Modele igłowe procesów wzrostu nierównowagowego

Non-equilibrium growth processes, such as electrodeposition, dielectric breakdown, viscous ngering, or even bacterial colonies formation, are often driven by instabilities. Accordingly, the resulting growth patterns are usually highly branched fractal structures. In all these processes the growth may be described in terms of a harmonic scalar eld Ψ, interpreted for instance as an electrostatic potential or pressure. Additionally, the front is assumed to grow with velocity proportional to the gradient of the eld. Such a growth problem is non-linear due to the boundary conditions the front is unstable under small perturbations. Therefore, even though the basic mechanisms of growth are well understood, the strongly non-linear character of the process makes the latter stages of evolution very complicated, with a strong competition between spontaneously formed dendrite-like structures, and tip-splitting e ects when dendrites bifurcate into secondary branches. In the thesis we considered a simple model of non-equilibrium growth in two spatial dimensions, in which the growth takes place only at the tips of long-and-thin ngers. The quantitative analysis of the model was 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. In spite of being considerably simpli ed, the model allows to describe a strong, nonlinear interaction between the ngers and their competition due to the long-range screening. In the rst part we applied the thin nger model to description of the growth processes in which the envelope of the rami ed structure grows in a highly regular manner, with the perturbations smoothed out over the course of time. We showed that the regularity of the envelope growth can be connected to small-scale instabilities leading to the tip splitting of the ngers at the advancing front of the structure. Whenever the growth velocity becomes too large, the nger splits into two branches. In this way it can absorb an increased ux and thus damp the instability. Hence, somewhat counterintuitively, the instability at a small scale results in a stability at a larger scale. In particular we analyzed the growth in a half-plane geometry, in which case the envelopes form perfect semi-circular shapes with non-uniform intensity of the splitting process along the interface. Interestingly, a similar e ect can be observed in some 2D combustion experiments. In the second part we studied patterns formed by viscous ngering in a rectangular network of micro uidic channels. Due to the strong anisotropy of such a system, the emerging patterns have a form of thin needle-like ngers, which interact with each other, competing for an available ow. We developed an upscaled description of this system in which only the ngers are tracked and the e ective interactions between them are 1 introduced, mediated through the evolving pressure eld. A complex two-phase ow problem was thus reduced to a much simpler task of tracking evolving shapes in a 2d complex plane. This description, although simpli ed, turned out to capture all the key features of the system's dynamics and allowed for the e ective prediction of the resulting growth patterns.