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

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

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School	Laney Graduate School
Department	Math and Computer Science
Degree	PhD
Submission	Dissertation
Language	English
Research Field	Mathematics
Keyword	Bingham fluids
Committee Chair / Thesis Advisor	Veneziani, Alessandro, Emory University
Committee Members	Borthwick, David, Emory University Nagy, James, Emory University

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