Mathematical modelling of subglacial bedform formation and dense granular flows

Subglacial bedforms,which include drumlins,ribbed moraine,and mega scale glacial lineations, are ubiquitous features in regions which were once covered by ice sheets, and their genesis is a long-standing and controversial problem in geophysics. In the first part of this thesis, a mathematical model for subglacial bedform formation is rigorously derived, which describes the coupled flow of ice, subglacial water, and sediment. We perform a linear stability analysis of the model and demonstrate that it can plausibly account for the formation of self-organised subglacial bedforms. We outline a novel numerical method to solve the model which is capable of providing three-dimensional simulations of the subglacial system for a restricted range of model parameters. The flow of granular material is a common occurrence in our everyday lives and understanding its behaviour is of critical importance for several industrial and geophysical applications. In part two of this thesis, we outline a popular local rheology for dense granular flows and present a recent compressible generalisation of this rheology. By using various analytical, asymptotic, and numerical techniques, we apply the model to two non-trivial flow phenomena; free-surface instability for inclined plane flows and spontaneous oscillations in plane shear flows, and compare model results to existing experimental and numerical data.