Study of growth phenomena via prototypical advection-diffusion problem in 2D

The simplest nontrivial problem in advection-diffusion involves a finite absorber of arbitrary cross section in a steady two-dimensional (2D) potential flow of concentrated fluid. The corresponding, ``classical'' boundary-value problem arises in different physical contexts, and has been studied in the theory of solidification from a flowing melt and of Advection-Diffusion-Limited Aggregation (ADLA), which is a discrete model describing the growth of fractal aggregates in a fluid flow via a stochastic conformal map. In ADLA the quantity of interest is the particle flux to a circular disk, which is obtained via conformal mapping from more complicated shapes, as a function of the local coordinate of the boundary and of a parameter, the Peclet number (Pe), which measures the relative strength of advection compared to diffusion.

My work focused on the derivation of apparently new, simple analytical formulae involving known functions for the flux on the absorber boundary, using methods from the theory of integral equations, in particular the Wiener-Hopf method of factorization. A surprising feature of these formulae is their remarkable accuracy uniformly in the local coordinate of the boundary for all values of Pe, with a maximum relative error of 1.75%.

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Related Papers:

1. J. Choi, D. Margetis, T. M. Squires, and M. Z. Bazant, Steady advection-diffusion around finite absorbers in two-dimensional potential flows (PDF), Journal of Fluid Mechanics, Vol. 536, pp. 155-184 (2005).

2. D. Margetis and J. Choi, Generalized iteration method for first-kind integral equations (PDF), Studies in Applied Mathematics, Vol. 117, pp. 1-25 (2006).