Pattern selection in a boundary-layer model of dendritic growth in the presence of impurities

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Abstract
We have analyzed, in the context of a boundary-layer model, the problem of pattern selection in dendritic growth in a situation where impurities are present in the undercooled liquid. We find that the tip-velocity selection criterion that has been proposed recently for the geometrical model and the boundary-layer model of a pure substance can be extended, in a nontrivial way, to this more complex situation where two coupled diffusion fields (temperature and solute) determine the interface dynamics. Our model predicts a sharp enhancement of tip velocity in good qualitative agreement with experiment. This agreement is consistent with the conjecture that a solvability condition can be used to determine the operating point of the dendrite in the full nonlocal problem.