Contracting bubbles in HeleShaw cells with a powerlaw fluid
McCue, Scott W. & King, John R. (2011) Contracting bubbles in HeleShaw cells with a powerlaw fluid. Nonlinearity, 24, pp. 613641.

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Abstract
The problem of bubble contraction in a HeleShaw cell is studied for the case in which the surrounding fluid is of powerlaw 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 shearthinning fluids the radially symmetric solution is stable, while for shearthickening 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 HeleShaw cell: for early times the leadingorder behaviour is onedimensional in this limit; however, as the bubble contracts its evolution is ultimately determined by the solution of a WienerHopf problem, the transition between the longthin 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 (slitlike) extinction behaviour for shearthickening fluids, the interface profiles that generalise the ellipses that characterise the Newtonian case being constructed by the WienerHopf calculation.
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ID Code:  39131 

Item Type:  Journal Article 
Refereed:  Yes 
Keywords:  HeleShaw flow, powerlaw fluid, bubble contraction, formal asymptotics, extinction problem, moving boundary problem, WienerHopf technique 
DOI:  10.1088/09517715/24/2/009 
ISSN:  09517715 (Print) 13616544 (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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