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 conformal 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 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 $Pe$ 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 $Pe\,{\approx}\,60$.