Asymmetric collapse by dissolution or melting in a uniform flow
An advectiondiffusionlimited dissolution model of an object being eroded by a twodimensional 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 timedependent 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 nonAnalytic 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 nonAnalytic function is examined for three different test cases, and a practical approach to determine the collapse point using a generalized NewtonRaphson rootfinding 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 pinchoff in fluid mechanics). In conclusion, the model raises fundamental mathematical questions about broken symmetries in finiteTime singularities of both continuous and stochasticmore »
 Authors:

^{[1]}
;
^{[2]}
 Harvard Univ., Cambridge, MA (United States); Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
 Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States)
 Publication Date:
 Grant/Contract Number:
 AC0205CH11231
 Type:
 Accepted Manuscript
 Journal Name:
 Proceedings of the Royal Society. A. Mathematical, Physical and Engineering Sciences
 Additional Journal Information:
 Journal Volume: 472; Journal Issue: 2185; Journal ID: ISSN 13645021
 Publisher:
 The Royal Society Publishing
 Research Org:
 Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
 Sponsoring Org:
 USDOE Office of Science (SC)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 42 ENGINEERING; conformal mapping; interfaces; dissolution; finitetime singularity; broken symmetry
 OSTI Identifier:
 1378713
Rycroft, Chris H., and Bazant, Martin Z.. Asymmetric collapse by dissolution or melting in a uniform flow. United States: N. p.,
Web. doi:10.1098/rspa.2015.0531.
Rycroft, Chris H., & Bazant, Martin Z.. Asymmetric collapse by dissolution or melting in a uniform flow. United States. doi:10.1098/rspa.2015.0531.
Rycroft, Chris H., and Bazant, Martin Z.. 2016.
"Asymmetric collapse by dissolution or melting in a uniform flow". United States.
doi:10.1098/rspa.2015.0531. https://www.osti.gov/servlets/purl/1378713.
@article{osti_1378713,
title = {Asymmetric collapse by dissolution or melting in a uniform flow},
author = {Rycroft, Chris H. and Bazant, Martin Z.},
abstractNote = {An advectiondiffusionlimited dissolution model of an object being eroded by a twodimensional 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 timedependent 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 nonAnalytic 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 nonAnalytic function is examined for three different test cases, and a practical approach to determine the collapse point using a generalized NewtonRaphson rootfinding 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 pinchoff in fluid mechanics). In conclusion, the model raises fundamental mathematical questions about broken symmetries in finiteTime singularities of both continuous and stochastic dynamical systems.},
doi = {10.1098/rspa.2015.0531},
journal = {Proceedings of the Royal Society. A. Mathematical, Physical and Engineering Sciences},
number = 2185,
volume = 472,
place = {United States},
year = {2016},
month = {1}
}