Asymmetric collapse by dissolution or melting in a uniform flow
Name
Bazant_Asymmetric collapse.pdf
Size
3.81 MB
Format
Adobe PDF
Checksum (MD5)
4c5f34a6bdfad3b90f532b9636d1d08a
Author(s)
Rycroft, Chris H.
Bazant, Martin Z
Date Issued
January 2016
Journal
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science
Publisher
Royal Society
Citation
Rycroft, Chris H., and Martin Z. Bazant. “Asymmetric Collapse by Dissolution or Melting in a Uniform Flow.” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science 472.2185 (2016): 20150531.
Version
Original manuscript
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 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). The model raises fundamental mathematical questions about broken symmetries in finite-time singularities of both continuous and stochastic dynamical systems.
MIT Department
Massachusetts Institute of Technology. Department of Chemical Engineering
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1098/rspa.2015.0531
