C. Références and . Gedepeon, Assemblage Métallurgique dans la Construction des Turboréacteurs Saitoh et Y. Ishiwata, « Influence of High-Temperature Protective Coatings on the Mechanical Properties of Nickel-Based Superalloys, Superalliages pour Turbine : Alliages pour Aubes et pour Disques Evolution of Ni-Based Superalloys for Single Crystal Gas Turbine Blade Applications », Aerospace Science and Technology Effect of Crystal Orientation on Fatigue Failure of Single Crystal Nickel Base Turbine Blade Superalloys Journal of Engineering for Gas Turbines and Power Fonderie de Précision à Modèle Perdu _ Application aux Superalliages », Techniques de l'Ingénieur. Matériaux Métalliques, no. M3580, 2007. [9] D. Furrer et H. Fecht, « Ni-Based Superalloys for Turbine Discs10] A. Pineau, « Alliages Métalliques pour Conditions Extrêmes », 281 eme conférence de l'Université de tous les savoirs ème Congrès International de Métrologie Métallurgie des Alliages de Fonderie, pp.297-323, 1993.

. Matériaux-métalliques-production, R. Sponsored, ]. S. Applications-center14, G. Yanqing, D. Jingjie et al., Cooling Crucible Vacuum Induction Melting alfa_TiAl Based Alloys », The Transactions of Nonferrous Metals Society of China « Studies on Addition of Inclusions to Molten Aluminum Using a Novel Technique, Vacuum Induction Melting Technologies Coulée Continue de l'Acier: Aspects Métallurgiques », Techniques de l'Ingénieur. Matériaux Métalliques16] J. M. Masson, « Élaboration de l'Acier Moulé: Fours de Fusion », Techniques de l'Ingénieur. Matériaux Métalliques, no. M3623, 2005. [17] P. Gillon19] J. M. Masson, « Fonderie et Moulage de l'Acier: Cas Pratiques », Techniques de l'Ingénieur. Matériaux Métalliques21] A. Pokorny et J. Pokorny, « Inclusions non Métalliques dans l'Acier », Techniques de l'Ingénieur. Matériaux Métalliques, no. M220, 1998. [22] L. Holappa et A. Helle, « Inclusion Control in High-Performance Steels Payandeh et M. Soltanieh, « Oxide Inclusions at Different Steps of Steel Production », Journal of Iron and Steel Research, International Murakami et M. Endo, « Effects of Defects, Inclusions and Inhomogeneities on Fatigue Strength », pp.69-72, 1987.

M. Bu?ko, W. Pyda, M. Gäumann, R. Trivedi, W. Kurz-yang et al., Simulation of Stray Grain Formation During Single Crystal Seed Melt-Back and initial Withdrawal in the Ni-Base Superalloy CMSX-4 Mechanism of Competitive Grain Growth in Directional Solidification of a Ni-Base Superalloy « Stray Grain Formation in Single Crystal Ni-base Superalloy Welds Hyzak et I. M. Bernstein, « The Effect of Defects on the Fatigue Crack Initiation Process in Two p/m Superalloys : Part I « Continuous Separation of Nonmetallic Inclusions From Aluminum Melt Using Alternating Magnetic Field, « Electromagnetic Separation of Nonmetallic Inclusion from Liquid Metal by Imposition of High Frequency Magnetic Field, Effect of Inclusion Size on Mechanical Properties of Alumina Toughened Cubic Zirconia Nucleation Ahead of the Advancing Interface in Directional Solidification32] O. Elkoca et H. Cengizler, « Cracking During Cold Forming Process of Rear Brake Component », Engineering Failure Analysis33] T. Baker et J. Charles, « Effect of Second-Phase Particles on the Mechanical Properties of Steel », Iron and Steel Institute34] R. Kiessling, « Influence of Nonmetallic Inclusions on the Properties of Steel High Temperature Materials and Processes Magnetohydrodynamic Forces Experienced by Spherical and Symmetrically Oriented Cylindrical Particles Application of Pinch Force to the Separation of Inclusion Particles From Liquid Steel », Journal of the Iron and Steel Institute of Japan, International Filtration Mechanism of Non- Metallic Inclusions In Steel by Ceramic Loop Filter », Journal of the Iron and Steel Institute of Japan, International Investigation on Foam Ceramic Filter to Remove Inclusions in Revert Superalloy », Materials Letters State of the Art in Evaluation and Control of Steel Cleanliness Model for Inclusion Formation in Low Alloy Steel Welds Van Geertruyden et W. Z. Misiolek, « Characterization of Non-Metallic Inclusions in Superelastic NiTi Tubes » Reconstruction of Inclusions in Solids Using Ultrasonic Born Inversion » Application of Fluorescence X-Rays to, pp.5191-5198, 1954.

