University Sétif 1 FERHAT ABBAS Faculty of Sciences
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Ajouter le résultat dans votre panier Affiner la rechercheA Study of Various Free Boundary Value Problems in Fluid Mechanics : The Asymptotic Behavior of Weak Solution / Malek Abdelali
Titre : A Study of Various Free Boundary Value Problems in Fluid Mechanics : The Asymptotic Behavior of Weak Solution Type de document : document électronique Auteurs : Malek Abdelali, Auteur ; Saadallah ,Abdelkader, Directeur de thèse Editeur : Setif:UFA Année de publication : 2025 Importance : 1 vol (93 f.) Format : 29 cm Langues : Anglais (eng) Catégories : Thèses & Mémoires:Mathématique Mots-clés : Asymptotic approach
Bingham fluid
Herschel-Bulkley fluid
Temperature
Reynolds equationIndex. décimale : 510 - Mathématique Résumé :
This thesis investigates the asymptotic behavior of non-isothermal flows of viscoplastic fluids
(Bingham and Herschel-Bulkley) in thin domains with Tresca-type friction conditions.
An effective limit model is derived, combining a Reynolds-type variational inequality with
a simplified heat equation. The analysis rigorously proves the existence, uniqueness, and
convergence of solutions. These results establish a robust mathematical framework for
simulating confined flows, with significant applications in lubrication engineering and process
design.Note de contenu : Sommaire
0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1 Formulation of Boundary Problems and Preliminaries 8
1.1 Mathematical formulation of boundary value problems . . . . . . . . . . . 9
1.1.1 Constitutive Law of the Bingham Fluid . . . . . . . . . . . . . . . . 9
1.1.2 Constitutive Law of the Herschel–Bulkley Fluid . . . . . . . . . . . 13
1.1.3 Sliding with Tresca Friction Law . . . . . . . . . . . . . . . . . . . 15
1.1.4 Energy Conservation Equation and Fourier Boundary Conditions . 16
1.1.5 Steady-State Thermal Flow of a Bingham Fluid under Tresca-Type
Frictional Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.1.6 Steady-State Thermal Flow of a Herschel–Bulkley Fluid under Tresca-
Type Frictional Contact . . . . . . . . . . . . . . . . . . . . . . . . 19
1.2 Review of Mathematical Analysis . . . . . . . . . . . . . . . . . . . . . . . 21
1.2.1 Sobolev Spaces and Lp Spaces . . . . . . . . . . . . . . . . . . . . . 21
1.2.2 Minty’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.2.3 Fundamental Inequalities . . . . . . . . . . . . . . . . . . . . . . . 23
2 Stationary Solutions of Non-Isothermal Bingham Flow with Nonlinear
Boundary Conditions in a Thin Domain 25
2.1 Introduction and Problem Statement . . . . . . . . . . . . . . . . . . . . . 25
2.2 Variational Problem and Existence Results . . . . . . . . . . . . . . . . . . 29
2.2.1 Functional Framework and Variational Formulation . . . . . . . . . 31
2.3 Domain Transformation and A Priori Estimates . . . . . . . . . . . . . . . 35
2.3.1 Change of Reference Domain . . . . . . . . . . . . . . . . . . . . . 35
2.3.2 Variational Formulation on Ω . . . . . . . . . . . . . . . . . . . . . 36
2.3.3 A Priori Estimates for Velocity and Pressure . . . . . . . . . . . . . 38
2.3.4 A Priori Estimates for Temperature . . . . . . . . . . . . . . . . . . 42
2.4 Study of the Limit Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3 Asymptotic Behavior of Solutions for a Non-Isothermal Coupled Problem
With Nonlinear Boundary Conditions in a Thin Domain 58
3.1 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2 Weak formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3 Change of the domain and study of convergence . . . . . . . . . . . . . . . 64
3.3.1 A priori estimates on the velocity and the pressure . . . . . . . . . 68
3.3.2 A priori estimates on the temperature . . . . . . . . . . . . . . . . 69
3.3.3 Theorem of Convergence . . . . . . . . . . . . . . . . . . . . . . . 74
3.4 Study of the Limit Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.4.1 Limit Problem and the Generalized Reynolds Equation . . . . . . . 77
3.4.2 Uniqueness of Solutions to the Limit Problem . . . . . . . . . . . . 85
Bibliographie 88Côte titre : DM/0219 A Study of Various Free Boundary Value Problems in Fluid Mechanics : The Asymptotic Behavior of Weak Solution [document électronique] / Malek Abdelali, Auteur ; Saadallah ,Abdelkader, Directeur de thèse . - [S.l.] : Setif:UFA, 2025 . - 1 vol (93 f.) ; 29 cm.
