Asymmetric collapse by dissolution or melting in a uniform flow

TL;DR

This paper models the erosion of objects in a uniform flow using conformal invariance, revealing exact collapse points and exploring the mathematical nature of finite-time singularities in dissolution processes.

Contribution

It introduces a numerical method based on Laurent series for tracking boundary evolution and uncovers a surprising exact relationship for collapse points in dissolution models.

Findings

Collapse points are roots of a non-analytic function related to flow and shape.

The model can break down before complete collapse due to boundary overlap.

A practical root-finding algorithm for collapse points is proposed.

Abstract

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.