M. Microradiography, ». C. Shamblen, S. Culp, R. Lober, D. Cormier et al., Ultrasonic Technology for Measuring Molten Aluminium Quality Electromagnetic Visualization Technique for Non-Metallic Inclusions in a Melt Melting and Refining of Superalloys and Titanium Alloys Melt Convection Driven by Electrodynamic Forces From Induction Heating « Electron Beam Surface Melting- Numerical Calculation of the Melt Geometry and Comparison With Experimental ResultsSitu Observation of Collision, Agglomeration and Cluster Formation of Alumina Inclusion Particles on Steels Melts [70] J. Visser, « Van der Waals and Other Cohesive Forces Affecting Powder Fluidization « The Interaction of Colloidal Particles Collected at Fluid Interfaces Peukert, « A General Approach for the Characterization of Fragmentation Problems « Numerical Modelling of the Breakage of Loose Agglomerates of Fine Particles, « Numerical Analysis on a Flow Field of Liquid Metals Under a Magnetic Field, Using a Spectral Finite Difference Scheme, Superalloy Cleanliness Evaluation Using the EB Button Melt Test Vacuum Metallurgy Conference Development of Separation and Evaluation Technique of Non-metallic Inclusions in Steel by Electron Beam Melting Characterization of High Alloy Steel Produced via Electron Beam Melting », Rapid Prototyping Journal Visualization of Marangoni Convection in Simulated Weld Pools », Welding Journal63] ASM Handbook, Metallography and Microstructures64] P. A. Meury, « Alliages Métalliques Pour l'Etalon de Masse de la Balance du Watt et des Références Secondaires Thèse de l'Ecole Nationale Supérieure des Mines de Paris, 2005. [65] B. Lallement, « Effet de la Profondeur d'Immersion du Thermocouple sur la Mesure de la Température du Bain », Rapport Interne SNECMA Gennevilliers, service ITKFM, 2008. [66] D. Jerebtsov et G. Mikhailov, « Phase Diagram of CaO -Al2O3 System », Ceramics International The Clustering of Alumina Inclusions », Metallurgical and Materials Transactions B An Experimental Investigation of Particle Fragmentation Using Single Particle Impact Studies », Powder Technology Impact Breakage of Spherical Granules: Experimental Study and DEM Simulation Ciomocos et M. Adams, « Numerical Simulations of Agglomerate Impact Breakage », Powder Technology Mechanistic Analysis and Computer Simulation of Impact Breakage of Agglomerates: Effect of Surface Energy Impact Breakage of Spherical, Cuboidal and Cylindrical Agglomerates Modeling of the Turbulent Flow in Induction Furnaces Modélisation des Transferts de Matière et de la Précipitation Inclusionnaire Lors de L'Elaboration du Marphy 17 au Four à Induction Sous Vide Thèse de l'Institut National Polytechnique de Lorraine Liquid Metal Turbulent Flow Dynamics in a Cylindrical Container With Free Surface _ Experiment and Numerical Analysis88] R. Schwarze et F. Obermeier, « Modelling of Unsteady Electromagnetically Driven Recirculating Melt Flows », Modelling and Simulation in Materials Science and Engineering Matas, B. Ulrych et I. Dole?el, « Velocity Field in Molten Metal at Electromagnetic Stirring in Cylindrical Crucible Proceedings of V International Workshop « Computational Problems of Electrical Engineering90] L. Basquin-Petitnicolas, « Etude Expérimentale et Modélisation Mathématique des Cinétiques de Nitruration et de Dénitruration des Alliages Ni-Cr Liquides Thèse de l'Institut National Polytechnique de Lorraine Plasma-Refining Process to Provide Solar-Grade Silicon », Solar Energy Materials and Solar Cells Coupling of Magnetic and Fluid Flow Problems and its Application in Induction Melting Apparatus Hosokawa, K. Ayata et M. Morishita, « Numerical Simulation of Meniscus Shape Considering Internal Flow Effects », Magnetohydrodynamics in Process Metallurgy Metallurgical and Materials Transactions B [96] I. M. Cohen et P. K. Kundu, Fluid Mechanics On The Dispersion of Discrete Particles Moving in a Turbulent Shear Flow » [98] S. Lain, M. Sommerfeld et J. Kussin, « Experimental Studies and Modelling of Four- Way Coupling in Particle-Laden Horizontal Channel Flow », International Journal of Heat and Fluid Flow99] G. Lipari, D. D. Apsley et P. K. Stansby, « Numerical Particle Tracking Studies in a Turbulent Round Jet », Particle-Laden Flow, ERCOFTAC Series, pp.1217-1222, 1952.

