In this thesis, we first propose the finite element methods for simulating the moving contact line problems. The model that we used is the coupled Cahn-Hilliard and Navier-Stokes equations with the generalized Navier boundary condition(GNBC). The system is solved in a decoupled way. For the Cahn-Hilliard equations, a convex splitting finite element scheme is used. It has been shown that the scheme is unconditionally stable. For the Navier-Stokes equations, the standard mixed finite element scheme and the pressure stabilization scheme are employed. To improve the efficiency of simulation, the adaptive mesh refinement(AMR) technique has been employed for our system. We use the residual type a posteriori error estimates for our adaptive algorithm. Since the phase field variable decays much...[ Read more ]

In this thesis, we first propose the finite element methods for simulating the moving contact line problems. The model that we used is the coupled Cahn-Hilliard and Navier-Stokes equations with the generalized Navier boundary condition(GNBC). The system is solved in a decoupled way. For the Cahn-Hilliard equations, a convex splitting finite element scheme is used. It has been shown that the scheme is unconditionally stable. For the Navier-Stokes equations, the standard mixed finite element scheme and the pressure stabilization scheme are employed. To improve the efficiency of simulation, the adaptive mesh refinement(AMR) technique has been employed for our system. We use the residual type a posteriori error estimates for our adaptive algorithm. Since the phase field variable decays much faster away from the interface than the velocity variables, we use the adaptive strategy which will take into account of such difference. Some numerical experiments show that our algorithm is both efficient and reliable.

Then the algorithm is applied to the moving contact line problems on topologically rough surfaces. By considering the Gibbs free energy of the system, the Cassie and Wenzel states transition is studied on rough hydrophobic surfaces. It has been found that the crucial conditions for the two different states depend on the equilibrium contact angle, the droplet size and its initial position. Moreover, the contact angle hysteresis of droplet spreading on roughness induced superhydrophobic surfaces is investigated. We find that there are multiple local minimums and one global minimum for the Gibbs free energy landscape, which can reveal the mechanism of stick-slip phenomena. When the scale of roughness becomes smaller, the stick-slip becomes weaker and the apparent contact angle will get closer to the Cassie-Baxter's angle. Numerical simulations of our phase field model agree very well with the analytical results.

Lastly, based on the Cahn-Hilliard and Navier-Stokes system, the idea of generalized Navier boundary condition is extended to problems of flows containing three components. We show that our model, including boundary condition, is consistent with the two component model. Numerically, similar to the two phase system, we use the semi-implicit finite element method for the Cahn-Hilliard equations, and the pressure stabilization method for the Navier-Stokes equations. The adaptive mesh refinement technique is also employed to improve the efficiency of simulation. Several interesting fluid phenomena of three phase flows are investigated.