Steady-state solutions for an array of strongly-interacting needle crystals in the limit of small undercooling

We consider the free-boundary problem for the steady-state solidification of a pure undercooled liquid in the form of an array of three-dimensional needle crystals. We neglect surface energy, consider the limit of small undercooling, and solve for the crystal shape analytically using slender body theory. The solutions have two degrees of freedom which determine the growth velocity as a function of the tip radius and the array spacing. For large array spacings we recover the Ivantsov similarity solution for an isolated dendrite, while for small array spacings the strong interactions between neighboring dendrites cause the Peclet number of the dendrite tip to be determined by an array-modified undercooling. Our leading-order results are valid for any space-filling array pattern and apply to solidification in channels of various shapes. The results can also be adapted to describe solidification of a two-component supersaturated solution at uniform temperature by making the one-sided approximation.