Solvability conditions for dendritic growth in the boundary-layer model with capillary anisotropy

Abstract

This paper is concerned primarily with the development of an analytic approach to the theory of steady-state velocity selection in the boundary-layer model of dendritic solidification. We consider the two-dimensional version of this model with a fourfold crystalline anisotropy \ensuremath{\alpha} in the surface tension. By extending a WKB method introduced in an earlier paper, we are able to determine the \ensuremath{\alpha} dependence of the selected growth rate in the limit of small \ensuremath{\alpha}; and we are also able to study this rate for larger \ensuremath{\alpha}'s in the limit in which the dimensionless undercooling approaches unity. Portions of the paper are devoted to a reinterpretation of the mathematical structure of the solvability condition in problems of this kind.