Rayleigh-Taylor instability under curved substrates: An optimal transient growth analysis

Abstract

We investigate the stability of thin viscous films coated on the inside of a horizontal cylindrical substrate. In such a case, gravity acts both as a stabilizing force through the progressive drainage of the film and as a destabilizing force prone to form droplets via the Rayleigh-Taylor instability. The drainage solution, derived from lubrication equations, is found asymptotically stable with respect to infinitesimally small perturbations, although in reality, droplets often form. To resolve this paradox, we perform an optimal transient growth analysis for the first-order perturbations of the liquid's interface, generalizing the results of Trinh et al. [Phys. Fluids 26, 051704 (2014)]. We find that the system displays a linear transient growth potential that gives rise to two different scenarios depending on the value of the Bond number (prescribing the relative importance of gravity and surface tension forces). At low Bond numbers, the optimal perturbation of the interface does not generate droplets. In contrast, for higher Bond numbers, perturbations on the upper hemicircle yield gains large enough to potentially form droplets. The gain increases exponentially with the Bond number. In particular, depending on the amplitude of the initial perturbation, we find a critical Bond number above which the short-time linear growth is sufficient to trigger the nonlinear effects required to form dripping droplets. We conclude that the transition to droplets detaching from the substrate is noise and perturbation dependent.

Fingering instability on curved substrates: optimal initial film and substrate perturbations

“…The algorithm for the optimal transient growth of the optimal initial conditions is based on the one described in Balestra et al (2016), and only the main steps are presented hereafter.…”

“…Under the assumption of a small film aspect ratio δ = √ A/R 1 we can use the long-wavelength approximation to describe this flow (Oron et al 1997). Given the small Reynolds number Re = gρ 2 A 3/2 /(3µ 2 ), inertial effects can be neglected and the Stokes equations in polar coordinates (r * , θ, z * ) can be integrated over the radial direction to obtain, using mass conservation, the lubrication equation for the film thicknessH * (θ, z * , t * ) (see Balestra et al (2017) for a detailed derivation)…”

“…The termH * 3 is the mobility of the liquid and results from the integration of the half-Poiseuille velocity profile in the thin film. The only difference with the lubrication equation for a thin film coating the inside of a cylindrical substrate used in Balestra et al (2017) is the sign of the hydrostatic pressure variation. In the present geometry, the variation of the hydrostatic pressure is stabilizing for θ ∈ [0, π/2[ and destabilizing for θ ∈ ]π/2, π].…”

“…The spatial variation of the film thickness in the outer region can be found by a regular perturbation expansion as described in Balestra et al (2017). Under the assumption of negligible capillary pressure and hydrostatic pressure contributions, the approximate solution for the film thickness up to the fourth order in θ reads Figure 3.…”

“…Given the spatial variations of the forces at play, the characteristics of the draining flow, as the length of the capillary ridge, vary in space and time. Nonmodal approaches have been proven to be effective in predicting stability properties of timeevolving flows (Schmid 2007), also in the context of thin films (Balestra et al 2016). In the present work, we consider two distinct optimization problems.…”