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The Hele-Shaw injection problem for an extremely shear-thinning fluid

Published online by Cambridge University Press:  23 July 2015

G. RICHARDSON  and
J. R. KING
G. RICHARDSON
Affiliation:
Mathematical Sciences, University of Southampton, Southampton, SO17 1BJ, UK email: [email protected]
J. R. KING
Affiliation:
School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK email: [email protected]
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Abstract

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We consider Hele-Shaw flows driven by injection of a highly shear-thinning power-law fluid (of exponent n) in the absence of surface tension. We formulate the problem in terms of the streamfunction ψ, which satisfies the p-Laplacian equation ∇·(|∇ψ|p−2∇ψ) = 0 (with p = (n+1)/n) and use the method of matched asymptotic expansions in the large n (extreme-shear-thinning) limit to find an approximate solution. The results show that significant flow occurs only in (I) segments of a (single) circle centred on the injection point, whose perimeters comprise the portion of free boundary closest to the injection point and (II) an exponentially small region around the injection point and (III) a transition region to the rest of the fluid: while the flow in the latter is exponentially slow it can be characterised in detail.

Keywords

power law fluidsmatched asymptotic expansionsfree boundary problemsp-Laplace equation
Type
Papers
Information
European Journal of Applied Mathematics , Volume 26 , Special Anniversay Issue 5: Celebrating 75 years , October 2015 , pp. 563 - 594
Copyright
Copyright © Cambridge University Press 2015 

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