A Second Order Accuracy Preserving Method for Moving Contact Lines with Stokes Flow

Abstract

The immersed interface method (IIM) has been widely used in simulations of multiphase flows with closed interfaces. We generalize IIM to simulate the moving contact line problems, which are modeled by the Stokes equation with the Navier-slip boundary condition and the contact angle condition. 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 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. Various numerical experiments are presented in the study of droplet motion and contact angle hysteresis.