the existence and stability of solutions for a coupled electro-kinetic flow model involving Navier-Stokes, Nernst-Planck, and Poisson equations, relevant for micro- and nanofluidics.

Contribution

It establishes local and global well-posedness and stability results for the full electro-kinetic system without assuming electroneutrality.

Findings

Existence of unique local strong solutions in bounded domains for any dimension.

Existence of unique global strong solutions in two dimensions.

Exponential convergence to steady states in two-dimensional cases.

Abstract

We consider a coupled system of Navier-Stokes and Nernst-Planck equations, describing the evolution of the velocity and the concentration fields of dissolved constituents in an electrolyte solution. Motivated by recent applications in the field of micro- and nanofluidics, we consider the model in such generality that electrokinetic flows are included. This prohibits employing the assumption of electroneutrality of the total solution, which is a common approach in the mathematical literature in order to determine the electrical potential. Therefore we complement the system of mass and momentum balances with a Poisson equation for the electrostatic potential, with the charge density stemming from the concentrations of the ionic species. For the resulting Navier-Stokes-Nernst-Planck-Poisson system we prove the existence of unique local strong solutions in bounded domains in $\R^n$ for any $n\geq2$ as well as the existence of unique global strong solutions and exponential convergence to uniquely determined steady states in two dimensions.