Numerical methods and simulations of moving contact line problems

Abstract

We develop a second-order accurate sharp interface method to simulate moving contact line (MCL) problems. Based on the principle of total free energy dissipation, we derive the boundary conditions including the interfacial conditions, the Navier-slip boundary condition, and the contact angle condition. These conditions together with either the incompressible Navier-Stokes equations or the incompressible Stokes equations form a continuum model. Our models relieve the contact line singularity. The immersed interface method (IIM) has been widely used in simulations of multiphase flows with closed interfaces. We generalize the IIM to solve for the velocity field in the MCL problems. With the help of variational formulation, the contact angle condition can be combined with the interfacial kinematics in a weak form. A parametric finite element method (parametric FEM) is applied to solve for the interface motion as well as the curvature, which are in turn used to update the correction terms for the irregular points in the IIM. The hybrid IIM-parametric FEM method is Cartesian grid-based, and achieves second-order accuracy not only in the velocity field but also in the interface and the contact line motion. This is validated by numerical results. Moreover, we generalize the method to account for discontinuous viscosity and topological changes, where the order of accuracy is preserved. Various numerical experiments are presented in the study of droplet motion and contact angle hysteresis (CAH). We observe periodically stick-slip behavior and find that the velocity dependence of the CAH can be symmetric and asymmetric in different cases. We investigate the merging and collision of droplets and learn that the inertial effect concerns not only the interface motion but also the appearance of topological changes.