A falling film down a slippery inclined plane

A gravity-driven film flow on a slippery inclined plane is considered within the framework of long-wave and boundary layer approximations. Two coupled depth-averaged equations are derived in terms of the local flow rate q(x,t) and the film thickness h(x,t). Linear stability analysis of the averaged equations shows good agreement with the Orr-Sommerfeld analysis. The effect of a slip at the wall on the primary instability has been found to be non-trivial. Close to the instability onset, the effect is destabilising whereas it becomes stabilising at larger values of the Reynolds number. Nonlinear travelling waves are amplified by the presence of the slip. Comparisons to direct numerical simulations show a remarkable agreement for all tested values of parameters. The averaged equations capture satisfactorily the speed, shape and velocity distribution in the waves. The Navier slip condition is observed to significantly enhance the backflow phenomenon in the capillary region of the solitary waves with a possible effect on heat and mass transfer.