## 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 et al. [Phys. Rev. Lett. 75, 2132 (1995)] 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.

Original language | English (US) |
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Pages (from-to) | R4536-R4539 |

Journal | Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics |

Volume | 54 |

Issue number | 5 |

DOIs | |

State | Published - 1996 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Statistics and Probability
- Condensed Matter Physics