Assessing late-time singular behaviour in models of three dimensional Euler fluid flow

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Title: Assessing late-time singular behaviour in models of three dimensional Euler fluid flow
Authors: Mulungye, Rachel M.
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Date: 2016
Online since: 2019-05-23T07:32:44Z
Abstract: The open question of regularity of the fluid dynamical equations is considered one of the most fundamental challenges of mathematics and physics [C. L. Fefferman. Existence and smoothness of the Navier-Stokes equation. The millennium prize problems, pages 57-67 (2000)]. While the viscous Navier-Stokes equations have more physical relevance, the inviscid Euler equations present the greatest challenge and exhibit the most extreme behaviours. For this reason, the numerical study of possible finite-time blowup is typically concerned with these inviscid equations. Extensive numerical assessment of finite-time blow up of 3D Euler has been carried out, albeit with conflicting yes and no conclusions with regard to the existence of finite time singularity. The fundamental difficulty of this important problem is the lack of analytic solutions or any a priori knowledge of asymptotic behaviour. A secondary obstacle is that the spatial collapse associated with intense vortex stretching results in numerical solutions becoming unresolved beyond a certain time. It is therefore imperative to devise a framework with nontrivial blowup dynamics and where analytic solutions are known in order to validate and compare various numerical methods, for the purposes of accurately solving the system and diagnosing blowup. In this regard, I have proposed investigating the issue of Euler finite-time blowup using a novel approach where the original system of equations is bijectively transformed to a new mapped system which is globally regular in time [M. D. Bustamante. 3D Euler equations and ideal MHD mapped to regular systems: Probing the finite-time blowup hypothesis. Physica D: Nonlinear Phenomena, 240(13):1092-1099 (2011)]. Since no known analytical solution for the full 3D Euler equations exist, I have studied the robustness of the proposed novel approach using the one-dimensional Burgers equation and a proposed new one-parameter family of models of the 3D Euler equations on a 2D symmetry plane. The proposed 2D symmetry plane model equations were motivated by the work on stagnation-point-type exact solution of 3D Euler equations by Gibbon et al. [J. Gibbon, A. Fokas, and C. Doering. Dynamically stretched vortices as solutions of the 3D Navier-Stokes equations. Physica D: Nonlinear Phenomena, 132(4):497-510 (1999)]. I have shown that the mapped system’s numerical solution leads to more accurate estimates of the blowup quantities compared with the original system. I also established that only by using the mapped system can certain late-time behaviours be observed and asymptotic trends be established.
Type of material: Doctoral Thesis
Publisher: University College Dublin. School of Mathematics & Statistics
Qualification Name: Ph.D.
Copyright (published version): 2016 the Author
Keywords: Euler fluid equationsFinite-time singularityStagnation-point-type solutions of 3D Euler flows
Language: en
Status of Item: Peer reviewed
Appears in Collections:Mathematics and Statistics Theses

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