Hele-Shaw flow and pattern formation in a time-dependent gap

Michael Shelley, Fei Ran Tian, Krzysztof Wlodarski

Research output: Contribution to journalArticle

Abstract

We consider flow in a Hele-Shaw cell for which the upper plate is being lifted uniformly at a specified rate. This lifting puts the fluid under a lateral straining flow, sucking in the interface and causing it to buckle. The resulting short-lived patterns can resemble a network of connections with triple junctions. The basic instability - a variant of the Saffman-Taylor instability - is found in a version of the two-dimensional Darcy's law, where the divergence condition is modified to account for the lifting of the plate. For analytic data, we establish the existence, uniqueness and regularity of solutions when the surface tension is zero. We also construct some exact analytic solutions, both with and without surface tension. These solutions illustrate some of the possible behaviours of the system, such as cusp formation and bubble fission. Further, we present the results of numerical simulations of the bubble motion, examining in particular the distinctive pattern formation resulting from the Saffman-Taylor instability, and the effect of surface tension on a bubble evolution that in the absence of surface tension would fission into two bubbles.

Original languageEnglish (US)
Pages (from-to)1471-1495
Number of pages25
JournalNonlinearity
Volume10
Issue number6
DOIs
StatePublished - Nov 1997

Fingerprint

Hele-Shaw Flow
Pattern Formation
Surface Tension
Bubble
Surface tension
interfacial tension
bubbles
Taylor instability
fission
Hele-Shaw
Darcy's Law
Existence-uniqueness
Regularity of Solutions
Uniqueness of Solutions
Cusp
uniqueness
cusps
Analytic Solution
regularity
Existence of Solutions

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics
  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Hele-Shaw flow and pattern formation in a time-dependent gap. / Shelley, Michael; Tian, Fei Ran; Wlodarski, Krzysztof.

In: Nonlinearity, Vol. 10, No. 6, 11.1997, p. 1471-1495.

Research output: Contribution to journalArticle

Shelley, M, Tian, FR & Wlodarski, K 1997, 'Hele-Shaw flow and pattern formation in a time-dependent gap', Nonlinearity, vol. 10, no. 6, pp. 1471-1495. https://doi.org/10.1088/0951-7715/10/6/005
Shelley, Michael ; Tian, Fei Ran ; Wlodarski, Krzysztof. / Hele-Shaw flow and pattern formation in a time-dependent gap. In: Nonlinearity. 1997 ; Vol. 10, No. 6. pp. 1471-1495.
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