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Fluid Dynamics with Incompressible Schrödinger Flow

Citation

Chern, Albert Ren-Haur (2017) Fluid Dynamics with Incompressible Schrödinger Flow. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/Z98050N3. http://resolver.caltech.edu/CaltechTHESIS:06052017-102338732

Abstract

This thesis introduces a new way of looking at incompressible fluid dynamics. Specifically, we formulate and simulate classical fluids using a Schrödinger equation subject to an incompressibility constraint. We call such a fluid flow an incompressible Schrödinger flow (ISF). The approach is motivated by Madelung's hydrodynamical form of quantum mechanics, and we show that it can simulate classical fluids with particular advantage in its simplicity and its ability of capturing thin vortex dynamics. The effective dynamics under an ISF is shown to be an Euler equation modified with a Landau-Lifshitz term. We show that the modifying term not only enhances the dynamics of vortex filaments, but also regularizes the potentially singular behavior of incompressible flows.

Another contribution of this thesis is the elucidation of a general, geometric notion of Clebsch variables. A geometric Clebsch variable is useful for analyzing the dynamics of ISF, as well as representing vortical structures in a general flow field. We also develop an algorithm of approximating a "spherical" Clebsch map for an arbitrarily given flow field, which leads to a new tool for visualizing, analyzing, and processing the vortex structure in a fluid data.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Fluid Dynamics; Incompressible Schrödinger Flow; Clebsch Variables; Physical Simulation; Differential Geometry and Dynamics
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Applied And Computational Mathematics
Awards:The W.P. Carey & Co. Inc. Prize in Applied Mathematics, 2017.
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Schroeder, Peter
Thesis Committee:
  • Schroeder, Peter (chair)
  • Desbrun, Mathieu
  • Owhadi, Houman
  • Pinkall, Ulrich
Defense Date:23 May 2017
Record Number:CaltechTHESIS:06052017-102338732
Persistent URL:http://resolver.caltech.edu/CaltechTHESIS:06052017-102338732
DOI:10.7907/Z98050N3
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1145/2897824.2925868DOIArticle adapted for Part I and Part III
http://dx.doi.org/10.1145/3072959.3073591DOIArticle adapted for Part II
ORCID:
AuthorORCID
Chern, Albert Ren-Haur0000-0002-9802-3619
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:10278
Collection:CaltechTHESIS
Deposited By: Albert Chern
Deposited On:05 Jun 2017 23:37
Last Modified:16 Jun 2017 23:05

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