A fundamental example of pattern formation under far-from-equilibrium
conditions is given by the famous diffusion-limited-aggregation (DLA)
model, introduced by Witten and Sander in 1981. This growth model was shown to be the basic mechanism for the formation of complicated, fractal
patterns in a variety of chemical, biological and geological systems.
In this talk I will introduce a generalization of DLA to a wide family of
2d growth models, limited by transport processes which are not necessarily
diffusive. This progress is enabled by combining two recent developments:
(1) The Hastings-Levitov formalism (1998), expressing DLA as an iteration
of conformal maps , and (2) The application of conformal map methods to
solve a wide family of transport equations (Bazant 2003).
The theoretical and computational advantages of our formalism will be
demonstrated through a new growth model, namely an aggregation process
limited by an advective-diffusive transport (where advection is controlled
by a potential flow). This model is important in particular for studying
aggregation processes in porous media exposed to underground flow over
geological time scales, and may also be useful in capturing certain
features of growth of bacterial colonies.
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