Spreading of viscous fluids and granular materials on slopes
Materials can flow down a slope in a wide range of geophysical and industrial contexts, including lava flows on volcanoes and thin films on coated surfaces. The aim of my research is to provide quantitative insight into these forms of motion and their dependence on effects of the topography, the volume and the rheology of the flowing structure. Numerous different problems are investigated through mathematical models, which are developed analytically and confirmed by laboratory experiments. The initial advance of long lava flows is studied by considering the flow of viscous fluid released on sloping channels. A scaling analysis, in agreement with analog experiments and field data, offers a practical tool for predicting the advance of lava flows and conducting hazard analysis. A simple and powerful theory predicts the structure of flows resulting from any time-dependent release of fluid down a slope. Results obtained by the method of characteristics reveal how the speed of the advancing front depends importantly on the rate of fluid supplied at an earlier time. Viscous flows on surfaces with different shapes are described by similarity solutions to address problems motivated by engineering as well as geophysical applications. Pouring viscous fluid out of a container can be a frustratingly slow process depending on the shape and the degree of tipping of the container. The discharge rate of the fluid is analysed in simple cases, shedding light on how containers can be emptied most quickly in cosmetic and food industries. In a separate study motivated by coating industries, thin films are shown to evolve with uniform thickness as they drain near the top of a horizontal cylinder or sphere. The leading edge eventually splits into rivulets as predicted theoretically and confirmed by experiments. Debris flows can develop levees and trigger avalanches which are studied by considering dense granular flows down a rough inclined plane. Granular materials released down a slope can produce a flowing structure confined by levees or trigger avalanches at regular intervals, depending on the steady rate of supply. The experimental results are discussed using theoretical ideas of shallow granular flows. Finally, materials flowing in long and slender ducts are investigated theoretically to better understand the digestive and urinary systems in biology. The materials are pumped in an elastic tube by translating waves of muscular contraction and relaxation. The deformation of the tube is predicted by solving a free-boundary problem, a similar mathematical exercise to predicting the moving boundaries of materials spreading on slopes.