Sculpting of landforms by fluid flow

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The interaction between fluid flow and surfaces which evolve in response to erosion, abrasion, chemical dissolution or freezing/melting effects is, in general, a challenging nonlinear free boundary problem: the flow field depends on the surface shape which, in turn, depends on the local fluid transport properties near the fluid-solid boundary. There is much interest in these processes, especially in a geological context where the interaction leads to striking features and patterns spanning lengthscales from centimetres to tens of kilometres. Some specific examples include ramified stream networks, ice-stars, ventifacts and the surface scalloping of caves, beneath ice floes and the Martian landscape. The flowing fluid in these examples can be either ocean, surface- or ground-water, or wind, and the shaping of the landscape can be through abrasion, erosion, sediment transport, precipitation or dissolution.In two-dimensions, and assuming irrotational flow of inviscid fluid and that the surface evolves slowly in comparison to the timescale associated with the fluid flow, the free boundary problems described above can be formulated in terms of a conformal map from some canonical domain with fixed boundaries to the physical domain with an evolving boundary representing the fluid-solid interface. The mathematical task then becomes one of determining the time-dependent map. It can be shown that the map satisfies a Polubarinova-Galin type equation with forcing dependent on the 'physics' of the erosion process e.g. abrasion, melting.This PhD project will formulate and solve a series of problems using a combination of analytic, asymptotic (e.g. early/late times) and numerical methods (e.g. reducing the Polubarinova-Galin to a high-order system of coupled ODEs). A variety of initial surfaces will be considered e.g. circular/elliptical objects, flat surfaces; and forcing types e.g. normal velocity of the interface being proportional to the fluid pressure.The effect of shed and trapped vortices is of special interest: their inclusion in the flow about arbitrary shaped bodies will be computed using Laplacian 'fast' solvers of the type recently pioneered by Trefethen and co-workers. Such methods are also capable of computing conformal maps to canonical domains, opening up the possibility of computing the evolution of evolving bodies in the presence of vortices. It will be interesting to determine if the presence of trapped vortices gives rise to sculpted surfaces with 'sharp' edges as are commonly observed in some desert landscapes and the Martian surface.The project will be of interest to researchers in applied complex analysis, fluid mechanics, vortex dynamics and applied mathematicians working on free boundary problems. The project is also likely to have interdisciplinary appeal to geographers, Earth and planetary scientists.The research falls under the EPSRC theme 'Fluid dynamics and Aerodynamics'.