Shear flow of a Newtonian fluid over a quiescent generalized Newtonian fluid

The flow of an upper shear-driven Newtonian fluid above an otherwise still non-Newtonian fluid is considered. The lower fluid is modelled as a generalized Newtonian fluid and set into motion by interfacial shear. By means of similarity transformations, the governing partial differential equations for the two-fluid problem transform exactly into two sets of ordinary differential equations coupled only at the interface. The successful transformation of the two-fluid problem is applied to the particular case when the lower fluid obeys power-law rheology. The resulting three-parameter problem is solved numerically for some different parameter combinations by means of a direct integration approach with the density ratio fixed to unity. We observed that the interfacial velocities decreased with increasing values of the power-law index n in the range from 0.6 to 1.4 whereas the shear-induced motion of the lower fluid penetrates far deeper into a shear-thinning (n < 1) than into a shear-thickening (n > 1) fluid. This phenomenon is ascribed to a corresponding increase of the non-linear viscosity function with lower n-values.