Suppression of channel convection in solidifying Pb-Sn alloys via an applied magnetic field
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I. INTRODUCTION
CHANNEL convection, also known as chimney convection or freckling, is common during the unidirectional solidification of off-eutectic metallic alloys, certain aqueous salt solutions on both sides of the eutectic, and some organic system.[1,2] It is of interest in metallurgy, because it can lead to defects in castings;[3] in oceanography, where brine channels are largely responsible for the heat flux between the polar oceans and the atmosphere;[4] and in solid-earth geophysics, where chimneys at the inner-outer core boundary may drive compositional convection in the outer core and help sustain the geomagnetic field.[5] It is also a fascinating fluid-mechanical phenomenon. Channel convection results because of two instabilities. First, during solidification of an alloy, the solid grows as a solvent-rich phase, thus enriching the solute fraction in the fluid boundary layer near the solid. For a binary eutectic system (Figure 1), this depresses the liquidus temperature (Ts) in the fluid boundary layer so that, in spite of the temperature being warmer further from the solid, a perturbation of the solid into the fluid may continue to grow.[6] This condition is known as constitutional supercooling and requires that the temperature gradient (dT/dz) be less steep than the product of the liquidus (G 5 dTs /dC ) times the compositional gradient (dC/dz). The morphological instability leads to dendritic growth of the solid and results in a region between MICHAEL I. BERGMAN, Assistant Professor, is with the Physics Department, Simon’s Rock College, Great Barrington, MA 01230. DAVID R. FEARN, Professor, is with the Department of Mathematics, University of Glasgow, Glasgow, Scotland G12 8QW. JEREMY BLOXHAM, Professor, is with the Department of Earth and Planetary Sciences, Harvard University, Cambridge, MA 02138. Manuscript submitted October 9, 1998. METALLURGICAL AND MATERIALS TRANSACTIONS A
the solid and liquid that is known as the mushy zone, in which the temperature and composition are linked by the liquidus. The second instability that leads to channels is the convective instability, whereby liquid in the mushy zone that is enriched in the solute may be gravitationally unstable and overcome the retarding Darcy viscous force in the porousmedium mushy zone.[7] Channel flow is a nonlinear phenomenon whereby variations in the permeability (P), which is a function of the porosity (f), lead to variations in the Darcy viscosity.[8,9] In particular, because heat diffuses more readily than mass, liquid rejected during solidification warms up more quickly than it loses its excess solute, so it becomes supersaturated in the solute. As it convects away from the solid, it thus melts solvent-rich dendrites, creating a channel that has a high permeability and a low effective viscosity. This leads to narrow, high-velocity channel flow away from the solid and a broad, low velocity flow toward the solid. The linear problem leading to porosity variations has been studied by, among others, Fowler[10] and Worster.[11] The latter
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