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

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 $\eta$ 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 $\eta$ on the trajectories of asymmetric bifurcating fingers is also studied.

Figure 1

Mapping between the slit $z$ plane and the upper half of the $w$ plane. The slits in the form of straight needles have tips at $z={\gamma }_i$ which map to real parameters $w=a_i, i=1,2,3$. In the $z$ plane $\varphi =0$ on the needles and real $z$ axis, and $\varphi \to y$ as $y\to \infty$. The needles make angles ${\mu }_1\pi , {\mu }_2\pi$, and ${\mu }_3\pi$ with the real axis. The solution to $\mathrm{\Delta }\varphi =0$ in the $w$ plane is $\varphi =\mathrm{Im}\left(w\right)=v$. Contours of $\varphi$ in the $w$ plane are dashed lines parallel to the real axis.

Figure 2

Growth of two needles of equal length $h_1=h_2=h\left(t\right)$ (solid line). Superimposed are asymptotic length functions for small time, $h\sim t^2/4$, and large time, $h\sim t^2/4\sqrt{2}+\text{const}$ (both dashed lines).

Figure 3

Lengths of two needles as functions of time with initial lengths 1.0 and 1.05.

Figure 4

(a) Snapshot of a pair of needles at $t=6$. The initial lengths of the needles were both 0.05. One grows at right angles to the real axis and the other inclined at angle $\mu \pi =3\pi /5$. (b) The needle lengths as functions of time.

Figure 5

Contours of the stream-function field at $t=2.5$ for the same two needle experiment shown in Fig. 4. The stream-function values on the contours are uniformly spaced.

Figure 6

(a) Snapshot of a pair of needles at $t=6$. The initial lengths of the needles were both 0.05; both grow at angle $\mu \pi =3\pi /5$. (b) The needle lengths as functions of time.

Figure 7

(a) Nine evenly spaced needles growing vertically in a half plane. The initial lengths of the needles is 0.05. The needle lengths maintain a symmetric distribution. (b) Lengths of the needles as function of time.

Figure 8

(a) Nine evenly spaced needles growing vertically in a semi-infinite strip with zero flux conditions on the vertical sidewalls (shown as thick dark lines). The initial length of the needles is 0.05. The needle lengths maintain a symmetric distribution. (b) Length of the needles as function of time.

Figure 9

Mapping between the piecewise straight-line discretized finger in the physical $z$ plane and the upper half of the $w$ plane: ${\gamma }_j$ maps to $a_j$ on the $\mathrm{Re}\left(w\right)$ axis. The set of points $z^{*}$ lie on an arc of radius $\delta l$ from ${\gamma }_j$. The point $z^{*}=z^{**}$ is such that $\mathrm{Re}\left(w^{**}\right)=a_j$, where $w^{**}=g_t\left(z^{**}\right)$.

Figure 10

(a) Two symmetric fingers growing in a half plane computed using the numerical method. Nine iterations are shown with each step of length 0.1 showing as a different color. The dark line shows the initial starting configuration with fingers 0.2 apart. (b) Shape of the computed right finger for two different step sizes $\delta l=0.05$ (dotted line), 0.1 (dashed line) with 44 and 22 iterations respectively, compared with the exact solution (26) (solid line).

Figure 11

The average error of the two symmetric finger computation compared to the exact solution as a function of step size $\delta l=0.05,0.075,0.1,0.125,0.15.$

Figure 12

(a) Shape of finger starting at $x=-0.5$ and growing in a semi-infinite strip of width 2. Shown are numerically computed fingers for two different step sizes 0.15 (dashed line) and 0.5 (dotted line) and the exact solution given by (28) (solid line). (b) The average error compared to the exact solution as a function of step size.

Figure 13

Snapshots of the trajectories taken by three fingers which are symmetric about the middle finger. Three different numerical experiments are shown corresponding to $\eta =-2$ (solid line), 1 (dashed line), 4 (dotted line). The tip of the middle finger is denoted in each case by $\eta =-2$ ($\circ$); $\eta =1$ ($*$); $\eta =4$ ($\times$). The small vertical solid black ticks on the real axis represent the initial heights of the fingers.

Figure 14

(a) Comparison of exact finger trajectories (solid line) obtained by conformal mapping of (26) for $t=2.4$, and those obtained by the numerical method (dashed lines). The initial secondary bifurcation and the primary semi-infinite needle are the thick solid lines. (b) The angle between the tangents of the finger tips as a function of time. The dashed line is the $2\pi /5$ angle.

Figure 15

Screening of fingers from an initial right-angled bifurcation (thick solid line) from a semi-infinite needle for $\eta =-2$ (solid line), 1 (dashed line), 4 (dotted line). Shown are the shape of the fingers in each case after 100 iterations. In the case $\eta =4$ the finger growing from the right-angle starting at $z=-i$ is barely noticeable.