Keywords: dendritic growth, thermoelectricity, magnetohydrodynamics.

Experimental work on material processing under moderate to high magnetic field conditions has shown that the microstructure can be significantly altered. There is evidence that these effects can be attributed to Lorentz forces created through thermoelectric magnetohydrodynamic interactions [1,2]. Preliminary theoretical calculations have been conducted [3,4] and a plausible mechanism has been proposed for the observed morphological changes. So far two-dimensional models have only considered the fluid dynamics caused by these forces when the magnetic field is either directed perpendicular or parallel to the preferential direction of growth. The focus of this paper is to investigate the fluid dynamics of an arbitrarily orientated magnetic field to equiaxed dendritic growth.

By adopting an enthalpy based method [5], the dendrite morphology can be tracked. From the surface temperature the solution to the thermoelectric currents can be calculated using boundary conditions posed by Shercliff [6]. These currents circulate between the hot root of the crystal and the cold tip of the crystal. When a magnetic field is applied the interaction with these currents produces Lorentz forces which are the driving force of convection. Initially a low magnetic field strength approximation is taken; this has the effect of causing the induced currents, resistive terms and non-linear terms to become negligible allowing the principle of linear superposition to be applied and the flow field for any arbitrary orientation can be calculated from a single solution of the magnetic field in the (001) direction. Solutions for the (011) and (111) are also given to provide an insight into the complex nature of the flow fields that can be generated. Some of the features exhibited are a global circulation around the dendrite and circulations at the dendrite tips.

Using a quasi three-dimensional approximation, a two-dimensional model is used to calculated the morphological changes when the magnetic field strength is increased and convective transport effects become important. Rotation of the preferential direction of growth and an increase in secondary branching are observed and are a consequence of circulations altering surface free energy. Adopting a moving mesh technique that tracks the tip of a full three-dimensional dendrite, similar morphological changes are observed.

1

X. Li, Y. Fautrelle, Z.M. Ren, "Influence of an axial high magnetic field on the liquid-solid transformation in Al-Cu hypoeutectic alloys and on the microstucture of the solid", Acta materialia, 55(4), 1377-1386, 2007. doi:10.1016/j.actamat.2006.10.007

2

R. Moreau, O. Laskar, M. Tanaka, "Thermoelectric magnetohydrodynamic effects on solidification of metallic alloys in the dendritic regime", Mat. Sci. and Eng., 173, 93-100, 1993. doi:10.1016/0921-5093(93)90194-J

3

A. Kao, K. Pericleous, M. Patel, V. Voller, "The Effects of Thermo-Electrically Induced Convection in Alloy Solidification", in M. Papadrakakis, B.H.V. Topping, (Editors), "Proceedings of the Sixth International Conference on Engineering Computational Technology", Civil-Comp Press, Stirlingshire, UK, Paper 46, 2008. doi:10.4203/ccp.89.46

4

A. Kao, G. Djambazov, K.A. Pericleous, V. Voller, "Thermoelectric MHD in Dendritic Solidification", Magnetohydrodynamics, 45(3), 305-315, 2009.

5

V.R. Voller, "An enthalpy method for modelling dendritic growth in a binary alloy", International Journal of Heat and Mass Transfer, 51, 823-834, 2008. doi:10.1016/j.ijheatmasstransfer.2007.04.025

6

J.A. Shercliff, "Thermoelectric magnetohydrodynamics", Journal of Fluid Mechanics, 91(2), 231-251, 1978. doi:10.1017/S0022112079000136

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