Dendritic growth in a channel

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Abstract
We study the growth of dendritic crystals from a supersaturated solution in a channel geometry. This model provides a continuous interpolation between the Saffman-Taylor problem and the (two-dimensional) free-space dendrite. We derive an integral equation for the shape of steady-state propagating fingers, which we treat by discretizing the curve and solving the resulting set of nonlinear algebraic equations by Newton’s method. For general Peclet numbers, in the absence of anisotropy the finger widths are always greater than half the channel width, but with anisotropy all widths are obtained. The numerical results are in rough agreement with an approximate WKB treatment for small anisotropy and zero Peclet number. Finally, we speculate on the emergence of sidebranches in this system.