Inspired by river networks and other structures formed by Laplacian growth, we use the Loewner equation to investigate the growth of a network of thin fingers in a diffusion field. We first review previous contributions to illustrate how this formalism reduces the network's expansion to three rules, which respectively govern the velocity, the direction, and the nucleation of its growing branches. This framework allows us to establish the mathematical equivalence between three formulations of the direction rule, namely geodesic growth, growth that maintains local symmetry, and growth that maximizes flux into tips for a given amount of growth. Surprisingly, we find that this growth rule may result in a network different from the static configuration that optimizes flux into tips.

Pattern-forming processes, such as electrodeposition, dielectric breakdown, or viscous fingering, are often driven by instabilities. Accordingly, the resulting growth patterns are usually highly branched fractal structures. However, in some of the unstable growth processes the envelope of the structure grows in a highly regular manner, with the perturbations smoothed out over the course of time. In this paper we show that the regularity of the envelope growth can be connected to small-scale instabilities leading to the tip splitting of the fingers at the advancing front of the structure. Whenever the growth velocity becomes too large, the finger splits into two branches. In this way it can absorb an increased flux and thus damp the instability. Hence, somewhat counterintuitively, the instability at a small scale results in a stability at a larger scale. The quantitative analysis of these effects is provided by means of the Loewner equation, which one can use to reduce the problem of the interface motion to that of the evolution of the conformal mapping onto the complex plane. This allows an effective analysis of the multifingered growth in a variety of different geometries. We show how the geometry impacts the shape of the envelope of the growing pattern and compare the results with those observed in natural systems.

We study patterns formed by viscous fingering in a rectangular network of microfluidic channels. Due to the strong anisotropy of such a system, the emerging patterns have a form of thin needle-like fingers, which interact with each other, competing for an available flow. We develop an upscaled description of this system in which only the fingers are tracked and the effective interactions between them are introduced, mediated through the evolving pressure field. Due to the quasi-2d geometry of the system, this is conveniently accomplished using conformal mapping techniques. A complex two-phase flow problem is thus reduced to a much simpler task of tracking evolving shapes in a 2d complex plane. This description, although simplified, turns out to capture all the key features of the system's dynamics and allows for the effective prediction of the resulting growth patterns.

We consider viscous fingers created by injection of low viscosity fluid into the network of capillaries initially filled with a more viscous fluid (motor oil). Due to the anisotropy of the system and its geometry, such a setup promotes the formation of long-and-thin fingers which then grow and compete for the available flow, interacting through the pressure field. The interaction between the fingers is analyzed using the branched growth formalism of Halsey and Leibig (Phys. Rev. A 46, 7723, 1992) using a number of simple, analytically tractable models. It is shown that as soon as the fingers are allowed to capture the flow from one another, the fixed point appears in the phase space, corresponding to the asymptotic state in which the growth of one of the fingers in hindered by the other. The properties of phase space flows in such systems are shown to be remarkably insensitive to the details of the dynamics.