Q. Yuan, B. G. Thomas, and S. Vanka, Study of transient flow and particle transport in continuous steel caster molds: Part II. Particle transport, Metallurgical and Materials Transactions B, vol.37, issue.304, pp.703-714, 2004.
DOI : 10.2355/isijinternational.37.654

Z. Meijie, G. Huazhi, H. Ao, Z. Hongxi, and D. Chengji, Numerical simulation and industrial practice of inclusion removal from molten steel by gas bottom-blowing in continuous casting tundish, Journal of Mining and Metallurgy, Section B: Metallurgy, vol.47, issue.2, pp.137-147, 2011.
DOI : 10.2298/JMMB110120006M

H. Lei and J. C. , He, « A Dynamic Model of Alumina Inclusion Collision Growth in the Continuous Caster, Journal of Non-Crystalline Solids, vol.352, pp.36-37, 2006.

T. Toh, H. Yamamura, M. Wakoh, and E. Takeuchi, « Inclusion Behavior in Cold Crucible Levitation Melting and its Application to Cleanliness Evaluation, Proceedings of the 4 th International Conference on Electromagnetic Processing Materials, pp.226-231, 2003.

M. Kirpo, A. Jakovics, B. Nacke, and E. Baake, Particle transport in recirculated liquid metal flows, COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, vol.27, issue.2, pp.377-386, 2008.
DOI : 10.1017/S0022112085003275

V. Bojarevics, K. Pericleous, and R. Brooks, Dynamic Model for Metal Cleanness Evaluation by Melting in a Cold Crucible, Metallurgical and Materials Transactions B, vol.123, issue.4, pp.328-336, 2009.
DOI : 10.1007/978-3-642-82781-5_29

D. I. Graham and P. W. James, Turbulent dispersion of particles using eddy interaction models, International Journal of Multiphase Flow, vol.22, issue.1, pp.157-175, 1996.
DOI : 10.1016/0301-9322(95)00061-5

C. Crowe, R. Gore, and T. Troutt, PARTICLE DISPERSION BY COHERENT STRUCTURES IN FREE SHEAR FLOWS, Particulate Science and Technology, vol.29, issue.3, pp.3-4, 1985.
DOI : 10.1017/S0022112067000941

P. Salamon, D. Fernàndez-garcia, and J. J. , Gómez-Hernández, « A Review and Numerical Assessment of the Random Walk Particle Tracking Method, Journal of Contaminant Hydrology, vol.87, pp.3-4, 2006.

