Diffusively Driven Shear Flows in Stratified Fluids
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Harabin, George. Diffusively Driven Shear Flows In Stratified Fluids. 2016. https://doi.org/10.17615/exdt-q635APA
Harabin, G. (2016). Diffusively Driven Shear Flows in Stratified Fluids. https://doi.org/10.17615/exdt-q635Chicago
Harabin, George. 2016. Diffusively Driven Shear Flows In Stratified Fluids. https://doi.org/10.17615/exdt-q635- Last Modified
- March 20, 2019
- Creator
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Harabin, George
- Affiliation: College of Arts and Sciences, Department of Mathematics
- Abstract
- It is only recently that O.M. Phillips showed that in the presence of an impermeable, non-vertical plane inserted into a stratified fluid creates a parallel shear flow up the plane. The present dissertation combines theoretical, numerical, and experimental extensions of Phillips' work on diffusively driven flows such as time dependence, three-dimensional effects, and optimal shapes. We first develop the governing equations for time-dependent three-dimensional diffusively driven flows in cylindrical pipe geometries. Using Phillips' time independent solutions we non-dimensionalize this system and identify the Schmidt number as the single free parameter. We then consider a simple extension of Phillips' solutions to the time dependent case; the Laplace transform is used to recast the governing equations in the frequency domain, where the role of the Schmidt number can be understood. Long time asymptotics for the time dependent system are derived and a principal temporal frequency $\\frac{1}{\\sqrt{\\mathrm{Sc}}}$ is identified. A numerical integration of the governing equations reveals that in systems with small Schmidt numbers, this oscillations at this frequency are prominent in the time evolution. \\indent We then consider the time independent problem for non-planar geometries where the governing system becomes a pair of coupled Poisson's equations. Two methods of solution are developed for this system: first, the system is recast as a pair of uncoupled Fredholm equations of the second kind. The solution to these equations may be expressed as a Neumann series, which converges under certain criteria. In the case that the cross section of the domain is a circle, we are able to separate the solution into radial and angular components which allows us to solve the system exactly, as well as express the general term in the series solution. The qualitative features of both solutions are investigated and their comparison this discussion sheds light on the convergence criteria for the series solution. Another aspect of the three-dimensional problem is the role of geometry in the behavior of the system--- one important quantity which depends on the geometry of the pipe is the mass flux. We investigate the geometric optimization of integral functionals depending on the velocity, density. A criterion is derived for a geometry to be critical (a possible optimizer), and a gradient descent technique based on this criterion is developed and applied to the mass flux integral. We also show that for a certain "golden" radius, the circular cylinder is a critical pipe geometry for the mass flux under the constraint of a fixed cross sectional area. There is also an experimental component to this thesis which will aim to verify the theoretical considerations presented in the first four chapters. Building upon previous work by O.M. Phillips and Peacock et al., we present experimental verification of these flows for three dimensional circular cylinder geometries. We attempt to demonstrate the dependence of these flows on the polar angle of the cylinder using high molecular weight blue dextran as a tracer for the flow. Based on current experimental observations, we suggest interesting directions for future research in this area.
- Date of publication
- August 2016
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- Rights statement
- In Copyright
- Advisor
- Marzuola, Jeremy
- Williams, Mark
- McLaughlin, Richard
- Camassa, Roberto
- Passagia, Pierre-Yves
- Degree
- Doctor of Philosophy
- Degree granting institution
- University of North Carolina at Chapel Hill Graduate School
- Graduation year
- 2016
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