Augmented methods for Navier Stokes equations involving free boundary and moving interface and applications

该建议涉及的发展和分析的新的数值方法的Navier-Stokes方程，涉及不连续，自由边界，移动界面，和不规则域的问题。新方法将被应用于多相流、流体结构与界面、流体或固体的相互作用等几个重要应用领域。在该方案中，提出了增强沉浸式界面方法来解决这些困难的问题。新的方法是基于尖锐的界面模型（无涂抹），二阶和隐式离散。提出了新的方法来实现模拟几个重要的应用，如移动接触线问题，使用一个和两相流模型;不可压缩流中的不可扩展接口，开放式接口，移动接口与群众。 建议书中描述的项目在数学生物学、材料科学、空气动力学和机械工程中有广泛的应用。在生命科学中，不可扩展的接口被用来模拟红细胞的运动。有效的数值模拟将对早期发现与红细胞相关的疾病产生积极影响。动接触线问题在材料科学中有着直接的应用。使用计算机进行有效的数值模拟可以避免昂贵的实验。对于提案中的其他项目也是如此。这项建议对教育亦有正面影响，可吸引本科生和研究生在这方面进行研究。项目的一些组成部分将被设计为本科项目。此外，研究结果还可为涉及自由边界和运动界面的计算科学以及复杂域上的问题提供有用的计算机软件包。

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 inmulti-phase flows, fluid structure interactions with interfaces, fluids, or solids. In the proposal, augmented immersed interface methods are proposedto 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 movingcontact line problem has direct applications in the material science. Efficient numerical simulations using computers may avoid expensive experiments. Thisis 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 theprojects will be designed as undergraduate projects. Furthermore, the research results can lead to a computer software package that would be useful to thecomputational science involving free boundary and moving interface, and problems defined on complicated domain.