J. Barbier, Y. Fautrelle, J. Evans, and P. Cremer, « Numerical Simulation of Induction Heated Furnaces, Journal de Mécanique Théorique Appliquée, pp.533-556, 1982.

E. Taberlet and Y. Fautrelle, Turbulent stirring in an experimental induction furnace, Journal of Fluid Mechanics, vol.5, issue.-1, pp.409-440, 1985.
DOI : 10.1017/S0022112081002711__S0022112081002711

Y. I. Dutchak, V. Y. Prokhorenko, and E. Ratushnyak, Resistivity of the ternary In-Ga-Sn eutectic system near the melting point, Soviet Physics Journal, vol.12, issue.No. 3, pp.937-939, 1970.
DOI : 10.1007/BF01879849

C. Vives and R. Ricou, Fluid flow phenomena in a single phase coreless induction furnace, Metallurgical Transactions B, vol.17, issue.10, pp.227-235, 1985.
DOI : 10.1007/BF02679714

T. Quatravaux, « Evolution de la Modélisation du Procédé VAR -Contribution à la Description de la Dispersion Inclusionnaire Dans le Puits Liquide et à la Prévention de Défauts de Solidification», Thèse de l'Institut National Polytechnique de Lorraine, 2004.

W. Xu, Q. Chen, and «. Two, A two-layer turbulence model for simulating indoor airflow, Energy and Buildings, vol.33, issue.6, pp.613-625, 2001.
DOI : 10.1016/S0378-7788(00)00129-8

P. G. Saffman and J. Turner, On the collision of drops in turbulent clouds, Journal of Fluid Mechanics, vol.33, issue.01, pp.16-30, 1956.
DOI : 10.1175/1520-0469(1949)006<0134:TGOCDI>2.0.CO;2

A. Pétrissans, A. Taniere, and B. Oesterlé, Effects of Nonlinear Drag and Negative Loop Correlations on Heavy Particle Motion in Isotropic Stationary Turbulence Using a New Lagrangian Stochastic Model, Aerosol Science and Technology, vol.50, issue.9, pp.963-971, 2002.
DOI : 10.1016/S0065-2687(08)60106-5

S. Morsi and A. Alexander, An investigation of particle trajectories in two-phase flow systems, Journal of Fluid Mechanics, vol.none, issue.02, pp.193-208, 1972.
DOI : 10.1017/S0022112072001806

P. G. Saffman, The lift on a small sphere in a slow shear flow, Journal of Fluid Mechanics, vol.6, issue.02, pp.385-400, 1965.
DOI : 10.1007/BF01432457

R. Mei and . An, An approximate expression for the shear lift force on a spherical particle at finite reynolds number, International Journal of Multiphase Flow, vol.18, issue.1, pp.145-147, 1992.
DOI : 10.1016/0301-9322(92)90012-6

C. Yin, L. Rosendahl, S. Knudsen-kaer, and H. Sorensen, Modelling the motion of cylindrical particles in a nonuniform flow, Chemical Engineering Science, vol.58, issue.15, pp.3489-3498, 2003.
DOI : 10.1016/S0009-2509(03)00214-8

A. Hölzer and M. Sommerfeld, « Lattice Boltzmann Simulations to Determine Drag, Lift and Torque Acting on Non-Spherical Particles, pp.572-589, 2009.

S. Karanfilian and T. Kotas, Drag on a sphere in unsteady motion in a liquid at rest, Journal of Fluid Mechanics, vol.42, issue.01, pp.85-96, 1978.
DOI : 10.1115/1.3423549

E. E. Michaelides and . Particles, Bubbles & Drops: Their Motion, Heat And Mass Transfer, 2006.
DOI : 10.1142/6018

A. Xayasenh, L. Joly, and H. Duval, « Direct Numerical Simulation of Inclusion Turbulent Deposition at Liquid Metal/Slag Interface, Symposium CFD Modeling and Simulation in Materials Processing, TMS Annual meeting

