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.