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The Saffman-Taylor problem for an extremely shear-thinning fluid

The Saffman-Taylor problem for an extremely shear-thinning fluid
The Saffman-Taylor problem for an extremely shear-thinning fluid
We consider a steady flow driven by pushing a finger of gas into a highly shear-thinning power-law fluid, with exponent n, in a Hele-Shaw channel. We formulate the problem in terms of the streamfunction , which satisfies the p-Laplacian equation (with ), and investigate travelling wave solutions in the large-n (extreme shear-thinning) limit. We take a Legendre transform of the free-boundary problem for , which reduces it to a linear problem on a fixed domain. The solution to this problem is found by using matched asymptotic expansions and the resulting shape of the finger deduced (being, to leading order, a semi-infinite strip). The nonlinear problem for the streamfunction is also treated using matched asymptotic expansion in the physical plane. The finger-width selection problem is briefly discussed in terms of our results.
0033-5614
161-200
Richardson, Giles
3fd8e08f-e615-42bb-a1ff-3346c5847b91
King, J.
c28e1d73-7f6a-4e59-9354-bbba329f1b01
Richardson, Giles
3fd8e08f-e615-42bb-a1ff-3346c5847b91
King, J.
c28e1d73-7f6a-4e59-9354-bbba329f1b01

Richardson, Giles and King, J. (2007) The Saffman-Taylor problem for an extremely shear-thinning fluid. Quarterly Journal of Mechanics and Applied Mathematics, 60 (2), 161-200. (doi:10.1093/qjmam/hbm003).

Record type: Article

Abstract

We consider a steady flow driven by pushing a finger of gas into a highly shear-thinning power-law fluid, with exponent n, in a Hele-Shaw channel. We formulate the problem in terms of the streamfunction , which satisfies the p-Laplacian equation (with ), and investigate travelling wave solutions in the large-n (extreme shear-thinning) limit. We take a Legendre transform of the free-boundary problem for , which reduces it to a linear problem on a fixed domain. The solution to this problem is found by using matched asymptotic expansions and the resulting shape of the finger deduced (being, to leading order, a semi-infinite strip). The nonlinear problem for the streamfunction is also treated using matched asymptotic expansion in the physical plane. The finger-width selection problem is briefly discussed in terms of our results.

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Published date: 2007
Organisations: Applied Mathematics

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Local EPrints ID: 156347
URI: http://eprints.soton.ac.uk/id/eprint/156347
ISSN: 0033-5614
PURE UUID: b98dc054-df58-42e8-9467-61cf239648f0

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Date deposited: 03 Jun 2010 09:29
Last modified: 07 Jan 2022 21:02

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Author: J. King

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