Mathematical modelling of fluid flow

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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.