A pressure Poisson equation-based second-order method for solving two-dimensional moving contact line problems with topological changes

We develop a second-order Cartesian grid based numerical method to solve two-dimensional moving contact line problems, which are modeled by the incompressible Navier-Stokes equations with the Navier-slip condition and the contact angle condition (CAC). The solutions of the flow field and the interface motion are decoupled in an alternating way. For a given interface, the velocity field is solved via a pressure Poisson equation formulation of the incompressible Navier-Stokes equations, which is numerically discretized by the immersed interface method. Once the velocity field is obtained, the interfacial kinematics together with the CAC is reformulated into a variational system, which is solved by the parametric finite element method (FEM). With this hybrid method, we detect topological changes in the interface by the inconsistency of neighboring normal vectors, which are directly computed through the parametric FEM. Second-order accuracy of the proposed method in both the interface and the contact line positions before and after topological changes has been numerically validated. Moreover, with the help of the numerical method, the merging and collision dynamics of droplets on the substrates are easily investigated.