Solidification cells at low velocity: The moving symmetric model

This paper is the first in a series of theoretical studies of singular cells in the small solute Peclet number limit (P{→}0) of two-dimensional models of directional solidification. In this limit solute diffusion in the frame of the moving front is nearly Laplacian in which case solidification cells and Saffman-Taylor fingers are closely related. Here Langer's moving symmetric model in the absence of temperature gradient (which also describes solidification in a channel of width {λ} at unit undercooling) is considered. A boundary integral equation describing steady-state cells is derived and it is shown that in the P{→}0 limit this equation can be expressed in terms of a single dimensionless parameter {σ}{∝}d₀l/λ². The endpoint singularity is studied analytically and physically admissible solutions are found numerically to only exist for a discrete set of values of {σ}. The small P dependence of {σ} is also examined.