Steady-state cellular growth during directional solidification

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Abstract
We study cellular structures moving at constant velocity, in a symmetric model of directional solidification. This is done by employing Newton’s method to solve a discretized version of the integro-differential equation for the solid-liquid boundary. Our results indicate that there is a continuous band of allowed wavelengths and that there is a generic fold in the solution diagram. This fold provides a maximum allowed wavelength for any given velocity. Also, we argue that the mechanism of microscopic solvability serves to fix the tip shape at fixed wavelength. The implications of our results for wavelength and shape selection are briefly discussed.