Analytic Solutions to Determine Critical Magnetic Fields for Thermoelectric Magnetohydrodynamics in Alloy Solidification

During alloy solidification, it has been observed that the morphology of microstructures can be altered by applying an external DC magnetic field. This structural change can be attributed to solutal convective transport introduced by thermoelectric magnetohydrodynamics (TEMHD) which drives fluid motion within the inter-dendritic region. Complex numerical models with grid resolutions on the microscopic scale have been constructed to solve the equations governing TEMHD. To complement these computationally intensive numerical models, analytic solutions were sought. Specifically, the analytic solutions presented herein are asymptotic solutions derived for TEMHD under low and high magnetic field intensities. Combination of these asymptotic solutions leads to simple formulae for estimating critical magnetic fields which can be readily evaluated in terms of characteristic lengths of materials that have been identified in experiments as key parameters of critical fields. Indeed, the critical magnetic fields predicted with the asymptotic solutions exhibit magnitudes consistent with those applied in current ongoing experiments where significant changes in microstructure have been observed. The capability to predict accurate results indicates that the analytic solutions described herein are valuable precursors not only for detailed numerical simulations but also for experimental design to study critical magnetic fields in alloy solidification.