Langues : Anglais (eng)
Catégories : Thèses & Mémoires:Mathématique Mots-clés : Asymptotic approach
Bingham fluid
Herschel-Bulkley fluid
Temperature
Reynolds equationIndex. décimale : 510 - Mathématique Résumé :
This thesis investigates the asymptotic behavior of non-isothermal flows of viscoplastic fluids
(Bingham and Herschel-Bulkley) in thin domains with Tresca-type friction conditions.
An effective limit model is derived, combining a Reynolds-type variational inequality with
a simplified heat equation. The analysis rigorously proves the existence, uniqueness, and
convergence of solutions. These results establish a robust mathematical framework for
simulating confined flows, with significant applications in lubrication engineering and process
design.Note de contenu : Sommaire
0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1 Formulation of Boundary Problems and Preliminaries 8
1.1 Mathematical formulation of boundary value problems . . . . . . . . . . . 9
1.1.1 Constitutive Law of the Bingham Fluid . . . . . . . . . . . . . . . . 9
1.1.2 Constitutive Law of the Herschel–Bulkley Fluid . . . . . . . . . . . 13
1.1.3 Sliding with Tresca Friction Law . . . . . . . . . . . . . . . . . . . 15
1.1.4 Energy Conservation Equation and Fourier Boundary Conditions . 16
1.1.5 Steady-State Thermal Flow of a Bingham Fluid under Tresca-Type
Frictional Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.1.6 Steady-State Thermal Flow of a Herschel–Bulkley Fluid under Tresca-
Type Frictional Contact . . . . . . . . . . . . . . . . . . . . . . . . 19
1.2 Review of Mathematical Analysis . . . . . . . . . . . . . . . . . . . . . . . 21
1.2.1 Sobolev Spaces and Lp Spaces . . . . . . . . . . . . . . . . . . . . . 21
1.2.2 Minty’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.2.3 Fundamental Inequalities . . . . . . . . . . . . . . . . . . . . . . . 23
2 Stationary Solutions of Non-Isothermal Bingham Flow with Nonlinear
Boundary Conditions in a Thin Domain 25
2.1 Introduction and Problem Statement . . . . . . . . . . . . . . . . . . . . . 25
2.2 Variational Problem and Existence Results . . . . . . . . . . . . . . . . . . 29
2.2.1 Functional Framework and Variational Formulation . . . . . . . . . 31
2.3 Domain Transformation and A Priori Estimates . . . . . . . . . . . . . . . 35
2.3.1 Change of Reference Domain . . . . . . . . . . . . . . . . . . . . . 35
2.3.2 Variational Formulation on Ω . . . . . . . . . . . . . . . . . . . . . 36
2.3.3 A Priori Estimates for Velocity and Pressure . . . . . . . . . . . . . 38
2.3.4 A Priori Estimates for Temperature . . . . . . . . . . . . . . . . . . 42
2.4 Study of the Limit Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3 Asymptotic Behavior of Solutions for a Non-Isothermal Coupled Problem
With Nonlinear Boundary Conditions in a Thin Domain 58
3.1 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2 Weak formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3 Change of the domain and study of convergence . . . . . . . . . . . . . . . 64
3.3.1 A priori estimates on the velocity and the pressure . . . . . . . . . 68
3.3.2 A priori estimates on the temperature . . . . . . . . . . . . . . . . 69
3.3.3 Theorem of Convergence . . . . . . . . . . . . . . . . . . . . . . . 74
3.4 Study of the Limit Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.4.1 Limit Problem and the Generalized Reynolds Equation . . . . . . . 77
3.4.2 Uniqueness of Solutions to the Limit Problem . . . . . . . . . . . . 85
Bibliographie 88Côte titre : DM/0219 Exemplaires (1)
Code-barres Cote Support Localisation Section Disponibilité DM/0219 DM/0219 Thèse Bibliothèque des sciences Anglais Disponible
Disponible
