Abstract

(i) The Laplacian growth of thin two-dimensional protrusions in the form of either straight needles or curved fingers satisfying Loewner's equation is studied using the Schwarz-Christoffel (SC) map. Particular use is made of Driscoll's numerical procedure, the SC Toolbox, for computing the SC map from a half-plane to a slit half-plane, where the slits represent the needles or fingers. Since the SC map applies only to polygonal regions, in the Loewner case, the growth of curved fingers is approximated by an increasing number of short straight line segments. The growth rate of the fingers is given by a fixed power of the harmonic measure at the finger or needle tips and so includes the possibility of ‘screening’ as they interact with themselves and with boundaries. The method is illustrated by examples of needle and finger growth in half-plane and channel geometries. Bifurcating fingers are also studied and application to branching stream networks discussed. (ii) Solutions are found for the growth of infinitesimally thin, two-dimensional fingers governed by Poisson's equation in a long strip. The analytical results determine the asymptotic paths selected by the fingers which compare well with the recent numerical results of Cohen and Rothman (2017) for the case of two and three fingers. The generalisation of the method to an arbitrary number of fingers is presented and further results for four finger evolution given. The relation to the analogous problem of finger growth in a Laplacian field is also discussed.