Release of a viscous power-law fluid over an inviscid ocean

We consider the two- and three-dimensional spreading of a finite volume of viscous power-law fluid released over a denser inviscid fluid and subject to gravitational and capillary forces. In the case of gravity-driven spreading, with a power-law fluid having strain rate proportional to stress to the power n, there are similarity solutions with the extent of the current being proportional to t(1/n) in the two-dimensional case and t(1/2n) in the three-dimensional case. Perturbations from these asymptotic states are shown to retain their initial shape but to decay relatively as t(-1) in the two-dimensional case and t(-3/(n+3)) in the three-dimensional case. The former is independent of n, whereas the latter gives a slower rate of relative decay for fluids that are more shear-thinning. In cases where the layer is subject to a constraining surface tension, we determine the evolution of the layer towards a static state of uniform thickness in which the gravitational and capillary forces balance. The asymptotic form of this convergence is shown to depend strongly on n, with rapid finite-time algebraic decay in shear-thickening cases, large-time exponential decay in the Newtonian case and slow large-time algebraic decay in shear-thinning cases.