L. Kondic,
Michael J. Shelley and
Peter Palffy-Muhoray,
On Models of Non-Newtonian Hele-Shaw Flow,
Physical Review E 54, R4536, 1996.
Abstract:
We study the Saffman-Taylor instability of a non-Newtonian fluid in a
Hele-Shaw cell. Using a fluid model with shear 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 related 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 Saffman-Taylor instability of
an expanding bubble in a radial cell.