Moving contact line problem plays an important role in fluid-fluid interface motion on solid surfaces. The problem can be described by a phase-field model consisting of the coupled Cahn-Hilliard and Navier-Stokes equations with the generalized Navier boundary condition (GNBC). In this thesis, we generalize the GNBC to surfaces with complex geometry and introduce a finite element method on unstructured 3D meshes. In order to construct a stable and efficient solver for the case of large density and viscosity ratio, we combine the idea of convex-splitting [34] for the Cahn-Hilliard equation and the pressure stablization formulation [46] for the Navier-Stokes equations. Two efficient numerical methods are presented, including a linear decoupled scheme and a linearized coupled sche...[ Read more ]

Moving contact line problem plays an important role in fluid-fluid interface motion on solid surfaces. The problem can be described by a phase-field model consisting of the coupled Cahn-Hilliard and Navier-Stokes equations with the generalized Navier boundary condition (GNBC). In this thesis, we generalize the GNBC to surfaces with complex geometry and introduce a finite element method on unstructured 3D meshes. In order to construct a stable and efficient solver for the case of large density and viscosity ratio, we combine the idea of convex-splitting [34] for the Cahn-Hilliard equation and the pressure stablization formulation [46] for the Navier-Stokes equations. Two efficient numerical methods are presented, including a linear decoupled scheme and a linearized coupled scheme. Numerical experiments are carried out to validate the effectiveness and efficiency of the proposed schemes.

Accurate simulation of the interface and contact line motion requires very fine meshes, and the computation in 3D is even more challenging. Thus, the use of high performance computers and scalable parallel algorithms are indispensable. A highly parallel solution strategy using different solvers for different components of the discretization is presented. More precisely, we apply a restricted additive Schwarz preconditioned GMRES method to solve the systems arising from implicit discretization of the Cahn-Hilliard equation and the velocity equation, and an algebraic multigrid preconditioned CG method to solve the pressure Poisson system. Parallel performances show that the strategy is efficient and scalable for 3D problems on a supercomputer with a large number of processors.

We apply the proposed schemes and solution algorithms to study three important application problems, particularly for those phenomena that can not be achieved by 2D simulations. In the first application, we study numerically the dynamics of a droplet spreading on a rough solid surface. A mass compensation algorithm is introduced to preserve the mass of the droplet. On a surface with circular posts, we study how wettability of the rough surface depends on the geometry of the posts. The contact line motion for a droplet spreading over some periodic rough surfaces are also efficiently computed. Moreover, we study the motion of a droplet on an inclined surface with hydrophilic or hydrophobic properties.

In the second application, we investigate the contact line motion and flow features for a solid object impacting on a liquid surface. The motion of impact object is governed by Newton’s second law and the surface is imposed with the GNBC that accounts for the effect of interface tension. For 2D problems, we apply the technique of adaptive refinement based on a posteriori error estimation in order to achieve high numerical accuracy. We also present 3D simulations for impact objects with different flow conditions and different geometries.

In the last application, we study the capillary force hysteresis and pinning-depinning events by direct simulations of a fiber intersecting a liquid-air interface. A 3D cylindrical fiber is used to model the surface with real geometric feature as in the Atomic-Force-Microscope (AFM)-based experiments. In the numerical experiments, we investigate how the flow velocity and wetting property affect the capillary force that generated by a defect with chemical heterogeneity or geometrical roughness.