Analysis of Free Boundaries: Contact Lines and Viscous Traveling Waves

自由边界分析：接触线和粘性行波

• 批准号: 2204912
• 负责人: Ian Tice
• 金额: $ 36.85万
• 依托单位: Carnegie-Mellon University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-15 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2204912&HistoricalAwards=false
• 关键词: Analysis; Free; Boundaries; Contact; Lines

Fluids with free boundaries occur universally in nature. At a human scale, we encounter them daily in our cups of coffee or on the surface of a pond. Important examples at smaller scales include the flow of blood and air through our cardiovascular and respiratory systems, while examples at larger scales include the Earth's ocean and atmosphere or even the hot plasma on the surface of a star. Since free boundaries are so common and occur at so many scales, it is important to understand the role they play in fluid mechanics. The main scientific goal of this project is to contribute to the understanding of such free boundaries through the mathematical study of the partial differential equations that describe their dynamics. This involves not only the application of existing mathematical tools to these problems, but also the development of new tools and techniques, which in turn may be useful in the study of other problems. The project also aims to contribute to the development of the next generation of researchers at multiple levels: at the undergraduate level through the project's Summer Analysis Program, which will provide undergraduate research opportunities to eight students; and at the graduate level through graduate research and mentoring.This project aims at studying fundamental questions of well-posedness and stability for two long-standing open problems in interfacial mechanics. The first is the contact line problem, which deals with the dynamics of a triple interface between a viscous fluid, a solid, and a vapor phase. This component of the project aims to verify the soundness of a recently proposed continuum model of contact lines. This has the potential to make a serious impact in the science of triple-phase junctions, to open many new lines of research, and to further the applications of contact line dynamics. From an analytic perspective, it will also create and further develop bridges between energy methods, the functional calculus of differential operators, and elliptic regularity theory, which will be useful in many other contexts. The second problem is the viscous traveling wave problem. While the existence of traveling wave solutions to the free boundary, incompressible Euler equations has been known for nearly a century, progress on the corresponding Navier-Stokes problem only began recently with the work of the PI and collaborators. The proposal aims to build on this work to construct traveling wave solutions in more general contexts and to study their dynamical stability and vanishing viscosity limits. It is important to account for the viscous case because, while many fluids have small viscosity (or more precisely, the fluid configuration has large Reynolds number), small does not mean zero, so all fluids experience some viscous effects. Developing the viscous theory also opens the possibility of connecting the viscous and inviscid cases through vanishing viscosity limits, which could potentially yield insight into the zoo of known inviscid solutions. In particular, it could lead to a selection mechanism for physically relevant inviscid solutions. This work will also contribute fundamental new tools in the theory of function spaces, which will be useful in other traveling wave problems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

具有自由边界的流体在自然界中普遍存在。在人类的尺度上，我们每天都在咖啡杯里或池塘的水面上遇到它们。小尺度的重要例子包括血液和空气通过我们的心血管和呼吸系统的流动，而大尺度的例子包括地球的海洋和大气，甚至是星星表面的热等离子体。由于自由边界是如此普遍，并出现在这么多的尺度，重要的是要了解他们在流体力学中发挥的作用。该项目的主要科学目标是通过对描述其动力学的偏微分方程的数学研究，促进对这种自由边界的理解。这不仅涉及到现有的数学工具，这些问题的应用程序，但也发展新的工具和技术，这反过来可能是有用的研究其他问题。该项目还旨在为多层次的下一代研究人员的发展做出贡献：在本科阶段，通过该项目的夏季分析方案，该方案将为8名学生提供本科研究机会;在研究生水平，通过研究生的研究和指导。这个项目的目的是研究基本问题的适定性和稳定性的两个长期-界面力学中悬而未决的问题。第一个是接触线问题，它涉及粘性流体，固体和汽相之间的三重界面的动力学。该项目的这一部分旨在验证最近提出的接触线连续模型的合理性。这有可能在三相结科学中产生严重影响，开辟许多新的研究领域，并进一步推动接触线动力学的应用。从分析的角度来看，它还将创建和进一步发展能量方法，微分算子的泛函微积分和椭圆正则性理论之间的桥梁，这将在许多其他情况下有用。第二个问题是粘性行波问题。虽然存在的行波解的自由边界，不可压缩的欧拉方程已经知道了近世纪，相应的Navier-Stokes问题的进展最近才开始与PI和合作者的工作。该提案旨在建立在这项工作的基础上，在更一般的情况下构建行波解，并研究其动态稳定性和消失的粘度限制。考虑粘性情况很重要，因为虽然许多流体具有小的粘度（或者更准确地说，流体配置具有大的雷诺数），但小并不意味着零，因此所有流体都会经历一些粘性效应。粘性理论的发展也打开了通过消失的粘性极限将粘性和无粘性情况联系起来的可能性，这可能会使人们深入了解已知的无粘性解决方案。特别是，它可以导致物理相关的无粘性解决方案的选择机制。这项工作也将有助于基本的新工具，在理论的功能空间，这将是有用的，在其他行波problem.This奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。