Steady advection-diffusion around finite absorbers in two-dimensional potential flows

Abstract

We perform an exhaustive study of the simplest, nontrivial problem in advection-diffusion -- a finite absorber of arbitrary cross section in a steady two-dimensional potential flow of concentrated fluid. This classical problem has been studied extensively in the theory of solidification from a flowing melt, and it also arises in Advection-Diffusion-Limited Aggregation. In both cases, the fundamental object is the flux to a circular disk, obtained by conformal mapping from more complicated shapes. We construct the first accurate numerical solution using an efficient new method, which involves mapping to the interior of the disk and using a spectral method in polar coordinates. Our method also combines exact asymptotics and an adaptive mesh to handle boundary layers. Starting from a well-known integral equation in streamline coordinates, we also derive new, high-order asymptotic expansions for high and low Péclet numbers ($\Pe$). Remarkably, the `high' $\Pe$ expansion remains accurate even for such low $\Pe$ as $10^{-3}$. The two expansions overlap well near $\Pe = 0.1$, allowing the construction of an analytical connection formula that is uniformly accurate for all $\Pe$ and angles on the disk with a maximum relative error of 1.75%. We also obtain an analytical formula for the Nusselt number ($\Nu$) as a function of the Péclet number with a maximum relative error of 0.53% for all possible geometries. Because our finite-plate problem can be conformally mapped to other geometries, the general problem of two-dimensional advection-diffusion past an arbitrary finite absorber in a potential flow can be considered effectively solved.