Numerical Methods for the Moving Contact Line Problem with Applications

In this thesis, efficient numerical methods are designed for a phase field model for the moving contact line (MCL) problem which consists of a coupled system of the Cahn-Hilliard and Navier-Stokes equations with the generalized Navier boundary condition. Two separate cases are considered: (1) fluids with equal density and (2) fluids with large density ratio. For the equal density case, the scheme is based on a convex splitting of the Cahn-Hilliard free energy (including the boundary energy) together with a projection method for the Navier-Stokes equations. Under certain conditions, it is shown that the scheme has the total energy decaying property and is unconditionally stable. The linearized scheme is easy to implement and introduces only mild CFL time constraint. Then the phase field model is extended to describe the MCL problem with large density ratio. To overcome the difficulty due to large density ratio, a splitting method based on a pressure Poisson equation is adopted which requires only one elliptic solver with constant coefficient per time step. The method still preserves energy decaying property and is computationally efficient, making it possible to study the sharp interface limit numerically for extremely small interface thickness. Three dimensional simulations are also included to validate the efficacy of the scheme.;The numerical method is then used to study two important application problems numerically. In the first application, we study the behavior of the friction coefficient due to MCL. By properly setting up numerical configuration and extracting physical parameters from experiments, numerical simulations are carried out to obtain accurate velocity field around MCL, from which the friction coefficient can be evaluated. A universal behavior of friction coefficient regarding viscosity, density, and contact angle is obtained which is in excellent agreement with the experiments conducted in [3].;In the second application, we have carried out three dimensional simulations of spin coating process to investigate the effect of inertia, surface tension and gravity on the formation for thin films. Fingers, often seen in spin coating process due to inhomogeneity of the substrates, are simulated by chemically patterned surfaces and the growth of fingers is illustrated by numerical results.