Physics of Fluids
saffman-taylor instability, viscous fingering patterns, shear-thinning fluid, surface-tension, 2-phase displacement, viscoelastic fluid, polymer-solutions, liquid-crystals, cell, behavior
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.
Fast, P.; Kondic, L.; Shelley, M. J.; and Palffy-Muhoray, Peter (2001). Pattern Formation in Non-Newtonian Hele-Shaw Flow. Physics of Fluids 13(5), 1191-1212. doi: 10.1063/1.1359417 Retrieved from https://digitalcommons.kent.edu/cpippubs/239