Wavy regime of a power-law film flow

We consider a power-law fluid flowing down an inclined plane under the action of gravity. The divergence of the viscosity at zero strain rate is taken care of by introducing a Newtonian plateau at small strain rate. Two-equation models are formulated within the framework of lubrication theory in terms of the exact mass balance and an averaged momentum equation, which form a set of evolution equations for the film thickness h, a local velocity amplitude or the flow rate q. The models account for the streamwise diffusion of momentum. Comparisons with Orr-Sommerfeld stability analysis and with direct numerical simulation (DNS) show convincing agreement in both linear and nonlinear regimes. The influence of shear-thinning or shear-thickening on the primary instability is shown to be non-trivial. A destabilization of the base flow close to threshold is promoted by the shear-thinning effect, whereas, further from threshold, it tends to stabilize the base flow when the viscous damping of short waves becomes dominant. A reverse situation is observed in the case of shear-thickening fluids. Shear-thinning accelerates solitary waves and promotes a subcritical onset of travelling waves at larger wavenumber than the linear cut-off wavenumber. A conditional stability of the base flow is thus observed. This phenomenon results from a reduction of the effective viscosity at the free surface. When compared with DNS, simulations of the temporal response of the film based on weighted residual models satisfactorily capture the conditional stability of the film.