Three-dimensional Rayleigh–Taylor instability under a unidirectional curved substrate

Abstract

We 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. The formation of rivulets allows for a faster drainage of the liquid from top to bottom when compared to a uniform drainage. For higher Bond numbers, a two-dimensional stretched lattice of droplets is found to form on the top part of the cylinder. Rivulets and the lattice of droplets are inherently three-dimensional phenomena and therefore require a careful three-dimensional analysis. We found that the transition between the two types of pattern may be rationalized by a linear optimal transient growth analysis and nonlinear numerical simulations.

Experimental investigations of liquid falling films flowing under an inclined planar substrate

“…Three more regions relating to commonly observed interfacial features have been marked in case, before decreasing at higher Re. It is also worth noting that the rivulet structures appear to "invade" the imaging domain from the edge to the right (the rivulets on the right-hand side display larger amplitudes than those on the left), in accordance with the description by Balestra et al [68].…”

Instability of a thin viscous film flowing under an inclined substrate: steady patterns

“…We thus propose a new experimental set-up, combining a permanent liquid supply to a tilted and flat coated surface, in contrast to recent studies on cylindrical and spherical substrates displaying a stabilizing effect on the film thanks to Thin viscous film flowing under an inclined planar substrate 898 A6-3 drainage-induced thinning and stretching (Balestra et al 2018a;Balestra, Nguyen & Gallaire 2018b). In order to overcome the limitation of a transient experiment, we impose a constant flow rate so that the flow can reach a steady state or an asymptotic behaviour.…”

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

“…When the substrate is tilted with respect to the horizontal direction, the gravity component parallel to the substrate creates a flow. In this work, we consider a configuration with a permanent influx, in opposition to the case of cylindrical and spherical substrates in which a transient release of fluid is studied (Balestra et al 2018a;Balestra, Nguyen & Gallaire 2018b). A strong modulation of the thickness along the direction perpendicular to the flow (spanwise direction) is identified as rivulet formation (Charogiannis et al 2018).…”