Critical inclination for absolute/convective instability transition in inverted falling films

Abstract

1Critical inclination for A/C transition in inverted falling films Liquid films flowing down the underside of inclined plates are subject to the interaction between the hydrodynamic and the Rayleigh-Taylor (R-T) instabilities causing a patterned and wavy topology at the free surface. The R-T instability results from the denser liquid film being located above a less dense ambient gas, and deforming into an array of droplets, which eventually drip if no saturation mechanism arises. Such saturation mechanism can actually be provided by a fluid motion along the inclined plate. Using a weighted integral boundary layer model, this study examines the critical inclination angle, measured from the vertical, that separates regimes of absolute and convective instability. If the instability is of absolute type, growing perturbations stay localized in space potentially leading to dripping. If the instability is of convective type, growing perturbations move downwards the inclined plate, forming waves and eventually, but not necessarily, droplets. Remarkably, there is a minimum value of the critical angle below which a regime of absolute instability cannot exist. This minimum angle decreases with viscosity : it is about 85• for water, about 70• for silicon oil, and reaches a limiting value for liquid with a viscosity larger than about 1000 times the one of water. It results that for any fluid, absolute dripping can only exist for inclination angle (taken from the vertical) larger than 57.4• .

Instability and dripping of electrified liquid films flowing down inverted substrates

“…promotes convective instability (spatial stability) of the flat film solution. Using a WIBL model, we obtain a minimum critical angle depending on the electric field strength, below which only convective instabilities occur, extending the findings in [38]. Comparisons with results for the full stability problem at zero Reynolds number are also provided.…”

“…In the case that the curves of ω r (ξ) = 0 do not connect the points 0 and ξ c on R + , then the instability is absolute. The Briggs-Bers pinching criterion can then be applied to our problem -see [19,38] for its application to the non-electrified flow.…”

“…It is noticeable from their results that only a small amount of dripping was observed just beyond the absolute-convective (A/C) threshold, with much more intense dripping further into the absolute instability regime. Inertial effects and higher order terms were included by Scheid et al [38] in their study of the A/C transition for WIBL models. The authors reported a large discrepancy with the results of the inertialess Benney equation away from zero Reynolds numbers, and found a fluid-independent critical angle below which only convective instabilities exist for their models.…”

“…The authors reported a large discrepancy with the results of the inertialess Benney equation away from zero Reynolds numbers, and found a fluid-independent critical angle below which only convective instabilities exist for their models. Kofman et al [39] employed DNS for the 2D problem with spatially periodic boundary conditions, finding that the dripping onset did not coincide closely with the A/C transition curve obtained from the WIBL models in [38]. They attribute dripping to a secondary instability of travelling wave solutions.…”

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

“…When inclining the solid substrate, the RTI is marginally modified for small slopes, but can be annihilated if the slope is large enough [14]. The distinction between these two regimes can be rationalized using an absolute/convective instability analysis [14,15]. Similarly, when the inclination of the substrate varies uniformly, i.e., the solid surface has a finite curvature, Trinh et al [5] showed that the Rayleigh-Taylor instability can be suppressed for a thin film if surface tension forces are strong enough.…”