Study of directionally solidified eutectics with emphasis on oscillatory instabilities
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Two dimensional binary lamellar eutectics are studied through the process of mathematically modeling the Stefan-type problem associated with directionally solidified eutectic growth. A mathematical model of the boundary value problem is presented and a linear stability analysis is performed in which the triple points of the eutectic are perturbed tangential to the interface. This analysis indicates the presence of an "acoustic" mode of wavelength twice that of the lamellar spacing, generally found at low Peclet numbers. and an "optical" mode of wavelength equal to the lamellar spacing, generally found at higher Peclet numbers. Then the non-linear equations which are a result of the linear stability analysis are asymptotically analyzed and it is found that the results for the "acoustic" oscillations can be simplified; but that the asymptotics applied to the "optical" oscillations are not valid. A revision of the mathematical model is contained in a high Peclet number model for eutectics, involving slopes and partition coefficients dependent on the Peclet number. The linear stability analysis is repeated on the high Peclet number model and it is found that the earlier results change for high Peclet number oscillations such that an oscillation with a fixed wavelength occurs at a smaller velocity and a smaller concentration than originally predicted. Finally, a random walk simulation is developed from the solidification model and the boundary value problem is rewritten as a probabilistic process. The simulation results are used to demonstrate the movement of the triple points under given perturbation conditions.