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1 K Vc gl # m

B02 1 m0 dind

at some depth within the mushy zone where dendrite arms are sufficiently developed, typically 8 2, where 2 is the final secondary dendrite arm spacing, K is the permeability of the mushy zone, gl is the volume fraction of liquid,  is the dynamic viscosity, B0 is the magnetic field, 0 is the permeability of vacuum, and dind is the distance between the inductor and the liquidus front.

I. INTRODUCTION

IN order to increase the homogeneity of the properties of alloys[1] and limit the formation of internal defects and surface exudations,[2] fine grain structures are often desired in solidification processes. Several methods can be used for grain refinement: (1) the addition of inoculation substances such as TiB2 or TiC in aluminum alloys,[3,4] (2) the use of rapid solidification conditions[5,6] and (3) stirring of the melt by mechanical means or by electromagnetic stirring (EMS) fields.[7,8,9] The basic principle of EMS is to create an inhomogeneous Lorentz force, F, in the metal by an alternating magnetic field, B. In many cases, this field is created by an electrical current density, j, flowing in an induction coil and inducing a current, j ind, in the metal, opposite to j. This induced current is localized within a certain skin depth, 
(0)0.5, of the metal, where 0 is the magnetic permeability of vacuum (in cases where the magnetic permeability of metal is equivalent to 0),  is the electrical conductivity of the metal, and  is the angular frequency of the current. The Lorentz force, F, is then given by F
jind B

[1]

T. CAMPANELLA, formerly Ph.D. Student, Computational Materials Laboratory, Institut des Materiaux, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland, is Project Manager, Research and Development, Rolex S.A., Geneva, Switzerland. C. CHARBON, Head “Processes and Materials,” is with Swissmetal, Plant Dornach, CH-4143 Dornach, Switzerland. M. RAPPAZ, Professor, is with the Institute of Materials, Computational Materials Laboratory, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland. Contact e-mail: michel.rappaz@ epfl.ch Manuscript submitted December 17, 2003. METALLURGICAL AND MATERIALS TRANSACTIONS A

In the case of an axisymmetric induction coil surrounding a cylindrical specimen, the force is normal to the surface of the alloy in the plane of the coil, but decreases with the depth. Furthermore, the polar magnetic field makes the force also diverge in orientation as one moves away from the plane of the coil. Both variations in space contribute to the forced convection field observed in EMS (case of a pulsating field[10]). Please note that, in other situations where fluid flow is undesirable, d.c. magnetic fields can be used to dampen natural convection.[11] Melt stirring induces two effects that contribute to grain refinement: (1) the flow can penetrate the mushy zone and partially remelt dendrite arms; and (2) the thermal gradient in the liquid is lowered, thus extending the undercooled region and favoring a col