Dynamics and stability of a power-law film flowing down a slippery slope

A power-law fluid flowing down a slippery inclined plane under the action of gravity is deliberated in this research work. A Newtonian layer at a small strain rate is introduced to take care of the divergence of the viscosity at a zero strain rate. A low-dimensional two-equation model is formulated using a weighted-residual approach in terms of two coupled evolution equations for the film thickness h and a local velocity amplitude or the flow rate q within the framework of lubrication theory. Moreover, a long-wave instability is shown in detail. Linear stability analysis of the proposed two-equation model reveals good agreement with the spatial Orr-Sommerfeld analysis. The influence of a wall-slip on the primary instability has been found to be non-trivial. It has the stabilizing effect at larger values of the Reynolds number, whereas at the onset of the instability, the role is destabilizing which may be because of the increase in dynamic wave speed by the wall slip. Competing impressions of shear-thinning/shear-thickening and wall slip velocity on the primary instability are captured. The impact of slip velocity on the traveling-wave solutions is discussed using the bifurcation diagram. An increasing value of the slip shows a significant effect on the traveling wave and free surface amplitude. Slip velocity controls both the kinematic and dynamic waves of the system, and thus, it has the profound passive impact on the instability.