Mathematical modelling of fluid flow

这个建议集中在广泛的主题统一的流体流动的数学建模一个共同的主题。大部分研究集中在复杂流体上，在工程和物理科学以及地球物理学中有着广泛的应用。首要目标是利用数学方法将问题的复杂性提炼成简化模型，方便地捕捉潜在的流体物理。这些模型提供了与互补的、理想化的实验室实验相比较的见解和预测。从数学的角度来看，分析结合了渐近，变分方法和滑动线理论与专门设计的数值技术。粘塑性流体具有广泛的微观结构，产生宏观内部强度；只有当材料受到足够强的压力时，结构才会屈服，允许类似流体的流动。实际上，这些材料包括浆液和聚合物溶液、结晶熔岩、泥浆、重油、化妆品面霜和发胶、液体巧克力和粘液。因此，粘塑性在从地球物理学到钻井、食品加工和化妆品工业以及生物学等领域都有应用。关键问题包括粘塑性流体如何在重力作用下流动和静止（如泥石流和矿山尾矿的破坏），经典流体动力稳定性问题的对应问题（如富含晶体的岩浆房中的热对流），以及粘塑性如何介导流体-结构相互作用（如穿过或在泥浆和粘液之上的运动）。一个潜在的问题是粘塑性是否定性地改变和控制流动行为，或者动力学是否可以用于探测未知环境下的材料特性（提供实用的流变仪）。有时，所讨论的介质是两相、流固混合物，流体在固体应力的驱动下通过多孔基质渗透，但流体阻力导致固体基质变形。多孔介质的流动引起的压实是土力学和许多工业问题中的经典问题，其中过滤或从悬浮液中去除水是至关重要的。虽然经典理论假设基体表现为弹性固体，但在许多问题中，变形是塑性的或粘性的，或两者兼而有之，导致与单相粘塑性流体模型的联系。例如，水凝胶球或纤维素纤维的悬浮液在缓慢压缩下都表现出塑性变形，但在快速压缩下则表现出部分粘性变形。然而，卸载材料显示出一些弹性恢复。因此，这些介质的压实需要一种孔隙-弹性-粘-塑性理论。虽然水凝胶悬浮液是一种方便的实验介质（具有简单的几何形状和易于观察的固体变形），但纤维素纤维悬浮液在微观尺度上更为复杂，但对纸浆和造纸工业至关重要。

This proposal focusses on a broad range of topics united by a common theme of mathematical modelling of fluid flow. A large part of the research focusses on complex fluids, with wide applications in the engineering and physical sciences and geophysics. An overarching goal is to exploit mathematical methods to distill the complexities of a problem down to reduced models that conveniently capture the underlying fluid physics. The models provide insights and predictions to be compared with complementary, idealized laboratory experiments. From a mathematical perspective, the analysis combines asymptotics, variational methods and slipline theory with specially designed numerical techniques. Viscoplastic fluids possess an extensive microstructure that creates macroscopic internal strength; only when the material is forced sufficiently strongly will the structure yield to allow fluid--like flow. Practically, such materials include slurries and polymer solutions, crystallizing lavas, mud, heavy oils, cosmetic creams and hair gel, liquid chocolate and mucus. Consequently, viscoplasticity has application in fields from geophysics to the drilling, food processing and cosmetics industries, and to biology. Key problems include how viscoplastic fluids flow and come to rest under gravity (as in mud slides and the failure of mine tailings), the counterparts of classical hydrodynamic stability problems (such as thermal convection in crystal--rich magma chambers), and how viscoplasticity mediates fluid- structure interaction (as for locomotion through or above mud and mucus). An underlying question is whether viscoplasticity qualitatively changes and controls flow behaviour, or if the dynamics can be used to probe the material properties in settings in which those are unknown (providing a practical rheometer). Sometimes, the medium in question is a two--phase, fluid--solid mixture, with fluid percolating through a porous matrix driven by the solid stresses, but the fluid drag causing the solid matrix to deform. Flow--induced compaction of a porous medium is classical in soil mechanics and many industrial problems in which filtration or removal of water from a suspension is paramount. Although classical theory assumes the matrix behaves like an elastic solid, in many problems the deformation is plastic or viscous, or both, leading to connections with models of single--phase viscoplastic fluids. For example, suspensions of both hydrogel spheres or cellulose fibres appear to deform plastically under slow compression, but partly viscously with faster compression. Unloading the materials, however, demonstrates some elastic recovery. The compaction of these media therefore demands a theory of poro--elasto--visco-plasticity. Although hydrogel suspensions constitute a convenient experimental medium (with simple geometry and easily observed solid deformations), cellulose fibre suspensions are more complex at the microscale, but central to the pulp and paper industry.