Pattern formation in non-Newtonian Hele-Shaw flow

We study theoretically the Saffman-Taylor instability of an air bubble expa nding into a non-Newtonian fluid in a Hele-Shaw cell, with the motivation o f understanding suppression of tip-splitting and the formation of dendritic structures observed in the flow of complex fluids, such as polymeric liqui ds or liquid crystals. A standard visco-elastic flow model is simplified in the case of flow in a thin gap, and it is found that there is a distinguis hed limit where shear thinning and normal stress differences are apparent, but elastic response is negligible. This observation allows formulation of a generalized Darcy's law, where the pressure satisfies a nonlinear ellipti c boundary value problem. Numerical simulation shows that shear-thinning al one modifies considerably the pattern formation and can produce fingers who se tip-splitting is suppressed, in agreement with experimental results. The se fingers grow in an oscillating fashion, shedding "side-branches" from th eir tips, closely resembling solidification patterns. A careful analysis of the parametric dependencies of the system provides an understanding of the conditions required to suppress tip-splitting, and an interpretation of ex perimental observations, such as emerging length-scales. (C) 2001 American Institute of Physics.