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

KING, JOHN; Richardson, G.

Abstract

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.

Citation

KING, J., & Richardson, G. (2015). The Hele-Shaw injection problem for an extremely shear-thinning fluid. European Journal of Applied Mathematics, 26(5), 563-594. https://doi.org/10.1017/S095679251500039X

• Journal Article Type: Article
• Acceptance Date: Jun 30, 2015
• Online Publication Date: Jul 23, 2015
• Publication Date: Oct 1, 2015
• Deposit Date: Aug 3, 2018
• Journal: European Journal of Applied Mathematics
• Print ISSN: 0956-7925
• Electronic ISSN: 1469-4425
• Publisher: Cambridge University Press (CUP)
• Peer Reviewed: Peer Reviewed
• Volume: 26
• Issue: 5
• Pages: 563-594
• DOI: https://doi.org/10.1017/S095679251500039X
• Keywords: Power law ?uids, Matched asymptotic expansions, Free boundary problems, p-Laplace equation
• Public URL: https://nottingham-repository.worktribe.com/output/1115349
• Publisher URL: https://www.cambridge.org/core/journals/european-journal-of-applied-mathematics/article/heleshaw-injection-problem-for-an-extremely-shearthinning-fluid/F08BE241A2D5C4B47720846B7BCCA186
• PMID: 00036099