Application of the Schwarz-Christoffel map to the Laplacian growth of needles and fingers.

Abstract

A numerical procedure based on the Schwarz-Christoffel map suitable for the study of the Laplacian growth of thin two-dimensional protrusions is presented. The protrusions take the form of either straight needles or curved fingers satisfying Loewner's equation, and are represented by slits in the complex plane. Particular use is made of Driscoll's numerical procedure, the SC Toolbox, for computing the Schwarz-Christoffel map from a half plane to a slit half plane. Since the Schwarz-Christoffel map applies only to polygonal regions, the growth of curved fingers is approximated by an increasing number of short straight line segments. The growth rate is given by a fixed power η of the harmonic measure at the finger or needle tips and so includes the possibility of "screening" as the needles of fingers interact with themselves and with boundaries. The method is illustrated with examples of multiple needle and finger growth in half-plane and channel geometries. The effect of η on the trajectories of asymmetric bifurcating fingers is also studied.