Mathematical modelling of fluid flow

点击翻译按钮获取中文摘要

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.