Superposed layers of fluid flowing down an inclined plane are prone to interfacial instability even in the limit of zero Reynolds number. This situation can be explored by making use of a lubrication-style approximation of the governing fluid equations. Two versions of the lubrication theory are presented for superposed layers of non-Newtonian fluid with power-law rheology. First, the fluids are assumed to have comparable effective viscosities. The approximation then furnishes a simplified model for which the linear stability problem can be solved analytically and concisely. Weakly nonlinear analysis and numerical computations indicate that instabilities saturate supercritically beyond onset and form steady wavetrains. Further from onset, secondary instabilities arise that destroy trains of widely spaced wave trains. Patterns of closely spaced waves, on the other hand, coarsen due to wave merger events. The two mechanisms select steady wavetrains with wavelengths lying within a prescribed range. The second lubrication theory assumes that the upper layer is far more viscous than the lower layer. As a result, the upper fluid flows almost rigidly, and extensional stresses can become promoted into the leading-order balance of forces. Interfacial instability still arises in Newtonian fluid layers, and the nonlinear dynamics is qualitatively unchanged. Significant complications arise when the upper fluid is non-Newtonian due to the behaviour of the viscosity at zero strain rate.
Co-authored with N. J. Balmforth and C. Toniolo