Thin film of non-Newtonian fluid on an incline

The slow flow of thin liquid films on solid surfaces is an important phenomenon in nature and in industrial processes, and an intensive effort has been made to investigate it. It is well known that the contact line of currents on an inclined surface may become unstable and then a pattern of "fingers" develops that affects the quality of the coatings. This instability has been intensively studied due to its relevance for the technology of various industrial processes. So far the theoretical and numerical research has been focused on Newtonian fluids, notwithstanding that often in the real situations as well as in the experiments, the rheology of the involved liquid is non-Newtonian. Using the lubrication approximation, we derive the governing equations for a current of a power law non-Newtonian fluid on an inclined plane under the action of gravity and the viscous stresses. We show that surface tension effects can be included in the theory by a slight modification of the governing equations, that can then be used as a starting point to investigate the influence of rheology on the fingering instability and other phenomena of interest. We consider the one-dimensional case and we present three families of traveling wave solutions: two running downwards and the other upwards.