Asymmetric collapse by dissolution or melting in a uniform flow
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 La...
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| Format: | Article |
| Language: | en_US |
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Royal Society
2017
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| Online Access: | http://hdl.handle.net/1721.1/108434 |
| _version_ | 1811084699405123584 |
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| author | Rycroft, Chris H. Bazant, Martin Z |
| author2 | Massachusetts Institute of Technology. Department of Chemical Engineering |
| author_facet | Massachusetts Institute of Technology. Department of Chemical Engineering Rycroft, Chris H. Bazant, Martin Z |
| author_sort | Rycroft, Chris H. |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T12:56:13Z |
| format | Article |
| id | mit-1721.1/108434 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T12:56:13Z |
| publishDate | 2017 |
| publisher | Royal Society |
| record_format | dspace |
| spelling | mit-1721.1/1084342022-10-01T12:00:47Z Asymmetric collapse by dissolution or melting in a uniform flow Rycroft, Chris H. Bazant, Martin Z Massachusetts Institute of Technology. Department of Chemical Engineering Massachusetts Institute of Technology. Department of Mathematics Bazant, Martin Z 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. United States. Department of Energy. Office of Science. Computational and Technology Research (Contract DE-AC02-05CH11231) 2017-04-26T20:33:08Z 2017-04-26T20:33:08Z 2016-01 2015-01 Article http://purl.org/eprint/type/JournalArticle 1364-5021 1471-2946 http://hdl.handle.net/1721.1/108434 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. en_US http://dx.doi.org/10.1098/rspa.2015.0531 Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Royal Society arXiv |
| spellingShingle | Rycroft, Chris H. Bazant, Martin Z Asymmetric collapse by dissolution or melting in a uniform flow |
| title | Asymmetric collapse by dissolution or melting in a uniform flow |
| title_full | Asymmetric collapse by dissolution or melting in a uniform flow |
| title_fullStr | Asymmetric collapse by dissolution or melting in a uniform flow |
| title_full_unstemmed | Asymmetric collapse by dissolution or melting in a uniform flow |
| title_short | Asymmetric collapse by dissolution or melting in a uniform flow |
| title_sort | asymmetric collapse by dissolution or melting in a uniform flow |
| url | http://hdl.handle.net/1721.1/108434 |
| work_keys_str_mv | AT rycroftchrish asymmetriccollapsebydissolutionormeltinginauniformflow AT bazantmartinz asymmetriccollapsebydissolutionormeltinginauniformflow |