Critical inclination for absolute/convective instability transition in inverted falling films

Liquid films flowing down the underside of inclined plates are subject to the interaction between the hydrodynamic and the Rayleigh-Taylor (R-T) instabilities causing a patterned and wavy topology at the free surface. The R-T instability results from the denser liquid film being located above a less dense ambient gas, and deforming into an array of droplets, which eventually drip if no saturation mechanism arises. Such saturation mechanism can actually be provided by a fluid motion along the inclined plate. Using a weighted integral boundary layer model, this study examines the critical inclination angle, measured from the vertical, that separates regimes of absolute and convective instability. If the instability is of absolute type, growing perturbations stay localized in space potentially leading to dripping. If the instability is of convective type, growing perturbations move downwards the inclined plate, forming waves and eventually, but not necessarily, droplets. Remarkably, there is a minimum value of the critical angle below which a regime of absolute instability cannot exist. This minimum angle decreases with viscosity: it is about 85° for water, about 70° for silicon oil 20 times more viscous than water, and reaches a limiting value for liquid with a viscosity larger than about 1000 times the one of water. It results that for any fluid, absolute dripping can only exist for inclination angle (taken from the vertical) larger than 57.4°.