Analysis and simulations of Bingham fluid problems with Papanastasiou-like regularizations: Primal and Dual formulations 公开

Svishcheva, Anastasia (2014)

Permanent URL: https://etd.library.emory.edu/concern/etds/dn39x1934?locale=zh
Published

Abstract

In this dissertation thesis I conduct analysis and simulation of Binghamfluid problems with Papanastasiou-like regularizations. I explain primal and dual formulations of the problems. I discuss the mixed (dual) formulation of Bingham-Papanastasiou problem, its well-posedness and show the numerical results. In general, common solvers for the regularized problem experience a performance degradation when the regularization parameter m gets greater. The mixed formulation enhanced numerical properties of the algorithm by introduction of an auxiliary tensor variable. I also introduce a new regularization for the Bingham equations, so called Corrected regularization. Corrected regularization demonstrates better accuracy than other ones. I show its well-posedness, and in addition, compare its numerical results with the results obtained with the applications of other regularizations.

Table of Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1

1.1 Non-Newtonian fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Bingham Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Outline of the thesis and motivation . . . . . . . . . . . . . . . . . .7

2 Mathematical Formulations of the Bingham problem and its
regularizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1 The Equations . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 10
2.2 Regularized Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 Bercovier-Engelman regularization . . . . . . . . . . . . 12
2.2.2 Papanastasiou's regularization . . . . . . . . . . . . . . .13
2.2.3 Corrected regularization . . . . . . . . . . . . . . . . . . . .14
2.2.4 Other approaches to the numerical solution of
the regularized Bingham model . . . . . . . . . . . . . . 15

2.3 The Mixed Formulation . . . . . . . . . . . . . . . . . . . . . . . .. . . 18

3 Well-Posedness analysis of the Bingham-Papanastasiou problem . . 23

3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23
3.2 Theoretical tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 Well-Posedness of the Primitive Formulation . . . . . . . . . . 26

3.3.1 Papanastasiou's regularization . . . . . . . . . . . . . . .28
3.3.2 Corrected regularization . . . . . . . . . . . . . . . . . . . 33

3.4 Well-Posedness of The Mixed Formulation . . . . . . . . . . . . 36

3.4.1 Papanastasiou's regilarization . . . . . . . . . . . . . . . 37
3.4.2 Corrected regularization . . . . . . . . . . . . . . . . . . . 39

4 Numerical Approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41


4.1 The Picard Linearization Method . . . . . . . . . . . . . . . . . . . .41

4.1.1 Well-Posedness of the Picard Iteration . . . . . .. . . 42
4.1.2 Error Estimates for the Picard Iterative Scheme . . 45

4.2 The Newton method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.3 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.3.1 Well-Posedness of the Discrete Mixed Problem . . 58

4.3.1.1 Papanastasiou's regularization . . . . . . . . 58
4.3.1.2.Corrected regularization . . . . . . . . . . . . . 61
4.3.1.3 Time dependent case . . . . . . . . . . . . . . 63

4.3.2 Algebraic properties . . . . . . . . . . . . . . . . . . . . . . 64

4.3.2.1 Papanastasiou's regularization . . . . . . . . 66
4.3.2.1 Corrected regularization . . . . . . . . . . . . .67

5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .69

5.1 An Analytical Test Case . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.2 Picard's Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.2.1 Comparison of the iteration numbers . . . . . . .. . . .78
5.2.2 Corrected regularization. General case . . . . . . . . .82

5.3 Newton Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .89
5.4 Lid Driven Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .95
5.5 Inertia Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .109

Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

About this Dissertation

Rights statement
  • Permission granted by the author to include this thesis or dissertation in this repository. All rights reserved by the author. Please contact the author for information regarding the reproduction and use of this thesis or dissertation.
School
Department
Degree
Submission
Language
  • English
Research Field
关键词
Committee Chair / Thesis Advisor
Committee Members
最新修改

Primary PDF

Supplemental Files