Mathematical modelling of fluid flow

这项提议要求资金支持广泛的主题，这些主题因流体流动的数学和实验建模这一共同主题而结合在一起。在某种程度上，这项研究具有地球物理倾向，但在工程和物理科学中也存在许多其他应用。下面概述了三个特定领域。粘塑性流体具有广泛的微观结构，可以产生宏观的内部强度；只有当材料受到足够大的压力时，结构才会屈服，从而允许类似流体的流动。实际上，这些材料包括泥浆、泥浆和粘土、重油、化妆品霜和发胶，以及液体巧克力。因此，粘塑性流体力学在石油和天然气工业、食品加工和地球物理等领域都有应用。我的研究探索了粘塑性流体如何在重力作用下流动(就像在经典的溃坝问题中一样)，经典流体动力稳定性问题的粘塑性对应问题(如滚波，或从下面加热的流体中对流的开始)，以及粘塑性如何调节流固相互作用(这是在泥浆和粘液等流体中或之上运动的关键)。在许多问题中的一个基本问题是，流动动力学是否可以用来探测材料的性质，从而作为一种实用的流变仪。或者，如果流体力学严重依赖于流变学，例如当屈服应力激活粘塑性薄膜上的运动时。最近，颗粒介质的半经验理论也与粘塑性流体模型密切相关，使得相关的许多其他应用在工程和地球物理中。这种联系引发了一个问题，即为粘塑性流体开发的理论是否适用于颗粒介质，反之亦然。例如，对于颗粒流来说，屈服的详细动力学并不是很清楚，但它是粘塑性的核心。地球物理流体动力学(GFD)传统上专注于形成海洋和大气的浅层旋转的流体。然而，GFD的数学方法也可以应用于地球科学的其他领域。例如，泥石流和岩石崩塌可以模拟为薄层粘塑性或颗粒状流体的突然释放。地貌学要求了解浅层流体流动与底层可侵蚀床的相互作用，以及与之相关的沉积物输送。熔岩呈现出一个特别复杂的流体问题，其结晶微结构，在很大的温度范围内冷却，表面凝固。冰盖和冰川底部的水薄膜的扩散似乎控制了滑动，这与冰川学有关。作为GFD的更传统的线条，我的研究是研究海洋中的波浪和不稳定性。例如，在世界上的一些海洋中，密度的分层被观察到采取长期存在的“阶梯”的形式，由缓冲均匀层的陡峭台阶组成。这些楼梯可以占据水柱的很大一部分，并水平延伸许多公里。在湖泊、太阳池和搅动的湍流流体中也观察到了类似的结构，并被认为存在于岩浆室和恒星中。楼梯与大尺度水流的相互作用可能导致相对未知的剪切流不稳定性，这不同于经典的Kelvin-Helmoltz原型，但可能控制楼梯的演化并限制其范围。

This proposal requests funds to support a broad range of topics united by the common theme of mathematical and experimental modelling of fluid flow. In part, the research has a geophysical slant, but many other applications exist in the engineering and physical sciences. Three particular areas are outlined below. Viscoplastic fluids possess an extensive microstructure that creates a 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, muds and clays, heavy oils, cosmetic creams and hair gel, and liquid chocolate. Consequently, viscoplastic fluid mechanics has application in fields ranging from the oil and gas industries, to food processing and geophysics. My research explores how viscoplastic fluids flow under gravity (as in the classical dam break problem), the viscoplastic counterparts of classical hydrodynamic stability problems (such as roll waves, or the onset of convection in a fluid heated from below), and how viscoplasticity mediates fluid-structure interaction (which is key for locomotion through or above fluids like mud and mucus). An underlying question in many problems is whether the flow dynamics can be used to probe the material properties and thereby act as a practical rheometer. Or, if the fluid mechanics is critically dependent on the rheology, such as when locomotion above a viscoplastic film is activated by the yield stress. Recent, semi-empirical theories of granular media are also closely connected to viscoplastic fluid models, making relevant many other applications in engineering and geophysics. This connection prompts the question of whether theory developed for viscoplastic fluids might be usefully employed for granular media, and vice versa. For example, the detailed dynamics of yielding is not well understood for granular flow, but is the centrepiece of viscoplasticity.Geophysical fluid dynamics (GFD) traditionally focusses on the shallow rotating layers of fluid forming the ocean and atmosphere. However, the mathematical methods of GFD can also be applied elsewhere in the geosciences. For example, mudslides and rock avalanches can be modelled as the sudden releases of thin layers of viscoplastic or granular fluids. Geomorphology demands an understanding of the interaction of shallow fluid flow with an underlying erodible bed, and the associated transport of sediment. Lava presents a particularly complicated fluids problem, with its crystalline microstructure, cooling through a vast temperature range and surface solidification. The spread of thin films of water at the base of ice sheets and glaciers appears to control sliding, drawing in connections with glaciology.Along more traditional lines of GFD, my research studies waves and instabilities in the ocean. For example, in some of the world's oceans, the stratification of density is observed to take the form of long-lived "staircases" consisting of sharp steps buffering homogeneous layers. These staircase can occupy a significant fraction of the water column and extend horizontally over many kilometres. Similar structures have been observed in lakes, solar ponds and stirred turbulent fluids, and suggested to exist inside magma chambers and stars. The interaction of a staircase with large-scale flow may lead to relatively unexplored kinds of shear flow instabilities, which are different to the classical Kelvin-Helmoltz archetype, yet may control the evolution of the staircase and limit its extent.