Magnetic permeability jumps

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Table of contents

1 Introduction 
1.1 Context and motivations
1.2 Thesis outline
1.3 Magnetohydrodynamic equations
1.3.1 Navier-Stokes Equations
1.3.2 Maxwell Equations
1.3.3 Magnetohydrodynamic Equations
2 SFEMaNS MHD-code 
2.1 Framework
2.2 Numerical approximation
2.2.1 Fourier discretization
2.2.2 Finite Element representation
2.3 SFEMaNS possibilities
2.3.1 Parallelization
2.3.2 Heat Equation
2.3.3 Magnetic permeability jumps in r and z
2.4 Extension to non axisymmetric geometry
2.4.1 Pseudo-penalization method and prediction-correction scheme for the Navier-Stokes equations
2.4.2 Numerical test with manufactured solutions
2.4.3 Flow past a sphere and drag coefficient
2.5 Extension to MHD problems with variable fluid and solid properties
2.5.1 Magnetic field based approximation for azimuthal dependent magnetic permeability
2.5.2 Momentum based approximation for multiphase flow problems
2.6 Outlook
3 Nonlinear stabilization method: entropy viscosity 
3.1 Context and method
3.1.1 On the need of models
3.1.2 Large Eddy Simulation models
3.1.3 Entropy viscosity as LES method
3.2 Entropy viscosity and SFEMaNS code
3.2.1 Numerical Implementation
3.2.2 Numerical tests
3.2.3 Outlook
4 Large Eddy Simulation with entropy viscosity 
4.1 Hydrodynamic study of a Von Kármán Sodium set-up
4.1.1 Experimental set-up
4.1.2 Numerical approximation
4.1.3 Hydrodynamic regimes for Re 2500
4.1.4 Numerical results with entropy viscosity method
4.1.5 Conclusion
4.2 Two spinning ways for precession dynamo
4.2.1 Introduction
4.2.2 Numerical settings
4.2.3 Hydrodynamic study
4.2.4 Dynamo action
4.2.5 Conclusion
4.2.6 Appendix: Stabilization method
5 Momentum-based approximation of incompressible multiphase fluid flows 
5.1 Introduction
5.2 The model problem
5.2.1 The Navier-Stokes system
5.2.2 Level-set representation
5.3 Semi-discretization in time
5.3.1 Constant matrix diffusion on a model problem
5.3.2 Pressure splitting
5.4 Full discretization and stabilization
5.4.1 Space discretization
5.4.2 Stabilization by entropy viscosity
5.4.3 Compression technique for the level-set
5.4.4 Extension of the algorithm to the MHD setting
5.4.5 Finite elements/Fourier expansion
5.5 Analytical tests
5.5.1 Manufactured solution
5.5.2 Gravity waves
5.6 Newton’s bucket
5.6.1 Physical setting
5.6.2 Influence of Strain rate tensor
5.6.3 Influence of the surface tension
5.7 Free surface flow in an open cylinder
5.7.1 Physical setting
5.7.2 Numerics vs. experiment
5.8 Bubbles
5.8.1 Rising bubbles
5.8.2 Oscillating bubbles
5.9 Liquid metal droplet falling in a vertical magnetic field
5.9.1 Physical configuration
5.9.2 Falling droplet under gravity
5.9.3 Lorentz force as an external force
5.9.4 Full MHD setting
5.10 Conclusion
6 Conclusion and prospects 
6.1 Outcome
6.2 Outlook
7 Résumé en français 
7.1 Introduction
7.1.1 Contexte et motivations
7.1.2 Rappel des équations adimensionnées de la MHD
7.2 Le code SFEMaNS
7.2.1 Description du code
7.2.2 Développements récents
7.3 Viscosité entropique
7.3.1 Nécessité de modélisation
7.3.2 La viscosité entropique comme modèle LES
7.3.3 La viscosité entropique dans SFEMaNS
7.4 Application aux Simulations des Grandes Echelles (LES)
7.4.1 Application à des écoulements de Von Kármán
7.4.2 Application à des récipients cylindriques en précession
7.5 Approximation d’écoulements multiphasique avec la quantité de mouvement
7.5.1 Approximation numérique
7.5.2 Récapitulatif de quelques test numériques
7.6 Conclusion
7.6.1 Résultats
7.6.2 Perspectives
Bibliography