Two Problems in Liquid-Solid Interaction

液固相互作用的两个问题

• 批准号: 2307811
• 负责人: Giovanni Galdi
• 金额: $ 29.98万
• 依托单位: University of Pittsburgh
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2307811&HistoricalAwards=false
• 关键词: Two; Problems; Liquid; Solid; Interaction

Over the past decades, the study of the motion of a viscous liquid interacting with solid bodies has become one of the main focuses of applied research. The objective of this project is to address two fundamental aspects in this area. The first regards the vibration-induced motion of a rigid body in a viscous liquid, when the vibration is produced by a time-periodically displaced mass inside the body. The wide area of applications of this phenomenon includes biomedical engineering and design of micro- and nano-technological equipment. The second aspect concerns the effect of flow-induced oscillations on a spring-mounted structure. These type of questions are of the utmost relevance in the study of the equilibrium and stability of suspension bridges and, in some cases, of their collapse, like in the notorious Tacoma Narrows Bridge disaster. This award will also provide opportunities for the involvement of graduate students in the project's research.Both projects fall into the realm of non-linear analysis and their study requires the use of fresh mathematical ideas that, possibly, will be employed also in other areas of applied nonlinear PDE's. First and foremost is the investigation of (local and global) Hopf bifurcation in a liquid-solid interaction problem in presence of a continuum spectrum of the relevant linearized operator. To date, this question represents an entirely uncharted territory. Another and not less important mathematical feature of the proposed research consists in the detailed study of the range of nonlinear elliptic operators (related to Navier-Stokes-like boundary-value problems) that are Fredholm of negative index. Interestingly enough, this kind of analysis may lead to the rigorous quantitative understanding of how and why a vibrating sphere can move in a viscous liquid.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.

在过去的几十年中，对粘性液体与固体相互作用的运动的研究已成为应用研究的主要重点之一。该项目的目的是解决该领域的两个基本方面。 当振动是由体内的时间周期移位的质量产生的时，第一个介绍了粘性液体中刚体的振动引起的运动。这种现象的广泛应用包括微型和纳米技术设备的生物医学工程和设计。第二个方面涉及流动诱导的振荡对弹簧安装结构的影响。这些类型的问题在研究悬架桥的平衡和稳定性方面至关重要，在某些情况下，它们的崩溃，例如在臭名昭著的塔科马（Tacoma Narrows）狭窄的桥梁灾难中。该奖项还将为研究生参与该项目的研究提供机会。两个项目都属于非线性分析领域，他们的研究需要使用新的数学思想，这些思想可能也将在应用非线性PDE的其他领域中使用。首先，在存在相关线性化操作员的连续频谱的情况下，在液体固体相互作用问题中对（局部和全局）HOPF分叉进行了研究。迄今为止，这个问题代表了一个完全未知的领域。 拟议的研究的另一个且同样重要的数学特征在于对非线性椭圆运算符（与Navier-Stokes样边界值问题有关的范围）的详细研究，即弗雷德（fredholm）的负指数。有趣的是，这种分析可能会导致对振动球如何以及为什么在粘性液体中移动的严格定量理解。该奖项反映了NSF的法定任务，并被认为是通过基金会的智力优点和更广泛影响的评估标准来通过评估来获得支持的。