Transient electrohydrodynamic flow with concentration-dependent fluid properties: Modelling and energy-stable numerical schemes

Abstract

Transport of electrolytic solutions under influence of electric fields occurs in phenomena ranging from biology to geophysics. Here, we present a continuum model for single-phase electrohydrodynamic flow, which can be derived from fundamental thermodynamic principles. This results in a generalized Navier-Stokes-Poisson-Nernst-Planck system, where fluid properties such as density and permittivity depend on the ion concentration fields. We propose strategies for constructing numerical schemes for this set of equations, where the electrochemical and the hydrodynamic subproblems are decoupled at each time step. We provide time discretizations of the model that suffice to satisfy the same energy dissipation law as the continuous model. In particular, we propose both linear and non-linear discretizations of the electrochemical subproblem, along with a projection scheme for the fluid flow. The efficiency of the approach is demonstrated by numerical simulations using several of the proposed schemes.have often aimed for the steady-state solution to the governing equations [15,24]. To this end, commercial multi-physics software packages (e.g. Comsol) are available, and have long been successfully applied to simulate a variety microfluidic systems. With regard to the transient development of streaming potential, detailed simulations have often been limited to two-dimensional or axisymmetric geometries such as finitelength symmetric channels [25,26]. In studies of electroconvection near permselective membranes [27], both finite element [28] and (pseudo-) spectral methods [29][30][31] have proven efficient. Recently, a spectral method was also applied in a study of the interaction between electrokinetics and turbulent drag [32]. In simulations of electrokinetic flow, the electrolyte solutions are usually assumed to be dilute enough for density, viscosity and permittivity to be independent of the local ion concentrations. The ion mobilities are usually taken to be proportional to the concentrations.For the separate subproblems comprising the NSPNP problem, there exists many efficient numerical methods. For the Poisson-Nernst-Planck (PNP) problem, efficient approaches have been demonstrated for semi-conductors [33] and biological ion channels [34], where e.g. dispersion and size effects of ions can be included. For transient simulation of the Navier-Stokes equations, projection methods that date back to Chorin [35,36] (see also Guermond, Minev, and Shen [37]), have imparted speedup compared to solving the monolithic problem, since it effectively decouples the computation of velocity and pressure (although at the cost of some reduced accuracy). For the full NSPNP problem, however, less is certain, but it seems clear that succesful numerical schemes should aim to decouple, at least, the fluid mechanical subproblem from the electrochemical subproblem, and thus take advantage of the progress made in numerically resolving each of these, although a direct combination does not necessarily yield a successful scheme.In the field of diff...

Transient Simulation of Scanning Electrochemical Microscope

“…The numerical model of the SECM is based on Poisson-Nernst-Planck equations coupled with the description of electrochemical reactions. Poisson's equation describes the electric field as well as the effects of ions charge density (7):…”

Bernaise: A Flexible Framework for Simulating Two-Phase Electrohydrodynamic Flows in Complex Domains

“…This gives a chemical energy G j = α(c j ) + β ij c j = c j (ln(c j /c ref,i j ) − 1) which has a minimum at c j = c ref,i j (see also [76]).…”

“…In Ref. [76] the authors augmented the Taylor-Green vortex with electrohydrodynamics, and in this work we supplement the latter with a phase field and non-matching densities of the two phases.…”

“…However, in the case of two-phase electrohydrodynamics, the Navier-Stokes equations couple to both the electrochemical and the phase field subproblems. In Ref [76]. the authors augmented the Taylor-Green vortex with electrohydrodynamics, and in this work we supplement the latter with a phase field and non-matching densities of the two phases.We consider the full set of equations on the domain Ω = [0, 2π] × [0, 2π], where all quantities may differ in the two phases.…”

Phase Field Modeling and Numerical Algorithm for Two-Phase Dielectric Fluid Flows

“…Another diffuse interface model is proposed in [52] for electrowetting on dielectric with different densities for the two fluids, which however appears not to be Galilean invariant. In [42] a thermodynamically consistent continuum model for single-phase electrohydrodynamic flows has been described. The model combines the Navier-Stokes equations and the Poisson-Nernst-Planck (PNP) equations, in which the fluid properties depend on the ion concentration fields.…”