In this thesis, I study a number of issues related to two-phase fluid flows and spike simulations of a biological model. In the first part of the thesis, I simulate the moving contact line in two-dimensional chemically patterned channels using a diffuse-interface model with the generalized Navier boundary condition (GNBC). A remarkable agreement with molecular dynamics (MD) simulations is obtained. Numerical results from continuum simulations are presented for the relaxational dynamics of fluid-fluid interface and an interesting phenomenon of interface breaking was observed for high wettability contrast....[ Read more ]

In this thesis, I study a number of issues related to two-phase fluid flows and spike simulations of a biological model. In the first part of the thesis, I simulate the moving contact line in two-dimensional chemically patterned channels using a diffuse-interface model with the generalized Navier boundary condition (GNBC). A remarkable agreement with molecular dynamics (MD) simulations is obtained. Numerical results from continuum simulations are presented for the relaxational dynamics of fluid-fluid interface and an interesting phenomenon of interface breaking was observed for high wettability contrast.

In the second part of the thesis, I study the dynamics of dripping-to-jetting transition for two immiscible coflowing liquid streams numerically. Two different classes of transition are identified. In both cases, nonlinear dynamical phenomena such as period doubling and chaos are observed between simple dripping and jetting. Extensive numerical calculations show that the first class of dripping-to-jetting transition is determined by the Weber number of the inner fluid W_in, and the second class of dripping-to-jetting transition is controlled by capillary number of the outer fluid C_out.

In the last part of the thesis, an adaptive numerical method is proposed to solve the Gierer-Meinhardt (GM) system on irregular domain. The method works for domains defined by level sets of implicit functions and the generated mesh is of high quality. The method is shown to be effective by comparing with asymptotic result. Boundary spike solutions of the GM system are obtained and studied numerically, including stability of boundary spike and spike motion along the boundary.