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Preprint-No.: <   432   >   Published in: October 2015   PDF-File: IGPM432.pdf
Title:Finite Element Methods for a Class of Continuum Models for Immiscible Flows with Moving Contact Lines
Authors:Arnold Reusken, Xianmin Xu, Liang Zhang
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 at wall and a spherical droplet on a at 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.
Keywords:moving contact line, General Navier Boundary Condition(GNBC), sharp interface, level set, Nitsche's method, extended finite element method