Mathematical modelling of fluid flow

本提案要求资金支持流体流动的数学和实验建模这一共同主题所结合的广泛主题。在某种程度上，这项研究具有地球物理学的倾向，但在工程和物理科学中存在许多其他应用。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，从而产生宏观内部强度；只有当材料受到足够强的强迫时，结构才会屈服，允许类似流体的流动。实际上，这些材料包括泥浆、泥浆和粘土、重油、化妆品面霜和发胶，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学等领域都有应用。我的研究探索粘塑性流体如何在重力作用下流动（如在经典的溃坝问题中），经典流体动力稳定性问题的粘塑性对应（如翻滚波，或从下面加热的流体中对流的开始），以及粘塑性如何介导流体-结构相互作用（这是通过或以上流体（如泥浆和粘液）运动的关键）。在许多问题中，一个潜在的问题是流动动力学是否可以用来探测材料的性质，从而作为一个实用的流变仪。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。近年来，颗粒介质的半经验理论也与粘塑性流体模型密切相关，在工程和地球物理中有许多其他应用。这种联系提出了一个问题，即粘塑性流体的理论是否适用于颗粒介质，反之亦然。例如，对于颗粒流动，屈服的详细动力学还没有得到很好的理解，但这是粘塑性的核心。地球物理流体动力学（GFD）传统上关注的是形成海洋和大气的浅层旋转流体。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，泥石流和岩石雪崩可以被模拟为粘塑性薄层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下伏可蚀层的相互作用，以及与之相关的沉积物运输。熔岩呈现出一个特别复杂的流体问题，它的晶体微观结构，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着滑动，这与冰川学有关。沿着更传统的GFD路线，我的研究是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“楼梯”的形式，由尖锐的台阶缓冲均匀的层组成。这些楼梯可以占据水柱的很大一部分，水平延伸数公里。在湖泊、太阳池和搅拌湍流中也观察到类似的结构，并认为它们存在于岩浆室和恒星中。阶梯与大尺度流动的相互作用可能导致相对未知的剪切流动不稳定性，这种不稳定性不同于经典的Kelvin-Helmoltz原型，但可能控制着阶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.