Global Well-Posedness and Stability of Electrokinetic Flows
Dieter Bothe, André Fischer, Jürgen Saal · SIAM Journal on Mathematical Analysis · 2014
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 $\mathbb{R}^n$ 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.