Fingered growth in channel geometry: A Loewner-equation approach

A simple model of Laplacian growth is considered, in which the growth takes place only at the tips of long, thin fingers. Following Carleson and Makarov [L. Carleson and N. Makarov, J. Anal. Math. 87, 103 (2002)], the evolution of the fingers is studied with use of the deterministic Loewner equation. The method is then extended to study the growth in a linear channel with reflecting sidewalls. One- and two-finger solutions are found and analyzed. It turns out that the presence of the walls has a significant influence on the shapes of the fingers and the dynamics of the screening process, in which longer fingers suppress the growth of the shorter ones.

Figure 1

Fingering in the combustion experiments [3] and the theoretical model (bottom). The scale bar corresponds to 1 cm. (The photograph is reproduced from [3].)

Figure 2

Competition in the fingered growth in combustion experiments [3]. Initially equal sized fingers (bottom) evolve toward a state where every other finger stops growing (top). The scale bar corresponds to 1 cm.

Figure 3

Competitive dynamics of the channels in the dissolving rock fracture. The figures present the dissolution patterns at two different time points. Similar to Fig. 2, longer channels grow faster and suppress the growth of the shorter ones [8].

Figure 4

An example configuration of three fingers $\left({\Gamma }_1,{\Gamma }_2,{\Gamma }_3\right)$ extending from the real axis in the physical plane ($z$ plane). The mapping $g_t$ maps the exterior of the fingers onto the empty half-plane ($\omega$ plane). The images of the tips ${\gamma }_i$ are located on the real line at the points $x=a_i$. The gradient lines of the Laplacian field (dot-dashed) in the $z$ plane are mapped onto the vertical lines in the $\omega$ plane. In a given moment of time, the fingers grow along the gradient lines, the images of which pass through the points $a_i$.

Figure 5

Illustration of the composition of conformal maps described in the text. The mapping $g_{t+\tau }$ is obtained as the composition of $g_t$ and the elementary slit mapping (13).

Figure 6

Two fingers growing in the half-plane for $\eta =-2$ (left panel) and $\eta =4$ (right panel). The initial positions of the fingers are $a_1\left(0\right)=-a_2\left(0\right)=0.1$.

Figure 7

The geometry of the channel together with the boundary conditions.

Figure 8

Schematic picture of the mappings used in construction of elementary slit mapping $\varphi$ in the channel geometry.

Figure 9

The growth of a single finger in the channel with $a\left(0\right)=-0.5$.

Figure 10

The channel $A$ and its mirror reflection $A^{\prime }$.

Figure 11

Two fingers growing in the cylinder with $a_{1,2}\left(0\right)=\pm 0.25$ and $\eta =-2$ (stable symmetric solution).

Figure 12

The scaling of a Laplacian field $u$ in rectangular and wedgelike fjords.

Figure 13

Two fingers growing in the cylinder (left panel) and in the reflecting channel (right panel) for $\eta =1$. The initial positions of the fingers are $a_1\left(0\right)=-0.2$ and $a_2\left(0\right)=0.4$. Inset: the interaction of two fingers in the combustion experiments of Zik and Moses [3].

Figure 14

Two fingers growing in the reflecting channel at $\eta =1$ (solid line) and $\eta =-2$ (dashed). The initial positions of the fingers are the same as those in Fig. 13.

Figure 15

The velocities of the fingers as a function of time in the cylinder (left) and in the channel (right), corresponding to the fingers of Fig. 13.

Figure 16

The positions of the poles as a function of time, in the cylinder (left) and in the channel (right), corresponding to the fingers of Fig. 13.