## 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 »

- Authors:

- 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:

- 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}

}

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