Modelling and simulation of complex fluid flows with interfaces, multi-physics and multiple scales

这一建议涉及流体动力学中的问题，即潜在的流体流动与其他一些物理、生物或化学过程相互作用。感兴趣的流动来自各种工业、工程和生物医学应用，这些应用可以组织在两个主题下： 主题一：涉及浸没在不可压缩流体中的可变形、弹性结构的流固相互作用。应用包括木浆纤维悬浮液的流动；充满流体的器官的生物力学，如心脏或耳蜗(或内耳)；以及生物的游泳行为，如水母、鱿鱼或海洋蠕虫。 主题II：多孔介质中的多相流动，其中流动与热传递、相变或反应化学相耦合。应用包括树木中的树液流动；土壤或混凝土中的水分吸收；以及氢燃料电池中冷凝气体的流动。 我解决所有这些问题的方法遵循一个共同的策略：(1)基于仔细的物理推理建立一个详细的数学模型；(2)使用诸如渐近性、线性稳定性分析或周期齐化等数学技术来简化控制方程，并了解解的行为；以及(3)开发一种准确而高效的数值算法，该算法可用于执行广泛的模拟，并根据解析解和实验数据验证模型。 主题一和主题二下列出的问题包括各种物理现象和数值求解技术。然而，有几个共同的方面统一了这项工作中出现的建模和计算问题： *解组件之间具有强耦合的非线性偏微分方程组； *存在内部或边界层以及可受益于使用多尺度分析和/或自适应数值方法的各种空间尺度；以及 *由变化很大的时间尺度引起的数值刚性，需要使用隐式时间步进算法。 这一建议的一个决定性特征是数学分析和算法开发之间的紧密相互作用，其中利用关于解析解的知识来做出关于选择数值方法和具体算法改进的明智决定。这项工作将对开发新的求解算法以及推进复杂的多物理流动现象的知识产生重大影响。例如，对心脏血流不稳定性的详细了解可以提高对心律失常的理解，对枫树树液流动的新见解可能被用来优化树液收集实践。许多子项目涉及与公司或其他非学术合作伙伴的合作，这将有助于确保成果实际应用于行业。

This proposal is concerned with problems in fluid dynamics where an underlying fluid flow interacts with some other physical, biological or chemical process. The flows of interest originate from a variety of industrial, engineering and biomedical applications that can be organized under two themes: Theme I: Fluid-structure interaction involving deformable, elastic structures immersed in an incompressible fluid. Applications include flow of wood pulp fiber suspensions; biomechanics of fluid-filled organs such as the heart or cochlea (or inner ear); and swimming behavior of organisms such as jellyfish, squid or marine worms. Theme II: Multiphase flow in a porous medium, where the flow is coupled with heat transport, phase change, or reaction chemistry. Applications include sap flow in trees; water uptake in soil or concrete; and flow of condensing gases in hydrogen fuel cells. My approach to solving all of these problems follows a common strategy: (1) develop a detailed mathematical model based on careful physical reasoning; (2) employ mathematical techniques such as asymptotics, linear stability analysis or periodic homogenization to simplify the governing equations and to obtain insight into solution behavior; and (3) develop an accurate and efficient numerical algorithm, which can be used to perform extensive simulations and validate the model against both analytical solutions and experimental data. The problems listed under Themes I and II encompass a diverse range of physical phenomena and numerical solution techniques. Nevertheless, there are several common aspects that unify the modelling and computational issues arising in this work: * systems of nonlinear partial differential equations with strong coupling between solution components; * presence of interior or boundary layers and a wide range of spatial scales that can benefit from the use of multiscale analysis and/or adaptive numerical methods; and * numerical stiffness arising from widely varying time scales that necessitates use of implicit time-stepping algorithms. A defining characteristic of this proposal is the tight interplay between mathematical analysis and algorithm development, wherein knowledge about the analytical solution is exploited to make informed decisions about the choice of numerical method and specific algorithmic improvements. This work should have significant impact in terms of developing novel solution algorithms as well as advancing knowledge of complex multi-physics flow phenomena. For example, detailed knowledge of cardiac flow instabilities could lead to improved understanding of heart arrhythmias, and new insights into sap flow in maple trees may be used to optimize sap harvesting practices. Many sub-projects involve collaborations with companies or other non-academic partners, which will help to ensure that results are actually put into practical use in industry.