Fluid-elastic structure interaction with the Navier slip boundary condition

研究者研究粘性不可压缩流体（如水或血液）和弹性结构（如人类心脏瓣膜）之间的相互作用，使用称为Navier滑移条件的非经典边界条件进行建模。这种情况允许液体在表面上滑动，已知这种情况发生在疏水性表面、具有粗糙表面的人体组织结构和鲨鱼皮肤上。最近有研究表明，使用标准无滑移条件来研究浸入粘性不可压缩流体中的结构（例如刚性球）之间的接触，接触在有限时间内永远不会发生。利用复杂的数学方法，最近的结果还表明，如果采用纳维尔滑移条件来模拟浸入体的接触，允许流体在结构之间滑动导致结构在有限时间内相互接触。这些最近的结果都是针对刚性结构之间的接触。本项目进一步研究了流体与弹性结构之间的相互作用，并采用了Navier滑移边界条件来模拟问题的物理性质。目的是解决与人体心脏瓣膜闭合与血流相互作用的建模相关的问题，以及粗糙弹性表面（如与血液相互作用的组织结构）上的流动。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方法是基于一种分块流固耦合算法。首先对多物理场耦合问题在时间上进行半离散，然后采用算子分裂策略将流体与结构子问题分离，从而提出了一种研究该类问题弱解存在性的新方法。特别注意处理可能导致不稳定的众所周知的附加质量效应。本文提出的用于处理Navier滑移条件的分裂方法有望提供正确的分裂策略，从而实现统一的能量估计。通过构造基于Simon引理和Ehrling引理的紧性论证，通过计算逼近序列在时间上收敛于弱解的极限，得到了弱解的存在性。这种分析可以作为计算这类非线性移动边界问题解的计算算法的基础。这项工作的结果有望有助于理解所谓的“无碰撞悖论”，即在无滑移条件下，浸入不可压缩粘性流体中的光滑刚体不可能接触。

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.