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

这一建议涉及发展和分析新的数值方法的Navier-Stokes方程涉及不连续，自由边界，移动界面，和问题的不规则区域。新方法将应用于多相流、流体结构与界面的相互作用、流体或固体的几个重要应用。本文提出了增强浸入式界面方法来解决这些难点问题。新方法基于锐界面模型（无涂抹）、二阶和隐式离散化。提出了新的方法来模拟一些重要的应用，如用一相流和两相流模型模拟移动接触线问题；不可压缩流中的不可扩展接口、开放式接口和带有质量的移动接口。所述项目在数学生物学、材料科学、空气动力学和机械工程等领域具有广泛的应用。在生命科学中，不可扩展接口已被用来模拟红细胞的运动。高效的数值模拟将对红细胞相关疾病的早期发现产生积极的影响。运动接触线问题在材料科学中有直接的应用。利用计算机进行有效的数值模拟可以避免昂贵的实验。提案中的其他项目也是如此。该提案还将对教育产生积极影响，吸引本科生和研究生进行该领域的研究。项目的一些组成部分将被设计为本科项目。此外，研究结果还可以为涉及自由边界和移动界面的计算科学以及复杂域上定义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.