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Surface waves along liquid cylinders. Part 1. Stabilising effect of gravity on the Plateau–Rayleigh instability

Published online by Cambridge University Press:  18 March 2020

Chi-Tuong Pham Open the ORCID record for Chi-Tuong Pham [Opens in a new window] ,
Stéphane Perrard Open the ORCID record for Stéphane Perrard [Opens in a new window]  and
Gabriel Le Doudic Open the ORCID record for Gabriel Le Doudic [Opens in a new window]
Chi-Tuong Pham*
Affiliation:
Université Paris-Saclay, CNRS, LIMSI, rue John von Neumann, 91400Orsay, France
Stéphane Perrard
Affiliation:
Laboratoire de Physique de l’École normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, 24 rue Lhomond, 75005Paris, France
Gabriel Le Doudic
Affiliation:
Matière et Systèmes Complexes, Université de Paris, CNRS, 10 rue Alice Domon et Léonie Duquet, 75013Paris, France
*
†Email address for correspondence: pham@limsi.fr
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Abstract

We study the shape and the geometrical properties of sessile drops with translational invariance (namely ‘liquid cylinders’) deposited upon a flat superhydrophobic substrate. We account for the flattening effects of gravity on the shape of the drop using a pendulum rotation motion analogy. In the framework of the inviscid Saint-Venant equations, we show that liquid cylinders are always unstable because of the Plateau–Rayleigh instability. However, a cylindrical drop deposited upon a superhydrophobic non-flat channel (here, wedge-shaped channels) is stabilised beyond a critical cross-sectional area. The critical threshold of the Plateau–Rayleigh instability is analytically computed for various profiles of the channel. The stability analysis is performed in terms of an effective propagation speed of varicose waves. Experiments are performed in order to test these analytical results. We measure the critical drop size at which breakup occurs, together with the decreasing effective propagation speed of varicose waves as the threshold is approached. Our theoretical predictions are in excellent agreement with the experimental measurements.

JFM classification

Drops and Bubbles: Breakup/coalescence Interfacial Flows (free surface): Capillary flows Waves/Free-surface Flows: Surface gravity waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 891 , 25 May 2020 , A8
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Supplementary material: File

Pham et al. supplementary movie 1

Below a critical volume (close to 15 mL), a 49 centimeter long drop, pinned at both ends, breaks up after a slow pinch-off dynamics. Pinch-off occurs about the center of the drop and both halves end up retracting, owing to surface tension (real time).

Download Pham et al. supplementary movie 1(File)
File 1.2 MB
Supplementary material: File

Pham et al. supplementary movie 2

Close-up of the pinch-off region (real time) and remaining satellite drop after the break-up.

Download Pham et al. supplementary movie 2(File)
File 1.3 MB