Mathematical methods for fluid-solid interfaces: meandering streams and sand ripples
Date
2008
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Abstract
This thesis presents several mathematical methods for modeling free surfaces, interfaces, and fluid-solid interactions. This work is done in the context of two physical systems. In the first two sections, the focus will be to understand the the physics of streams flowing down inclined substrates. Models will be derived to investigate both steady state and dynamic meandering profiles. It will be shown that, through the right approximation techniques, many physical insights can be drawn about this system. These results include: a complete understanding of the steady states, transitions between steady states, mechanism of meandering, forces involved in meandering, and spectral scaling laws of long-time ensemble averaged meandering stream profiles. In the third section, the focus will shift to how one can model underlying physics when it becomes too complicated to address from first principles. Here, the power of symmetries and conservation laws are explored to derive an amplitude equation describing the interface between sand and water when the water is subjected to oscillatory flow. The thesis will then close by posing a novel way to study scaling laws with respect to parameters using Lie's prolongation algorithm. Through this work various tools will be combined from the fields of physics, engineering, applied and pure mathematics to develop approaches for reducing complex systems into tractable pieces which can be studied carefully.
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Subject
amplitude equations
braiding patterns
fluid dynamics
fluid-solid interfaces
inclined plane flows
meandering
sand ripples
mathematics