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

Adriana J. Pesci; Raymond E. Goldstein; Michael J. Shelley

doi:10.1063/1.870139

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

Adriana J. Pesci, Raymond E. Goldstein, Michael J. Shelley

Mathematics

Research output: Contribution to journal › Article › peer-review

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
DOIs	https://doi.org/10.1063/1.870139
State	Published - Oct 1999

ASJC Scopus subject areas

Computational Mechanics
Condensed Matter Physics
Mechanics of Materials
Mechanical Engineering
Fluid Flow and Transfer Processes

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10.1063/1.870139

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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.",

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

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