Some problems in low Reynolds number environmental flows
This thesis investigates some problems in environmental fluid dynamics in which the role of inertia is negligible relative to viscous stresses and buoyancy forces. Simple models are developed to provide insight into the physics governing the flows and to determine the dependencies on a few parameters. We study the fluid dynamics of carbon dioxide storage in heterogeneous aquifers, the migration of tracers in such flows and the interaction of lava flows with barriers.
First, in chapter 2, we examine the injection of fluid of one viscosity and density into a horizontal permeable aquifer initially saturated with a second fluid of different viscosity and density. The novel feature of the analysis is that we allow the permeability to vary vertically across the aquifer so that there is a shear in the flow. This leads to recognition that the interface may evolve as either a rarefaction wave that grows at a rate proportional to time, a shock-like front of fixed length or a mixture of shock-like regions and rarefaction-wave-type regions. In chapter 3, we study the migration of a tracer within the flows described in chapter 2. Owing to the shear flow, tracer in the high permeability regions moves substantially faster than the mean flow and eventually enters the interface region. The tracer may either remain in this region or cycle through it and be left behind. We then proceed to consider the role of diffusion of the tracer in the case that the interface has fixed extent (chapter 4). Cross-aquifer diffusion homogenises the tracer distribution, which becomes independent of depth but spreads longitudinally in this shear dispersion regime. This leads to much faster spreading than by diffusion alone. The shear dispersion of tracer in a growing interface (chapter 5) is more complicated because the tracer migrates into thin regions where the shear becomes dominated by the lateral spreading owing to the growth of the interface.
In the second part of the thesis, we consider the flow of lava down a slope with an obstruction. We assume that the lava is a low Reynolds number gravity-driven flow and we study the free-surface deformation owing to the obstruction. For small smooth mounds, it is shown in chapter 6 that the flow surmounts the obstacles, but for larger mounds the flow is deflected around it and can form dry zones in its wake into which fluid does not flow. In chapter 7, we theoretically and experimentally investigate the interaction of free-surface flows with cylinders of various cross-sections on an inclined plane. The cylinders are oriented with their axis perpendicular to the plane and are sufficiently tall so that they are not overtopped. For relatively shallow flows, there is a ‘pond’ of nearly stationary fluid upstream of the cylinder and a ‘dry’ region in which there is no fluid downstream of the cylinder. The investigation has direct relevance to the deflection of lava flows by barriers and buildings and the theory is employed to deduce simplified asymptotic expressions of the force exerted on the cylinders.