Math Seminar, Princeton
March 31, 2005
Title: Transport and Aggregation in Two Dimensions
Speaker: Martin Bazant
Associate Professor of Applied Mathematics
MIT
Abstract:
Perhaps the most ubiquitous equation in physics is Laplace's equation,
which is conformal invariant in two dimensions. This special property has
led to many theoretical advances, especially in nonlinear pattern
formation, such as (continuous) viscous fingering and (stochastic)
diffusion-limited aggregation (DLA). Here, we note that some other
equations are also conformally invariant and discuss new applications of
conformal mapping to advection-diffusion, gravity currents, and ion
transport. First, we present a variety of exact solutions to the
Navier-Stokes equations of fluid mechanics and the Nernst-Planck equations
of electrochemistry. Next, we generalize models of viscous fingering and
DLA (the Hastings-Levitov formulation) to a broad class of non-Laplacian
growth phenomena. By analyzing the case of advection-diffusion-limited
aggregation (DLA in a fluid flow), we shed light on the failure of the
mean-field approximation to describe the average shape of fractal
clusters.