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

We consider perhaps the simplest non-trivial problem in advection-diffusion - a finite absorber of arbitrary cross-section in a steady two-dimensional potential flow of concentrated fluid. This 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 con-formal mapping from more complicated shapes. Here, we construct an accurate numerical solution by an efficient method that involves mapping to the interior of the disk and using a spectral method in polar coordinates. The method combines exact asymptotics and an adaptive mesh to handle boundary layers. Starting from a well-known integral equation in streamline coordinates, we also derive high-order asymptotic expansions for high and low Peclet 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 Pe with a maximum relative error of 0.53 % for all possible geometries after conformal mapping. Considering the concentration disturbance around a disk, we find that the crossover from a diffusive cloud (at low Pe) to an advective wake (at high Pe) occurs at Pe approximate to 60.