Fluid Transport Models for Multi-Phase Flow Systems: Asymptotic Analysis, Homogenization, and Computation

多相流体输运是许多有趣的流体现象背后最重要的机制之一。对多相流体输运机理的研究直接或间接地推动了渐近分析、均匀化和科学计算等领域许多先进技术的产生。该项目介绍了两种流体传输模型和一种有效的数值算法，以研究可以在人体肺通道或多孔介质的孔隙和喉部中观察到的多相流体动力学。 本计画所考虑的多相流为气液两相管流及流固交互作用。对于两相管流，研究了一个多相流体模型，其中空气是湍流，薄液膜是粘性或粘弹性流体。通过匹配的渐近展开得到液体界面的演化方程，而湍流模型应用于气体流动。研究这种流动的动机来自于需要了解人类呼吸系统中控制粘液-气流耦合的流体动力学反馈机制。因此，主要研究者（PI）开发了一种隐式浸没界面法（IIM），用于三维两相管流。IIM利用了界面上法向应力的跳跃，避免了奇异表面张力的涂抹，从而保持了体积守恒，并提供了界面上数值解的高分辨率。除了研究两相管流的动力学外，PI还通过开发一种新的流体传输模型来探索肺通道中粘液清除的主要机制，该模型描绘了粘膜纤毛系统中发生的三维流体-固体相互作用。受流体-固体均质化问题的启发，这种新型模型的目标是了解粘液层中的多纤毛动力学。该模型不仅能够描述粘液纤毛相互作用的本质，而且对均匀化理论的发展具有重要的基础意义。为了研究多相流体输运，输运模型方程通常是从基本系统导出的。模型方程在数学上比基本方程简单。 它们是通过分离某种被认为在流体输送现象中起主导作用的物理机制而得出的。 虽然模型方程近似于更复杂的系统，但这些方程的数学简单性有利于分析和有效的数值计算，可以提高理论的预测能力。虽然这个项目涵盖了生物学或石油工程的基础理论和实际应用，但总体目标仍然简单明了。也就是说，为研究多相流体输运提供易于理解的模型以及最有效和最准确的数值算法。在流体动力学中，多相流体输送以多种方式发生。一个例子是人类呼吸系统中的粘液运输。在肺通路中，有气流、粘液层和纤毛层的地毯。这三个阶段相互作用，并创造了一个独特的运输模式，粘液清除。肺途径中转运机制的实现对于囊性纤维化患者肺气道内的药物递送具有极其重要的意义。提高石油采收率（EOR）过程是多相流体输送的另一个实例。在这个过程中，压缩二氧化碳被注入到老油井威尔斯，这导致多孔岩石中的油层重新排列。 这种多相输送增加了生产多年的威尔斯井的石油产量。 这种EOR技术最近已被引入到包括怀俄明州在内的几个州的石油工业中。理解这种技术背后的传输机制的数学理论和分析对于这种实践的成功至关重要。

Multi-phase fluid transport is one of the most important mechanisms behind many interesting fluid phenomena. Research on the mechanisms of multi-phase fluid transport has directly or indirectly motivated the creation of many advanced techniques in the fields of asymptotic analysis, homogenization, and scientific computing. This project introduces two fluid transport models and an efficient numerical algorithm to study multi-phase fluid dynamics that can be observed in human lung pathways or in pores and throats of a porous medium. The multi-phase flows under consideration in this project are the two-phase gas-liquid pipe flow and the fluid-solid interactions. For the two-phase pipe flow, a multi-phase fluid model is investigated, for which the air is turbulent and the thin liquid film is either a viscous or viscoelastic fluid. Evolution equations for the liquid interfaces is obtained through matched asymptotic expansions while turbulence models are applied to the gas flow. The motivation for studying this kind of flow comes from a need to understand the hydrodynamic feedback mechanisms that govern mucus-air flow coupling in the human respiratory system. Accordingly, the principle investigator (PI) has developed an implicit Immersed Interface Method (IIM) for this two-phase pipe flow in three dimensions. The IIM takes advantage of jumps of normal stresses across the interface, avoiding smearing of the singular surface tension force, and thus preserves volume-conservation and provides sharp resolution of the numerical solution across the interface. In addition to studying dynamics of the two-phase pipe flow, the PI probes the major mechanism for mucus removal in lung pathways by developing a novel fluid transport model that portrays the three-dimensional fluid-solid interaction occurring in the muco-ciliary system. Inspired by fluid-solid homogenization problems, this novel model targets an understanding of multi-ciliary dynamics in the mucus layer. This model is not only capable of describing the essence of mucus-ciliary interaction, but also bears great fundamental interest in the development of homogenization theory. To study multi-phase fluid transport, model equations for the transport are usually derived from the fundamental systems. The model equations are mathematically simpler than the fundamental ones. They are derived by isolating a certain physical mechanism that is thought to play the dominant role in the fluid transport phenomenon. Although model equations are approximate to the more complicated systems, the mathematical simplicity of these equations is advantageous to the analysis and efficient numerics that can enhance the predictive power of theories. While this project spans fundamental theories and practical applications in biology or petroleum engineering, the overall goal is, nonetheless, simple and clear. That is, to provide well-understood models as well as the most efficient and accurate numerical algorithms for studying multi-phase fluid transport. Multi-phase fluid transport occurs in fluid dynamics in myriad ways. One example is mucus transport in human respiratory systems. In lung pathways, there is air flow, a mucus layer, and a carpet of cilium layer. The three phases interact with each other and create a unique transport pattern for mucus clearance. The realization of the transport mechanism in lung pathways has been of immense importance for drug delivery inside the lung airways of cystic fibrosis patients. The Enhanced Oil Recovery (EOR) process is another example of multi-phase fluid transport. In the process, compressed carbon dioxide is injected into old oil wells, which induces a rearrangement of the oil layer in porous rocks. Such a multi-phase transport increases the oil production for wells that are in production for years. This EOR technique has been recently introduced to the oil industry in several states, including the state of Wyoming. The mathematical theory and analysis for understanding the transport mechanism behind this technique is crucial to the success of such a practice.