### Abstract

Recent work [Phys. Fluids 10, 2701 (1998)] has shown that for Hele-Shaw flows sufficiently near a finite-time pinching singularity, there is a breakdown of the leading-order solutions perturbative in a small parameter ∈ controlling the large-scale dynamics. To elucidate the nature of this breakdown we study the structure of these solutions at higher order. We find a finite radius of convergence that yields a new length scale exponentially small in ∈. That length scale defines a ball in space and time, centered around the incipient singularity, inside of which perturbation theory fails. Implications of these results for a possible matching of outer solutions to inner scaling solutions are discussed.

Original language | English (US) |
---|---|

Pages (from-to) | 2809-2811 |

Number of pages | 3 |

Journal | Physics of Fluids |

Volume | 11 |

Issue number | 10 |

State | Published - Oct 1999 |

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### ASJC Scopus subject areas

- Fluid Flow and Transfer Processes
- Computational Mechanics
- Mechanics of Materials
- Physics and Astronomy(all)
- Condensed Matter Physics

### Cite this

*Physics of Fluids*,

*11*(10), 2809-2811.

**Domain of convergence of perturbative solutions for Hele-Shaw flow near interface collapse.** / Pesci, Adriana J.; Goldstein, Raymond E.; Shelley, Michael.

Research output: Contribution to journal › Article

*Physics of Fluids*, vol. 11, no. 10, pp. 2809-2811.

}

TY - JOUR

T1 - Domain of convergence of perturbative solutions for Hele-Shaw flow near interface collapse

AU - Pesci, Adriana J.

AU - Goldstein, Raymond E.

AU - Shelley, Michael

PY - 1999/10

Y1 - 1999/10

N2 - Recent work [Phys. Fluids 10, 2701 (1998)] has shown that for Hele-Shaw flows sufficiently near a finite-time pinching singularity, there is a breakdown of the leading-order solutions perturbative in a small parameter ∈ controlling the large-scale dynamics. To elucidate the nature of this breakdown we study the structure of these solutions at higher order. We find a finite radius of convergence that yields a new length scale exponentially small in ∈. That length scale defines a ball in space and time, centered around the incipient singularity, inside of which perturbation theory fails. Implications of these results for a possible matching of outer solutions to inner scaling solutions are discussed.

AB - Recent work [Phys. Fluids 10, 2701 (1998)] has shown that for Hele-Shaw flows sufficiently near a finite-time pinching singularity, there is a breakdown of the leading-order solutions perturbative in a small parameter ∈ controlling the large-scale dynamics. To elucidate the nature of this breakdown we study the structure of these solutions at higher order. We find a finite radius of convergence that yields a new length scale exponentially small in ∈. That length scale defines a ball in space and time, centered around the incipient singularity, inside of which perturbation theory fails. Implications of these results for a possible matching of outer solutions to inner scaling solutions are discussed.

UR - http://www.scopus.com/inward/record.url?scp=0000107553&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000107553&partnerID=8YFLogxK

M3 - Article

VL - 11

SP - 2809

EP - 2811

JO - Physics of Fluids

JF - Physics of Fluids

SN - 1070-6631

IS - 10

ER -