Contracting bubbles in Hele-Shaw cells with a power-law fluid

McCue, Scott W. & King, John R. (2011) Contracting bubbles in Hele-Shaw cells with a power-law fluid. Nonlinearity, 24, pp. 613-641.

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The problem of bubble contraction in a Hele-Shaw cell is studied for the case in which the surrounding fluid is of power-law type. A small perturbation of the radially symmetric problem is first considered, focussing on the behaviour just before the bubble vanishes, it being found that for shear-thinning fluids the radially symmetric solution is stable, while for shear-thickening fluids the aspect ratio of the bubble boundary increases. The borderline (Newtonian) case considered previously is neutrally stable, the bubble boundary becoming elliptic in shape with the eccentricity of the ellipse depending on the initial data. Further light is shed on the bubble contraction problem by considering a long thin Hele-Shaw cell: for early times the leading-order behaviour is one-dimensional in this limit; however, as the bubble contracts its evolution is ultimately determined by the solution of a Wiener-Hopf problem, the transition between the long-thin limit and the extinction limit in which the bubble vanishes being described by what is in effect a similarity solution of the second kind. This same solution describes the generic (slit-like) extinction behaviour for shear-thickening fluids, the interface profiles that generalise the ellipses that characterise the Newtonian case being constructed by the Wiener-Hopf calculation.

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8 citations in Scopus
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5 citations in Web of Science®

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ID Code: 39131
Item Type: Journal Article
Refereed: Yes
Keywords: Hele-Shaw flow, power-law fluid, bubble contraction, formal asymptotics, extinction problem, moving boundary problem, Wiener-Hopf technique
DOI: 10.1088/0951-7715/24/2/009
ISSN: 0951-7715 (Print) 1361-6544 (Online)
Subjects: Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > APPLIED MATHEMATICS (010200) > Approximation Theory and Asymptotic Methods (010201)
Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > APPLIED MATHEMATICS (010200) > Theoretical and Applied Mechanics (010207)
Divisions: Past > QUT Faculties & Divisions > Faculty of Science and Technology
Past > Schools > Mathematical Sciences
Copyright Owner: Copyright 2011 Institute of Physics
Deposited On: 13 Dec 2010 01:22
Last Modified: 21 Jan 2011 01:04

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