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.

Geodesic Loewner paths with varying boundary conditions

“…As shown in figure 2, agreement between the numerical procedure [22] and that of the Loewner equation methods of the present work is reasonable. There is some expected loss of accuracy in the numerical procedure [22] when approximating a path by straight-line segments as curvature increases.…”

“…Since the method [22] does not solve any form of Loewner’s equation, nor an equation for the forcing function, it is a useful independent check on the paths calculated using the Loewner equation-based formulations of §3. As shown in figure 2, agreement between the numerical procedure [22] and that of the Loewner equation methods of the present work is reasonable. There is some expected loss of accuracy in the numerical procedure [22] when approximating a path by straight-line segments as curvature increases.…”

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Figure 2Slit trajectories for c = −1 for a range of starting location on the real axis computed by solving Loewner’s equation (3.4) with forcing (3.12) (solid line). Also shown are trajectories for the same selection of starting locations computed using the numerical method [22] (dashed lines).

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“…The SC Toolbox maps the new point in the w -plane back to the z -plane giving the next segment of the growing slit. Since the method [22] does not solve any form of Loewner’s equation, nor an equation for the forcing function, it is a useful independent check on the paths calculated using the Loewner equation-based formulations of §3. As shown in figure 2, agreement between the numerical procedure [22] and that of the Loewner equation methods of the present work is reasonable.…”

“…In the case when ϕ = 0 on all boundaries of a simply connected domain, geodesic paths are simply found by mapping from a half-plane to the slit domain of interest, with the paths being images under the conformal map. In this way, for example, the paths taken by two slits emanating from the tip of a semi-infinite needle are obtained [8,22].…”