skip to main content
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Asymmetric collapse by dissolution or melting in a uniform flow

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). In conclusion, the model raises fundamental mathematical questions about broken symmetries in finite-Time singularities of both continuous and stochasticmore » dynamical systems.« less

Authors:
ORCiD logo [1];  [2]
  1. Harvard Univ., Cambridge, MA (United States); Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
  2. Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States)
Publication Date:
Research Org.:
Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
Sponsoring Org.:
USDOE Office of Science (SC)
OSTI Identifier:
1378713
Grant/Contract Number:  
AC02-05CH11231
Resource 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 1364-5021
Publisher:
The Royal Society Publishing
Country of Publication:
United States
Language:
English
Subject:
42 ENGINEERING; conformal mapping; interfaces; dissolution; finite-time singularity; broken symmetry

Citation Formats

Rycroft, Chris H., and Bazant, Martin Z. Asymmetric collapse by dissolution or melting in a uniform flow. United States: N. p., 2016. 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. Wed . "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 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). In conclusion, the model raises fundamental mathematical questions about broken symmetries in finite-Time 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}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record

Citation Metrics:
Cited by: 4 works
Citation information provided by
Web of Science

Save / Share: