Boundary-layer approaches to dendritic growth

Abstract

We analyze the derivation of boundary-layer models for dendritic growth and investigate the extent to which they yield information about the existence of a continuous family of steady-state needle crystal solutions. Although recent work has established that there exists only a discrete set of solutions for viscous fingering in a Hele-Shaw cell, side walls play an important role in this system, and we argue on physical grounds that the same mechanism may not apply to free dendrites. After a discussion highlighting the physical differences in these two systems, we analyze the model equations for dendritic growth, which first suggested the breakup of the family. We develop a systematic, and in principle exact, boundary-layer formalism for diffusion-controlled dendritic growth starting from the full heat-conduction equation. A consistent application of the formalism generates an expansion of the smooth steady-state solutions in powers of (1-Δ), where Δ is the dimensionless undercooling, but gives no indication as to whether or not a family of such solutions exist. Different physically motivated approximations yield different model equations, including the boundary-layer model of Ben-Jacob and co-workers, with very different properties. Steady-state predictions of all such models are arbitrary. We show that a proper phase-space description requires an infinite-dimensional phase space, in which there are stable directions not found in the boundary-layer model.