P. Fast, L. Kondic,
Michael J. Shelley, and
Peter Palffy-Muhoray,
Pattern formation in Non-Newtonian Hele-Shaw flow
, in preparation
Abstract:
We study 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. The Johnson-Segalman-Oldroyd 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. 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.