Thin films flowing down inverted substrates: two dimensional flow

We consider free surface instabilities of films flowing on inverted substrates within the framework of lubrication approximation. We allow for the presence of fronts and related contact lines, and explore the role which they play in instability development. It is found that a contact line, modeled by a commonly used precursor film model, leads to free surface instabilities without any additional natural or excited perturbations. A single parameter D=(3Ca)^{1/3}\cot\alpha$, where $Ca$ is the capillary number and $\alpha$ is the inclination angle, is identified as a governing parameter in the problem. This parameter may be interpreted to reflect the combined effect of inclination angle, film thickness, Reynolds number and the fluid flux. Variation of $D$ leads to change of the wave-like properties of the instabilities, allowing us to observe traveling wave behavior, mixed waves, and the waves resembling solitary ones.