An Energy-stable Finite Element Method for the Simulation of Moving Contact Lines in Two-phase Flows

Abstract

We consider the dynamics of two-phase fluids, in particular the moving contact line, on a solid substrate. The dynamics are governed by the sharp-interface model consisting of the incompressible Navier-Stokes\slash Stokes equations with the classical interface conditions, the Navier boundary condition for the slip velocity along the wall and a contact line condition which relates the dynamic contact angle of the interface to the contact line velocity. We propose an efficient numerical method for the model. The method combines a finite element method for the Navier-Stokes/Stokes equations on a moving mesh with a parametric finite element method for the dynamics of the fluid interface. The contact line condition is formulated as a time-dependent Robin-type of boundary condition for the interface so it is naturally imposed in the weak form of the contact line model. For the Navier-Stokes equations, the numerical scheme obeys a similar energy law as in the continuum model but up to an error due to the interpolation of numerical solutions on the moving mesh. In contrast, for Stokes flows, the interpolation is not needed so we can prove the global unconditional stability of the numerical method in terms of the energy. Numerical examples are presented to demonstrate the convergence and accuracy of the numerical methods.

References in corpus (6)

• Molecular scale contact line hydrodynamics of immiscible flows
• Hydrodynamic theory of de-wetting
• A parametric finite element method for solid-state dewetting problems with anisotropic surface energies
• A Stable Parametric Finite Element Discretization of Two-Phase Navier--Stokes Flow
• On the distinguished limits of the Navier slip model of the moving contact line problem
• On the Stable Numerical Approximation of Two-Phase Flow with Insoluble Surfactant

Cited by in corpus (5)

• A Thermodynamically Consistent Model and Its Conservative Numerical Approximation for Moving Contact Lines with Soluble Surfactants
• Structure-preserving discretizations of two-phase Navier-Stokes flow using fitted and unfitted approaches
• A Finite Element Method for Electrowetting on Dielectric
• Arbitrary Lagrangian-Eulerian finite element approximations for axisymmetric two-phase flow
• A Level Set Method for the Simulation of Moving Contact Lines in Three Dimensions