A least-squares/finite element method for the numerical solution of the Navier-Stokes-Cahn-Hilliard system modeling the motion of the contact line

In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: A least-squares/finite element method for the numerical solution of the Navier–Stokes-Cahn–Hilliard system modeling the motion of the contact line Keywords: Navier–Stokes Cahn–Hilliard Operator-splitting Least squares Conjugate gradient Contact line a b s t r a c t In this article we discuss the numerical solution of the Navier–Stokes-Cahn–Hilliard system modeling the motion of the contact line separating two immiscible incompressible vis-cous fluids near a solid wall. The method we employ combines a finite element space approximation with a time discretization by operator-splitting. To solve the Cahn–Hilliard part of the problem, we use a least-squares/conjugate gradient method. We also show that the scheme has the total energy decaying in time property under certain conditions. Our numerical experiments indicate that the method discussed here is accurate, stable and efficient. Describing accurately an immiscible two-phase flow in the vicinity of the contact line, where the fluid–fluid interface intersects the solid wall, is a classical and challenging problem in hydrodynamics. In the past two decades, it has been shown through Molecular Dynamics (MD) simulations that, indeed, a near complete slip does occur at the moving contact line (MCL) [16,17,27,26]. Through the analysis of extensive MD data, it was recently discovered that there is indeed a differential boundary condition, denoted the generalized Navier boundary condition (GNBC), which resolves the MCL problem [19]. In [19], one gives a continuum formulation of the immiscible flow hydrodynamics, including the GNBC, the Navier–Stokes equations, and the Cahn–Hilliard interfacial free energy. It is shown that the numerical results based on the GNBC can reproduce quantitatively the results from the MD simulation. This indicates that the new model can accurately describe the behavior near the contact line. The model has also been used to study several problems involving moving contact line [4,20,28]. A numerical method for the solution of the above coupled system is discussed in [19]; it relies on a standard explicit finite difference scheme whose stability requires a quite demanding limitation of the time step because of the high (fourth) order