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

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 psi, which satisfies the p-Laplacian equation del center dot (vertical bar del psi vertical bar(p-2)del psi) = 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.