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

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 ψ, which satisfies the p-Laplacian equation ∇. ( ∇ψ p-2∇ψ)=0 (with p=(n+1)/n), and investigate travelling wave solutions in the large-n (extreme shear-thinning) limit. We take a Legendre transform of the free-boundary problem for ψ, 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. © The author 2007. Published by Oxford University Press; all rights reserved.