Bj. Spencer et He. Huppert, STEADY-STATE SOLUTIONS FOR AN ARRAY OF STRONGLY-INTERACTING NEEDLE CRYSTALS IN THE LIMIT OF SMALL UNDERCOOLING, Journal of crystal growth, 148(3), 1995, pp. 305-323
We consider the free-boundary problem for the steady-state solidificat
ion of a pure undercooled liquid in the form of an array of three-dime
nsional needle crystals. We neglect surface energy, consider the limit
of small undercooling, and solve for the crystal shape analytically u
sing slender body theory. The solutions have two degrees of freedom wh
ich determine the growth velocity as a function of the tip radius and
the array spacing. For large array spacings we recover the Ivantsov si
milarity solution for an isolated dendrite, while for small array spac
ings the strong interactions between neighboring dendrites cause the P
eclet number of the dendrite tip to be determined by an array-modified
undercooling. Our leading-order results are valid for any space-filli
ng array pattern and apply to solidification in channels of various sh
apes. The results can also be adapted to describe solidification of a
two-component supersaturated solution at uniform temperature by making
the one-sided approximation.