Finite element methods for a class of continuum models for immiscible flows with moving contact lines

TL;DR

This paper develops a finite element method for simulating two-phase incompressible flows with moving contact lines, incorporating surface tension, Marangoni effects, and contact line dynamics, validated through various test cases.

Contribution

It introduces a unified FEM framework with level-set, XFEM, and Nitsche techniques for accurately modeling complex contact line phenomena.

Findings

Validated for stationary droplets and spreading cases

Accurately captures contact line dynamics and surface tension effects

Demonstrates robustness on complex geometries

Abstract

In this paper we present a finite element method (FEM) for two-phase incompressible flows with moving contact lines. We use a sharp interface Navier-Stokes model for the bulk phase fluid dynamics. Surface tension forces, including Marangoni forces and viscous interfacial effects, are modeled. For describing the moving contact we consider a class of continuum models which contains several special cases known from the literature. For the whole model, describing bulk fluid dynamics, surface tension forces and contact line forces, we derive a variational formulation and a corresponding energy estimate. For handling the evolving interface numerically the level-set technique is applied. The discontinuous pressure is accurately approximated by using a stabilized extended finite element space (XFEM). We apply a Nitsche technique to weakly impose the Navier slip conditions on the solid wall. A unified approach for discretization of the (different types of) surface tension forces and contact line forces is introduced. The numerical methods are first validated for relatively simple test problems, namely a stationary spherical droplet in contact with a flat wall and a spherical droplet on a flat wall that spreads or contracts to a stationary form. A further validation is done for a two-phase Couette flow with contact lines. To illustrate the robustness of our FEM we also present results of simulations for a problem with a curved contact wall and for a problem with more complicated contact line dynamics.