An efficient and energy stable scheme for a phase‐field model for the moving contact line problem

111

In this paper, we present two efficient energy stable schemes to solve a phase field model incorporating moving contact line. The model is a coupled system that consists of incompressible Navier-Stokes equations with a generalized Navier boundary condition and Cahn-Hilliard equation in conserved form. In both schemes the projection method is used to deal with the Navier-Stokes equations and stabilization approach is used for the non-convex Ginzburg-Landau bulk potential. By some subtle explicit-implicit treatments, we obtain a linear coupled energy stable scheme for systems with dynamic contact line conditions and a linear decoupled energy stable scheme for systems with static contact line conditions. An efficient spectral-Galerkin spatial discretization method is implemented to verify the accuracy and efficiency of proposed schemes. Numerical results show that the proposed schemes are very efficient and accurate.

Section: Introductionmentioning

confidence: 99%

Efficient energy stable numerical schemes for a phase field moving contact line model

Shen

Yang²,

Yu³

2015

111

“…Here the concepts from the aforementioned papers for the discretization of the bulk equations are straightforwardly applied. For the case of equal densities, schemes are proposed, e.g., in [14][15][16][17] and for the case of different densities in [11,18]. The model from [8] contains an additional flux term in the momentum equation, that renders the model thermodynamically consistent.…”

Section: Introductionmentioning

confidence: 99%

Comparison of energy stable simulation of moving contact line problems using a thermodynamically consistent Cahn–Hilliard Navier–Stokes model

Bonart

Kahle

Repke

2019

Liquid droplets sliding along solid surfaces are a frequently observed phenomenon in nature, e.g., raindrops on a leaf, and in everyday situations, e.g., drops of water in a drinking glass. To model this situation, we use a phase field approach. The bulk model is given by the thermodynamically consistent Cahn-Hilliard Navier-Stokes model from [Abels et al., Math. Mod. Meth. Appl. Sc., 22 (3), 2012]. To model the contact line dynamics we apply the generalized Navier boundary condition for the fluid and the dynamically advected boundary contact angle condition for the phase field as derived in [Qian et al., J. Fluid Mech., 564, 2006]. In recent years several schemes were proposed to solve this model numerically. While they widely differ in terms of complexity, they all fulfill certain basic properties when it comes to thermodynamic consistency. However, an accurate comparison of the influence of the schemes on the moving contact line is rarely found. Therefore, we thoughtfully compare the quality of the numerical results obtained with three different schemes and two different bulk energy potentials. Especially, we discuss the influence of the different schemes on the apparent contact angles of a sliding droplet.

“…• Velocity boundary condition: On the velocity field u, we assign a general Navier slip condition (see e.g. [53,54]),…”

Section: Boundary Conditionsmentioning

confidence: 99%

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

Linga

Bolet

Mathiesen

2020

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...