Theory of creeping gravity currents of a non-Newtonian liquid

Recently several experiments on creeping gravity currents have been perform ed, using highly viscous silicone oils and putties. The interpretation of t he experiments relies on the available theoretical results that were obtain ed by means of the lubrication approximation with the assumption of a Newto nian rheology. Since very viscous fluids are usually non-Newtonian, an exte nsion of the theory to include non-Newtonian effects is needed. We derive t he governing equations for unidirectional and axisymmetric creeping gravity currents of a non-Newtonian liquid with a power-law rheology, generalizing the usual lubrication approximation. The equations differ from those for N ewtonian liquids, being nonlinear in the spatial derivative of the thicknes s of the current. Similarity solutions for currents whose volume varies as a power of time are obtained. For the spread of a constant volume of liquid , analytic solutions are found that are in good agreement with experiment. We also derive solutions of the waiting-time type, as well as those describ ing steady flows from a constant source to a sink. General traveling-wave s olutions are given, and analytic formulas for a simple case are derived. A phase plane formalism that allows the systematic derivation of self-similar solutions is introduced. The application of the Boltzmann transform is bri efly discussed. All the self-similar solutions obtained here have their cou nterparts in Newtonian flows, as should be expected because the power-law r heology involves a single-dimensional parameter as the Newtonian constituti ve relation. Thus one finds similarity solutions whenever the analogous New tonian problem is self-similar, but now the spreading relations are theolog y-dependent. In most cases this dependence is weak but leads to significant differences easily detected in experiments. The present results may also b e of interest for geophysics since the lithosphere deforms according to an average power-law rheology. [S1063-651X(99)09011-X].