  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 P\'eclet
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 P\'eclet 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.
