Boundary-layer model of pattern formation in solidification

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Abstract
We propose and investigate the properties of a model of pattern formation in crystal growth. The principal dynamical variables in this model are the curvature of the solidification front and the thickness (or heat content) of a thermal boundary layer, both taken to be functions of position along the interface. This model is mathematically much more tractable than the realistic, fully nonlocal version of the free-boundary problem, and still recaptures many of the features that seem essential for studying dendritic behavior, for example. In this paper we describe analytic properties of the model. Preliminary numerical solutions produce snowflakelike patterns similar to those seen in nature.