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.
