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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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).