Crystallization in forced flows: the Saffman–Taylor problem

We consider the two-dimensional problem of crystal growth in a forced flow. A dendrite is placed in a Hele-Shaw cell with insulating walls and grows due to undercooling. We neglect the surface energy in the Gibbs–Thomson relation. The problem is formulated in terms of analytic functions similarly to closely related work on the viscous fingering problem of Saffman and Taylor. We derive a solvability condition for the existence of a steady-state needle-like solidification front in the limit of small Peclet number, Pe = V∞ l/a, where V∞ is the characteristic velocity of the melt, 2l is the channel width, and a is the thermal diffusivity of the liquid. The velocity of the crystallization front is directly proportional to the hydrodynamic velocity V∞ and undercooling, while the dendrite width ld does not depend upon the physical parameters, and indeed, ld = l.