  An advection--diffusion-limited dissolution model of an object being eroded
by a two-dimensional potential flow is presented. By taking advantage of the
conformal invariance of the model, a numerical method is introduced that tracks
the evolution of the object boundary in terms of a time-dependent Laurent
series. Simulations of a variety of dissolving objects are shown, which shrink
and then collapse to a single point in finite time. The simulations reveal a
surprising exact relationship whereby the collapse point is the root of a
non-analytic function given in terms of the flow velocity and the Laurent
series coefficients describing the initial shape. This result is subsequently
derived using residue calculus. The structure of the non-analytic function is
examined for three different test cases, and a practical approach to determine
the collapse point using a generalized Newton--Raphson root-finding algorithm
is outlined. These examples also illustrate the possibility that the model
breaks down in finite time prior to complete collapse, due to a topological
singularity, as the dissolving boundary overlaps itself rather than breaking up
into multiple domains (analogous to droplet pinch-off in fluid mechanics). In
summary, the model raises fundamental mathematical questions about broken
symmetries in finite-time singularities of both continuous and stochastic
dynamical systems.
