Modelling and simulation of biofluid mechanics with multiphysics and multiple scales

我建议探索生物流体力学中流体流动与其他生物或物理过程相互作用的问题。我的目标是通过分析和计算方法的结合，对这些复杂的流动有更深入的了解，并发现新的见解。感兴趣的问题分为两个主题：(I)涉及浸入不可压缩流体中的可变形、弹性结构的流体-结构相互作用。应用范围包括悬挂活跃游泳者，如海洋蠕虫和水母；以及充满液体的耳蜗（或内耳）的生物力学。（二）微结构多孔介质中的多相流动与输运。主要研究的重点是树液流及其与营养物质运输和其他生物物理过程的耦合，更具体地说，是驱动糖枫液渗出的冻融过程。另一个影响范围更广的树木物种的相关问题是冷冻引起的栓塞。这两个主题包含了一个看似不同的物理和生物现象的集合，但一些共同的特征统一了我们的建模和计算方法：*动力学是由具有强耦合解分量的非线性偏微分方程系统控制的；*解决方案的特点是内层和边界层以及空间尺度的明确分离，自然受益于多尺度分析和自适应数值方法的应用；广泛变化的时间尺度会产生数值刚度，这需要使用隐式时间步进算法。我对所有这些问题都采用了类似的解决方法：(1)基于仔细的生物和物理推理推导出详细的数学模型；(2)利用渐近性、线性稳定性和周期均匀化等分析技术简化控制方程，并深入了解解的行为；(3)开发准确有效的数值方案，使广泛的参数研究能够根据分析解和实验数据验证模型。该建议的一个定义特征是数学分析和算法开发之间的紧密相互作用，其中对解决方案的数学结构的洞察力被利用来对数值方法做出明智的决策。这项工作将对解决生物流体动力学中复杂问题的新算法以及推进多物理场和多尺度流动现象的知识产生重大影响。例如，对内耳中活性细胞结构与耳蜗液之间相互作用的研究应有助于提高对哺乳动物听力机制的理解。此外，对控制枫树汁液流动的细胞过程的多尺度性质的新见解将导致树木生理学的进步以及提高枫树糖浆工业的汁液产量。

I propose to explore problems in biofluid mechanics where the fluid flow interacts with other biological or physical processes. My aim is to develop a deeper understanding of these complex flows and uncover new insights through a combination of analytical and computational methods. The problems of interest fall under two themes: (I) Fluid-structure interaction involving deformable, elastic structures immersed in incompressible fluids. Applications include suspensions of active swimmers such as marine worms and jellyfish; and biomechanics of the fluid-filled cochlea (or inner ear). (II) Multiphase flow and transport in micro-structured porous media. The primary focus is on the study of sap flow in trees and the coupling with nutrient transport and other bio-physical processes, more specifically the freeze-thaw process that drives sap exudation in sugar maple. Another related problem that impacts a much wider range of tree species is freeze-induced embolism.  These two themes comprise a seemingly diverse collection of physical and biological phenomena, but a few common features unify our modelling and computational approaches: * dynamics are governed by nonlinear systems of partial differential equations with strongly-coupled solution components; * solutions are characterized by interior and boundary layers and a clear separation of spatial scales, which naturally benefit from the application of multiscale analysis and adaptive numerical methods; and * widely varying time scales that give rise to numerical stiffness, and which require the use of implicit time-stepping algorithms. I apply a similar solution methodology to all of these problems: (1) deriving a detailed mathematical model based on careful biological and physical reasoning; (2) employing analytical techniques such as asymptotics, linear stability, and periodic homogenization to simplify the governing equations and obtain insight into solution behavior; and (3) developing accurate and efficient numerical schemes that enable extensive parameter studies to validate the model against analytical solutions and experimental data. A defining characteristic of this proposal is the tight interplay between mathematical analysis and algorithm development, where insight into the mathematical structure of solutions is exploited to make informed decisions on numerical methods. This work will have significant impact in terms of novel algorithms for solving complex problems in biofluid dynamics, as well as advancing knowledge of multi-physics and multiscale flow phenomena. For example, studies of interactions between active cell structures and cochlear fluid in the inner ear should lead to improved understanding of mammalian hearing mechanisms. Moreover, new insights into the multiscale nature of cellular processes governing sap flow in maple trees will lead to advances in tree physiology as well as improving sap yields in the maple syrup industry.