Pattern selection in fingered growth phenomena

TLDR

Fingered growth processes, such as dendritic snowflakes and fluid fingering in immiscible displacement, arise from morphological instabilities that destabilize planar or circular interfaces, with surface tension acting as a stabilizing influence. The study surveys recent theory to explain how fingered growth systems develop their characteristic patterns. The authors analyze dendritic solidification and Saffman‑Taylor fingering by first solving for tension‑free finger shapes, then adding surface tension non‑perturbatively, and finally assessing secondary instabilities and noise effects.

Abstract

Abstract A variety of non-equilibrium growth processes are characterized by phase boundaries consisting of moving fingers, often with interesting secondary structures such as sidebranches. Familiar examples are dendrites, as seen in snowflake growth, and fluid fingers often formed in immiscible displacement. Such processes are characterized by a morphological instability which renders planar or circular shapes unstable, and by the competing stabilizing effect of surface tension. We survey recent theoretical work which elucidates how such systems arrive at their observed patterns. Emphasis is placed upon dendritic solidification, simple local models thereof, and the Saffman-Taylor finger in two-dimensional fluid flow, and relate these systems to their more complicated variants. We review the arguments that a general procedure for the analysis of such problems is to first find finger solutions of the governing equations without surface tension, then to incorporate surface tension in a non-perturbative manner, and lastly to examine possible secondary instabilities and the effects of noise.