H. Brenner, The slow motion of a sphere through a viscous fluid towards a plane surface, Chemical Engineering Science, vol.16, issue.3-4, pp.3-4, 1961.
DOI : 10.1016/0009-2509(61)80035-3

A. Maude, The movement of a sphere in front of a plane at low Reynolds number, British Journal of Applied Physics, vol.14, issue.12, p.894, 1963.
DOI : 10.1088/0508-3443/14/12/316

H. Faxen, « Die Bewegung Einer Starren Kugel Langs der Asche Eines mit Zahrer Flussigkeit Gefullten Rohres, Arkiv för Matematik, Astronomi och Fysik, pp.1-28, 1923.

P. Warszynski, Coupling of hydrodynamic and electric interactions in adsorption of colloidal particles, Advances in Colloid and Interface Science, vol.84, issue.1-3, pp.3-47, 2000.
DOI : 10.1016/S0001-8686(99)00015-9

M. O. Neill and «. A. , A Slow motion of viscous liquid caused by a slowly moving solid sphere, Mathematika, vol.10, issue.01, pp.67-74, 1964.
DOI : 10.1098/rspa.1926.0053

R. G. Cox and H. Brenner, The slow motion of a sphere through a viscous fluid towards a plane surface???II Small gap widths, including inertial effects, Chemical Engineering Science, vol.22, issue.12, pp.1753-1777, 1967.
DOI : 10.1016/0009-2509(67)80208-2

Z. Adamczyk, M. Adamczyk, and T. , Resistance coefficient of a solid sphere approaching plane and curved boundaries, Journal of Colloid and Interface Science, vol.96, issue.1, pp.204-213, 1983.
DOI : 10.1016/0021-9797(83)90022-X

K. Malysa and T. Van-de-ven, Rotational and translational motion of a sphere parallel to a wall, International Journal of Multiphase Flow, vol.12, issue.3, pp.459-468, 1986.
DOI : 10.1016/0301-9322(86)90018-2

G. Segre and A. Silberberg, Behaviour of macroscopic rigid spheres in Poiseuille flow Part 1. Determination of local concentration by statistical analysis of particle passages through crossed light beams, Behavior of Macroscopic Rigid Spheres in Poiseuille Flow », p.115, 1962.
DOI : 10.1017/S002211206200110X

J. Militzer, J. Kan, F. Hamdullahpur, and P. , Drag coefficient for axisymmetric flow around individual spheroidal particles, Powder Technology, vol.57, issue.3, pp.193-195, 1989.
DOI : 10.1016/0032-5910(89)80075-0

R. Shail and D. Norton, On the slow broadside motion of a thin disc along the axis of a fluid-filled circular duct, Mathematical Proceedings of the Cambridge Philosophical Society, p.793, 1969.
DOI : 10.1017/S0022112066000648

A. Haider and O. Levenspiel, Drag coefficient and terminal velocity of spherical and nonspherical particles, Powder Technology, vol.58, issue.1, pp.63-70, 1989.
DOI : 10.1016/0032-5910(89)80008-7

T. L. Thompson, N. Clark, and «. A. , A holistic approach to particle drag prediction, Powder Technology, vol.67, issue.1, pp.57-66, 1991.
DOI : 10.1016/0032-5910(91)80026-F

G. H. Ganser, A rational approach to drag prediction of spherical and nonspherical particles, Powder Technology, vol.77, issue.2, pp.143-152, 1993.
DOI : 10.1016/0032-5910(93)80051-B

A. Hölzer and M. Sommerfeld, New simple correlation formula for the drag coefficient of non-spherical particles, Powder Technology, vol.184, issue.3, pp.361-365, 2008.
DOI : 10.1016/j.powtec.2007.08.021

A. Khan and J. Richardson, The Resistance To Motion of a Solid Sphere in a Fluid References 1. I, 3D Modelling of the Aggregation ofOxide Inclusions in a Liquid Steel Ladle:Two NwnericalApproaches, Adv.Eng.Matl!l, pp.1-6, 1987.

