Many of the patterns that can be observed in nature or in experiments are the result of surface instabilities. These structures typically arise when an interface is forced to move due to fluxes of heat and/or mass or because of a mechanical forcing that leads to a pressure gradient. A small perturbation of the moving boundary will either grow unstably or decay such that the boundary recovers its original shape. The stability of such a perturbation is determined by the geometry of the surface together with the thermodynamic and material properties relevant to the specific problem. If the conditions are such that the moving boundary is unstable, it will evolve into some kind of a pattern. The characteristics and morphology of the final structure will also depend on the parameters that determine the stability of the interface.
If the temperature, concentration or pressure field that controls the process satisfy the Laplace equation, it belongs to an important category of surface evolution processes called Laplacian growth problems. In such problems, the interface velocity is a function of the gradient of the Laplacian field. Laplacian growth is viewed as a fundamental model for pattern formation [Bazant and Crowdy, 2005]. Both radial and directional growth are considered in this thesis.
Laplacian growth problems can be solved by means of conformal mapping techniques. Viscous fingering, electrochemical deposition and the growth of bacterial colonies are examples of Laplacian growth processes that often occur in radial geometry. There are many different approaches to the study of these patterns, both discrete and continuous. In order to obtain a conformally invariant boundary condition in the model, surface tension must be zero or constant. If no other regularisation is introduced, this will lead to the formation of singularities on the boundary in finite time.
The techniques used in radial geometry involve analytic solutions that are harder to handle for directional growth. The Loewner differential equation for conformal maps is therefore used to study these problems. Examples of relevant processes are fluidisation experiments, channel formation in dissolving rocks and other types of experiments or field observations that involve fingered growth. Common for all of these examples, and for the model, is that the growth of the interface is concentrated at the tips of long, thin fingers. There is a growth competition between the fingers, with the longer ones growing faster than the shorter ones, leading to a broad distribution of finger lengths.