Instability of a thin viscous film flowing under an inclined substrate: the emergence and stability of rivulets

We study the pattern formation of a thin film flowing under an inclined planar substrate. The phenomenon is studied in the context of the Rayleigh-Taylor instability using the lubrication equation. Inspired by experimental observations, we numerically study the thin film response to a streamwise-invariant sinusoidal initial condition. The numerical response shows the emergence of predominant streamwise-aligned structures, modulated along the direction perpendicular to the flow, called rivulets. Oscillations of the thickness profile along the streamwise direction do not grow significantly when the inclination is very large or the liquid layer very thin. However, for small inclinations or thick films, streamwise perturbations grow on rivulets. A secondary stability analysis of one-dimensional and steady rivulets reveals a strong stabilization mechanism for large inclinations or very thin films. The theoretical results are compared with experimental measurements of the streamwise oscillations of the rivulet profile, showing a good agreement. The emergence of rivulets is investigated by studying the impulse response. Both the experimental observation and the numerical simulation show a marked anisotropy favouring streamwise-aligned structures. A weakly nonlinear model is proposed to rationalize the levelling of all but streamwise-aligned structures.