Fluid-elastic structure interaction with the Navier slip boundary condition

流弹性结构与纳维滑移边界条件的相互作用

• 批准号: 1613757
• 负责人: Suncica Canic
• 金额: $ 18.32万
• 依托单位: University of Houston
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-15 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1613757&HistoricalAwards=false
• 关键词: Fluid; elastic; structure; interaction; Navier

CanicDMS-1613757 The investigator studies the interaction between viscous, incompressible fluids such as water or blood, and elastic structures, such as human heart valves, modeled using a non-classical boundary condition called the Navier slip condition. This condition allows the fluid to slip over a surface, which is known to occur with hydrophobic surfaces, with human tissue constructs that have rough surfaces, and with shark's skin. Very recently it was shown that using the standard no-slip condition to study contact between structures (e.g., rigid balls) immersed in a viscous, incompressible fluid, contact can never occur in finite time. Using sophisticated mathematics, recent results also show that if the Navier slip condition is used to model contact of immersed bodies, allowing fluid to slip in between the structures leads to the structures touching each other in finite time. These recent results all address contact between rigid structures. This project goes a step further to study the interaction between fluids and elastic structures when the Navier slip boundary condition is used to model the physics of the problem. The aim is to resolve questions related to the modeling of closure of human heart valves interacting with blood flow, and flows over rough elastic surfaces such as tissue constructs interacting with blood. A series of novel mathematical and computational methods is proposed to understand these complex physical and physiological problems. Students and postdocs are involved in the project. The mathematical techniques to study these problems are based on a partitioned fluid-structure interaction algorithm. The investigator develops a novel approach to studying existence of weak solutions to this class of problems by first semi-discretizing the multi-physics coupled problem in time, and then using an operator splitting strategy to split the fluid from structure sub-problems. Particular care is used to deal with the well-known added mass effect that may lead to instabilities. The splitting approach developed here to deal with the Navier slip condition promises to provide the correct splitting strategy, which leads to the uniform energy estimates. With the construction of compactness arguments based on Simon's and Ehrling Lemmas, the existence of a weak solution is obtained by calculating the limits of the approximate sequences converging in time to a weak solution. This analysis can be used as a base for a computational algorithm in the calculation of solutions to this class of nonlinear, moving-boundary problems. The result of this work promises to contribute to the understanding of the so called "no-collision paradox," referring to the impossibility of a contact of smooth rigid bodies immersed in an incompressible, viscous fluid, modeled with the no-slip condition.

犬DMS-1613757 研究人员研究粘性不可压缩流体（如水或血液）与弹性结构（如人体心脏瓣膜）之间的相互作用，使用称为Navier滑动条件的非经典边界条件建模。 这种情况允许流体在表面上滑动，已知这发生在疏水表面、具有粗糙表面的人体组织构造和鲨鱼皮肤上。 最近的研究表明，使用标准的无滑动条件来研究结构之间的接触（例如，刚性球）浸没在粘性的、不可压缩的流体中，接触在有限时间内永远不会发生。 使用复杂的数学，最近的结果还表明，如果使用Navier滑动条件来模拟浸没体的接触，允许流体在结构之间滑动会导致结构在有限时间内相互接触。 这些最近的结果都解决了刚性结构之间的接触。 该项目进一步研究了流体和弹性结构之间的相互作用时，Navier滑移边界条件被用来模拟物理问题。 其目的是解决与血流相互作用的人类心脏瓣膜闭合建模相关的问题，以及在粗糙弹性表面上的流动，例如与血液相互作用的组织结构。 提出了一系列新的数学和计算方法来理解这些复杂的物理和生理问题。 学生和博士后都参与了这个项目。 研究这些问题的数学技术是基于分区流固耦合算法。 研究者发展了一种新的方法来研究这类问题的弱解的存在性，首先在时间上半离散化多物理耦合问题，然后使用算子分裂策略来分裂流体和结构子问题。 特别注意处理众所周知的可能导致不稳定性的附加质量效应。 这里开发的分裂方法来处理Navier滑移条件承诺提供正确的分裂策略，这导致统一的能量估计。 利用Simon引理和Ehrling引理构造紧性定理，通过计算逼近序列在时间上收敛到弱解的极限，得到了弱解的存在性. 这一分析可以作为基础的计算算法在计算这类非线性，移动边界问题的解决方案。 这项工作的结果有望有助于理解所谓的“无碰撞悖论”，指的是不可能的接触光滑的刚体浸入不可压缩的粘性流体，建模与无滑移条件。