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

Black American Millennials Coping with the Myth of a Post-Racist Society

For generations, Black Americans depended on religious coping strategies to counter the impact of racism; the idea of giving the problem to God was shown to alleviate powerlessness and racism-related stress among pre-Millennial generations of Black Americans. However, the shift to an allegedly post-racist society has complicated recognition of racism and the coping process for Black American Millennials (BAMs). The purpose of this quantitative study was to determine whether BAMs and pre-Millennials perceive racism as an ongoing problem and whether generation (i.e., pre-Millennial Blacks and BAMs) moderates the effectiveness of religious coping strategies to mitigate racism-related stress. The theory of cognitive appraisal and symbolic racism theory were used to frame the study. Online surveys were administered to 206 Black Americans who were at least 20 years old. Results of an independent sample t test indicated no significant generational differences in the perceptions of ongoing racism. Results of a hierarchical moderated regression analysis indicated no significant differences in the mean racism-related stress scores between BAMs and pre-Millennial Blacks. Findings may be used to promote discussion about generational differences in perceptions of racism as an ongoing problem and how those differences impact the use of religious coping strategies to mitigate racism-related stress

Interaction of viscous free-surface flows with topography

The interaction of gravitationally driven, free-surface flows of viscous fluid with topographic features is investigated theoretically. The motion is studied in the regime where the depth of the flow is much smaller than the streamwise extent of the topography. A lubrication model of the motion is developed, integrated numerically and analysed asymptotically. For small mounds, it is shown 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, as well as forming deeper ponded regions upstream. Which of these phenomena prevails is shown to depend upon the amplitude of the mound height and the thickness of the oncoming flow relative to the streamwise length scale over which the topography varies. By using numerical and asymptotic results, we demonstrate that relatively wide mounds lead to the development of deep ponds of material upstream, which may lead to flow overtopping if the mound is not sufficiently high. These insights can be used to inform the design of barriers that defend built infrastructures from lava flows, and it is shown how this model can also provide an upper bound on the force exerted by the flow on them.</jats:p

Shallow free-surface Stokes flow around a corner.

The steady lateral spreading of a free-surface viscous flow down an inclined plane around a vertex from which the channel width increases linearly with downstream distance is investigated analytically, numerically and experimentally. From the vertex the channel wall opens by an angle α to the downslope direction and the viscous fluid spreads laterally along it before detaching. The motion is modelled using lubrication theory and the distance at which the flow detaches is computed as a function of α using analytical and numerical methods. Far downslope after detachment, it is shown that the motion is accurately modelled in terms of a similarity solution. Moreover, the detachment point is well approximated by a simple expression for a broad range of opening angles. The results are corroborated through a series of laboratory experiments and the implication for the design of barriers to divert lava flows are discussed. This article is part of the theme issue 'Stokes at 200 (Part 1)'