In this article, we propose and analyze a fully coupled, nonlinear, and energy-stable virtual element method (VEM) for solving the coupled Poisson-Nernst-Planck (PNP) and Navier--Stokes (NS) equations modeling microfluidic and electrochemical systems (diffuse transport of charged species within incompressible fluids coupled through electrostatic forces). A mixed VEM is employed to discretize the NS equations whereas classical VEM in primal form is used to discretize the PNP equations. The stability, existence and uniqueness of solution of the associated VEM are proved by fixed point theory. Global mass conservation and electric energy decay of the scheme are also proved. Also, we obtain unconditionally optimal error estimates for both the electrostatic potential and ionic concentrations of PNP equations in the $H^{1}$-norm, as well as for the velocity and pressure of NS equations in the $\mathbf{H}^{1}$- and $L^{2}$-norms, respectively. Finally, several numerical experiments are presented to support the theoretical analysis of convergence and to illustrate the satisfactory performance of the method in simulating the onset of electrokinetic instabilities in ionic fluids, and studying how they are influenced by different values of ion concentration and applied voltage. These tests relate to applications in the desalination of water.

翻译：在本篇文章中,我们提议和分析一种完全结合的、非线性和能源稳定的虚拟元素方法(VEM),用于解决同时模拟微氟化和电化学系统的Poisson-Nernst-Planck(PNP)和Navier-Stokes(NS)等式(微氟化和电化学系统中充电物种通过静电力量结合压缩液体中的分解);混合的VEM用于分离NS等式,而传统的初级VEM用于分解PNP等式。相关的VEM的稳定性、存在性和独特性通过固定点理论得到证明。全球大规模保护和电能衰减也得到证明。此外,我们获得了无条件的最佳误差估计,即PNP等方在$H ⁇ 1}-诺姆中的电动潜力和电离子浓度,以及在$mathbf{H ⁇ 1}美元和$L ⁇ 2}和$L ⁇ 2}-noms的等式方程式的速和压力。最后,数项数字实验将分别用来支持这些正态和正态研究的精度的精度研究的精度的精度的精度的精度和精度的精度分析。