Laboratoire Matière et Systèmes Complexes

Dripping down the rivulet

François Gallaire

(LMFI) Laboratory of Fluid Mechanics and Instabilities École Polytechnique Fédérale de Lausanne, Suisse.

We consider the Rayleigh-Taylor instability of a thin film coating the underside of a curved or inclined surface. We first focus on the effect of curvature and investigate the Rayleigh-Taylor instability of a thin liquid film coated on the inside of a cylinder whose axis is orthogonal to gravity. We are interested in the effects of geometry on the instability, and contrast our results with the classical case of a thin film coated under a flat substrate. In our problem, gravity is the destabilizing force at the origin of the instability, but also yields the progressive drainage and stretching of the coating along the cylinder’s wall. We find that this flow stabilizes the film, which is asymptotically stable to infinitesimal perturbations. However, the short-time algebraic growth that these perturbations can achieve promotes the formation of different patterns, whose nature depends on the Bond number that prescribes the relative magnitude of gravity and capillary forces. Our experiments indicate that a transverse instability arises and persists over time for moderate Bond numbers. The liquid accumulates in equally spaced rivulets whose dominant wavelength corresponds to the most amplified mode of the classical Rayleigh-Taylor instability.

We then focus on the pattern formation in thin liquid flowing down along the underside an inclined plate. Rivulets parallel to the flow direction are the naturally selected patterns in this geometry too, and we demonstrate that this results from the effect of nonlinearity. Depending on the inclination, a secondary instability sets in which breaks the rivulet in a collection of drops draining or dripping down.