We study theoretically the Saffman-Taylor instability of an air bubble expanding into a non-Newtonian fluid in a Hele-Shaw cell, with the motivation of understanding suppression of tip-splitting and the formation of dendritic structures observed in the flow of complex fluids, such as polymeric liquids 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 distinguished 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 elliptic boundary value problem. Numerical simulation shows that shear-thinning alone modifies considerably the pattern formation and can produce fingers whose tip-splitting is suppressed, in agreement with experimental results. These fingers grow in an oscillating fashion, shedding "side-branches" from their 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 experimental observations, such as emerging length-scales. (C) 2001 American Institute of Physics.
Physics of Fluids
Copyright 2001 American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. The following article appeared in Phys. Fluids 13, 1191 (2001) and may be found at http://dx.doi.org/10.1063/1.1359417
Fast, P., Kondic, L., Shelley, M., & Palffy-Muhoray, P. (2001). Pattern Formation in Non-Newtonian Hele-Shaw Flow. Physics of Fluids. https://doi.org/10.1063/1.1359417
Fast, P., Ljubinko Kondic, Michael Shelley, and Peter Palffy-Muhoray. 2001. “Pattern Formation in Non-Newtonian Hele-Shaw Flow”. Physics of Fluids. https://doi.org/10.1063/1.1359417.
Fast, P., L. Kondic, M. Shelley, and P. Palffy-Muhoray. Pattern Formation in Non-Newtonian Hele-Shaw Flow. Physics of Fluids, 1 Jan. 2001, doi:10.1063/1.1359417.