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.