On Models of Non-Newtonian Hele-Shaw Flow
with Lou Kondic and Peter Palffy-Muhoray, in Physical Review E, Rapid
Communications, 54, page 4536 (1996).
Abstract
We study the Saffman-Taylor instability of a non-Newtonian fluid in a Hele-Shaw
cell. Using a fluid model with shear-rate dependent viscosity, we derive
a Darcy's law whose viscosity depends upon the squared pressure gradient.
This yields a natural, nonlinear boundary value problem for the pressure.
A model proposed recently by Bonn, Kellay, Ben Amar, and Meunier follows
from this modified law. For a shear-thinning liquid, our derivation shows
strong constraints upon the fluid viscosity -- strong shear-thinning does
not allow the construction of a unique Darcy's law, and is related to the
appearance of slip layers in the flow. For a weakly shear-thinning liquid,
we calculate corrections to the Newtonian instability of an expanding bubble
in a radial cell.
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Non-Newtonian Hele-Shaw flow and the Saffman-Taylor instability
with Lou Kondic and Peter Palffy-Muhoray, in Physical Review Letters
80, page 1433 (1997).
Abstract
We explore the Saffman-Taylor instability of an air bubble expanding into
a shear thinning liquid in a radial Hele-Shaw cell. Using our previously
derived generalization of Darcy's law (Phys. Rev. E, {\bf 54}, 4536, 1996)
for non-Newtonian fluids, we perform time-dependent numerical simulations
of the full dynamical problem. These simulations show that a shear-rate
dependent viscosity of the driven fluid significantly influences the developing
interfacial patterns. Shear thinning can suppress tip-splitting and produce
fingers which grow in an oscillating fashion, shedding ``side-branches''
from their tips. Emergent length-scales show reasonable agreement with
the results of a general linear stability analysis.
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Pattern formation in non-Newtonian Hele-Shaw flow
with P. Fast, L. Kondic, and P. Palffy-Muhoray, in Physics of Fluids
13, page 1191 (2001).
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. 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. 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.
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