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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Modelling and Simulation of Pattern formation in Non-Newtonian Hele-Shaw flow

with P. Fast, L. Kondic, and P. Palffy-Muhoray), submitted to Physics of Fluids, 1999.

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