O. Mirgaux, D. Ablitzer, E. P. Waz, and . Bellot, MathematicalModelling and Computer Simulation ofMolten Alwniniwn Purification by Flotation in Stirred Reactor, Metal. Mate~ Trans. B, pp.40-363, 2009.

H. Duval, C. Rivière, E. Lae, P. Le-brun, and . Guillot, Pilot-S cale Investigation of Liquid Aluminwn Filtration Through Cer:unic Foam Filters, Met. Trans. B, issue.2, pp.40-233, 2009.

J. P. Bellot, B. Foster, S. Hans, E. Hess, D. Ablitzer et al., Aluminium Volatilization and Inclusion Removal in the Electron Beam Cold Hearth Metting of Ti Alloys, Mati!!: Trans. B, pp.31-845, 2000.

E. Hénault, A Statistical Method to Assess the Reliability of Cleanness Measurements for High Quality Bearing Steels, J ASTM, issue.3, pp.10-15, 2006.

W. H. Sultan and I. Clark, Developrnent of an EB Metting Test to Evaluate the Cleanliness ofSuperalloys, Praceedings of the Conference on Electron Beam Me/ting and Refining-State of the Art, pp.36-61, 1983.

W. H. Sutton and S. O. Mancuso, Evaluation of Filter Performance in Purifying MM-200 PlusHfMETALS bytheEB-Button Test,Praceedings aftheCanferenceanElectranBeamMeltingandReftning-StateaftheArl, pp.330-355, 1983.

P. N. Quested, D. M. Hayes, and K. C. Mills, Factors affecting raft formation in electron beam buttons, Materials Science and Engineering: A, vol.173, issue.1-2, pp.173-371, 1993.
DOI : 10.1016/0921-5093(93)90247-C

M. Halali, M. Mclean, and D. R. West, Effect of Flux Additions on Inclusion Removal and Microstructure in Electron Beam Button Metting ofUdirnet 720, Mater. Sei. Technol, vol.16, issue.4, pp.457-462, 2000.

A. Mitchell, E. Samuelsson, and G. Silda, Assessment of Oxide Inclusions in Steels by EB Button Metting, Praceedings of the Conference an Electron Beam Me/ting and Refining -State of the Arl, Bakish Corp, 1983.

J. Goldak, A. Chakravart~, and M. Bibby, A New Finite Element Madel for Welding Heat Sources, Met. Trans. B, pp.15-299, 1984.

P. S. Wei and Y. Chow, Bearn Focussing Characteristics and Alloying Element Effects on High-Intensity Electron Beam Welding

Z. Li, K. Mukai, and M. Zele, Determination of the surface tension of liquid stainless steel, Journal of Materials Science, vol.25, issue.9-10, pp.40-2191, 2005.
DOI : 10.1007/BF02650084

H. Q. Yang, Boundary effect on the B??nard-Marangoni instability, International Journal of Heat and Mass Transfer, vol.35, issue.10, pp.24-37, 1992.
DOI : 10.1016/0017-9310(92)90083-5

P. Gillon and G. M. Homsy, Combined Thermocapillary-Buoyancy Convection in a Cavity: An Experimental Study, Phys. Fluid, pp.2953-2963, 1996.

J. P. Bellot, A. Jardy, and D. Ablitzer, Simulation numérique des transports couplés au sein du puits liquide d'un lingot de titane refondu par bombardement électronique, Revue de Métallurgie -C.I, Science et Génie des Matériaux, issue.12, pp.92-1399, 1995.
DOI : 10.1051/metal/199592121399

S. A. Morsi and A. J. Alexander, An Investigation afP article Trajectories in Two-Phase Flow Systems, J. Fluid Mech, pp.55-193, 1972.

A. Petris-san, Sur les modèles stochastiques lagrangiens de suivi de particuLes dans Wl champ turbulent, 2001.

T. Quatravaux, Evolution de la modélisation du procédé VAR, Thèse Institut National Polytechnique de Lorraine, 2004.