Time evolution of the velocity field in the magnetic oil case

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Table of contents

1 Introduction 
1.1 Context and motivation
1.2 Objectives
1.3 Outline of the manuscript
2 Bibliography study on ferrofluids 
2.1 Generalities
2.1.1 Definition
2.1.2 Composition
2.1.3 Stability
2.1.4 Applications
2.2 Modeling
2.2.1 Magnetic properties
2.2.2 Governing equations
2.2.3 Thermophysical properties
2.3 Thermomagnetic convection
2.3.1 Thermomagnetic instability
2.3.2 Literature review
2.3.3 Conclusive remarks
2.4 Transformer cooling
2.4.1 Cooling performance
2.4.2 Suitability as insulating liquid
2.5 Conclusion of the chapter
3 New developments in SFEMaNS: ferrohydrodynamics applications 
3.1 Physical setting
3.1.1 Geometry
3.1.2 Equations
3.2 Numerical method
3.2.1 Fourier representation
3.2.2 Finite element approximation spaces
3.2.3 Time-marching algorithm
3.3 Validation
3.3.1 Nondimensionalized equations
3.3.2 Magnetostatics
3.3.3 Temperature computation in solid regions
3.3.4 Kelvin magnetic body force
3.3.5 Robin boundary conditions
3.3.6 Temperature-dependent viscosity
3.3.7 Helmholtz magnetic body force
3.3.8 Pyromagnetic coefficient term in the temperature equation
3.4 Conclusion of the chapter
4 Thermomagnetic convection in an oil bath heated by a solenoid 
4.1 Article published in IEEE Transaction on Magnetics
4.2 Additional comments
4.2.1 Experimental setup
4.2.2 Governing equations
4.2.3 Physical properties
4.2.4 Mesh choice
4.2.5 Comparison with another experiment
4.2.6 Interface conditions on the magnetic field
4.2.7 Focus on the convective flow
4.2.8 Presence of oscillations in the regular oil case
4.2.9 Time evolution of the velocity field in the magnetic oil case
4.3 Complementary discussion
4.3.1 Visualization of the magnetic body force
4.3.2 Three-dimensional study
4.3.3 Use of a temperature-dependent viscosity
4.3.4 Comparison of the Kelvin and Helmholtz force models
4.4 Conclusion of the chapter
5 Realistic ferrofluid thermophysical properties 
5.1 Article published in Journal of Magnetism and Magnetic Materials
5.2 Complementary discussion
5.2.1 Experimental setup
5.2.2 Physical properties
5.2.3 Experiment vs. numerics using pure transformer oil
5.2.4 Numerical simulations with ferrofluid
5.2.5 Experiment vs. numerics using transformer oil-based ferrofluid
5.3 Improvement of the ferrofluid modeling
5.3.1 Temperature-dependent saturation magnetization of the magnetic nanoparticles
5.3.2 Complete temperature equation of ferrofluids
5.3.3 Limit of the linear magnetic material approximation
5.4 Conclusion of the chapter
6 Thermomagnetic convection in a transformer 
6.1 Transformer principle
6.1.1 Simplified transformer
6.1.2 Absence of load
6.1.3 Presence of a load
6.1.4 Superimposed coils
6.2 Physical problem and modeling
6.2.1 The considered electromagnetic system
6.2.2 Modeling
6.3 Regular oil vs. magnetic oil cooling
6.3.1 Time evolutions
6.3.2 Temperature and velocity fields
6.3.3 Magnetic field
6.3.4 Three-dimensional study
6.4 Variations of the model
6.4.1 Tank magnetic permeability
6.4.2 Curie temperature
6.4.3 Size of the tank
6.4.4 Distance between the coils
6.5 Conclusion of the chapter
7 Conclusion 
7.1 Outcome
7.2 Perspectives
8 Résumé en français 
8.1 Introduction
8.2 Etude bibliographique sur les ferrofluides
8.2.1 Généralités
8.2.2 Modélisation
8.2.3 Convection thermomagnétique
8.2.4 Refroidissement des transformateurs
8.3 Nouveaux développements dans SFEMaNS
8.3.1 Problème physique
8.3.2 Méthode numérique
8.4 Convection thermomagnétique dans un bain d’huile
8.4.1 Modélisation du problème
8.4.2 Avantage du refroidissement par ferrofluide
8.4.3 Comparaison des modèles de force de Kelvin et de Helmholtz
8.5 Vraies propriétés thermophysiques du ferrofluide
8.5.1 Adaptation du modèle
8.5.2 Résultats avec les propriétés modifiées
8.5.3 Amélioration de la modélisation du ferrofluide
8.6 Convection thermomagnétique dans un transformateur
8.6.1 Modélisation du problème
8.6.2 Refroidissement par huile classique versus par ferrofluide
8.7 Conclusion
8.7.1 Bilan
8.7.2 Perspectives
A Governing equations in fluid mechanics 
A.1 General equations
A.1.1 Continuity equation
A.1.2 Momentum equation
A.1.3 Energy equation
A.1.4 Theorem of the kinetic energy
A.1.5 Internal energy and temperature equations
A.2 Incompressible Navier-Stokes equations
A.2.1 Continuity equation
A.2.2 Momentum equation
A.2.3 Temperature equation
A.3 Newtonian fluid under Boussinesq approximation
A.3.1 Applicability of the approximation
A.3.2 Governing equations
B Approximations using Taylor expansions in SFEMaNS 
B.1 Backward Difference Formula of second order (BDF2)
B.2 Time extrapolation of second order
B.3 Time extrapolation of first order
C Additional convergence tests 
C.1 Magnetostatics
C.2 Kelvin magnetic body force
C.2.1 Linear law of magnetic susceptibility
C.2.2 Periodic solution
C.2.3 Periodic solution with non-zero pressure
C.2.4 Inverse law of magnetic susceptibility
C.3 Temperature-dependent viscosity
C.3.1 Polynomial temperature
C.3.2 Exponential temperature
D Preparatory study for the design of the experiment 
D.1 Modeling
D.2 Results
E Transformer case: estimation of the convection coefficient in the absence of heat transfer fins 
E.1 Theory
E.2 Computations
E.3 Conclusion
Bibliography