This proposal concerns the development and analysis of new numerical methods for Navier-Stokes equations involving discontinuities, free boundaries, moving interfaces, and problems on irregular domains. The new methods will be applied to several important applications in multi-phase flows, fluid structure interactions with interfaces, fluids, or solids. In the proposal, augmented immersed interface methods are proposed to tackle those difficulty problems. The new methods are based on sharp interface models (no smearing), second order and implicit discretization. The new methods are proposed to be implemented to simulate several important applications such as the moving contact line problem using one and two phase flow models; non-extensible interfaces in incompressible flows, open-ended interfaces, and moving interfaces with masses.
The projects described in the proposal have wide applications in mathematical biology, material science, aerodynamic and mechanical engineering. The non-extensible interfaces have been used to model the motion of red blood cells in the life science. Efficient numerical simulations will have positive impact on early detection of illness related to the red-blood cells. The moving contact line problem has direct applications in the material science. Efficient numerical simulations using computers may avoid expensive experiments. This is true for other projects in the proposal as well. This proposal will also have positive effect on education by attracting under and graduate students to conduct research in this area. Some components of the projects will be designed as undergraduate projects. Furthermore, the research results can lead to a computer software package that would be useful to the computational science involving free boundary and moving interface, and problems defined on complicated domain.
