MODELS OF NON-NEWTONIAN HELE-SHAW FLOW

We study the Saffman-Taylor instability of a non-Newtonian fluid in a Hele-Shaw cell. Using a fluid model with shear-ratz dependent viscosit y,we derive a Darcy's-law whose viscosity depends upon the squared pre ssure gradient. This yields a natural, nonlinear boundary value proble m for the pressure. A model proposed recently by Bonnet ai. [Phys. Rev . Lett. 75, 2132 (1995)] follows from this modified law. For a shear-t hinning liquid,our derivation shows strong constraints upon the fluid viscosity - strong shear-thinning does not allow the construction oi a unique Darcy's law, and is related to the appearance of slip layers i n the flow. For a weakly shear-thinning liquid, we calculate correctio ns to tile Newtonian instability of an expanding bubble in a radial ce ll.