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Surface waves along liquid cylinders. Part 2. Varicose, sinuous, sloshing and nonlinear waves

Published online by Cambridge University Press:  27 July 2021

Gabriel Le Doudic Open the ORCID record for Gabriel Le Doudic [Opens in a new window] ,
Stéphane Perrard Open the ORCID record for Stéphane Perrard [Opens in a new window]  and
Chi-Tuong Pham Open the ORCID record for Chi-Tuong Pham [Opens in a new window]
Gabriel Le Doudic
Affiliation:
Laboratoire Matière et Systèmes Complexes, Université de Paris, CNRS, 10 rue Alice Domon et Léonie Duquet, 75013Paris, France
Stéphane Perrard
Affiliation:
Laboratoire de Physique de l’École normale supérieure, École normale supérieure, Université PSL, CNRS, Sorbonne Université, Université de Paris, 24 rue Lhomond, 75005Paris, France
Chi-Tuong Pham*
Affiliation:
Laboratoire Interdisciplinaire des Sciences du Numérique, Université Paris-Saclay, CNRS, Bâtiment 507, rue John von Neumann, 91400Orsay, France
*
†Email address for correspondence: pham@limsi.fr
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Abstract

Gravity–capillary waves propagating along extended liquid cylinders in the inviscid limit are studied in the context of experiments on sessile cylinders deposited upon superhydrophobic substrates, with tunable geometries. In Part 1 of this work (Pham et al., J. Fluid Mech., vol. 891, 2020, A5), we characterised the non-dispersive regime of the varicose waves. In this second part, we characterise the varicose waves in the dispersive regime, as well as the sinuous and the sloshing modes. We numerically study the shape function of the system (the counterpart of the standard $\tanh$ function of the dispersion relation of a gravity–capillary wave in a rectangular channel) and the cutoff frequencies of the sloshing modes, and show how they depend on the geometry of the substrate. A reduced-gravity effect is evidenced and the transition between a capillary- and a gravity-dominated regime is expressed in terms of an effective Bond number and an effective surface tension. Semiquantitative agreement is found between the theoretical computations and the experiments. As a consequence of these results, resorting to the inviscid section-averaged Saint-Venant equations, we propose a Korteweg–de Vries equation with adapted coefficients that governs the propagation of localised nonlinear waves. We relate these results to the propagation of depression solitons observed in our experimental set-up and along Leidenfrost cylinders levitating above a hot substrate (Perrard et al., Phys. Rev. E, vol. 92, 2015, 011002(R)). We extend our derivation of the Korteweg–de Vries equation to solitary-like waves propagating along Plateau borders in soap films, evidenced by Argentina et al. (J. Fluid. Mech., vol. 765, 2015, pp. 1–16).

JFM classification

Waves/Free-surface Flows: Solitary waves Drops and Bubbles: Drops
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 923 , 25 September 2021 , A13
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

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Le Doudic at al. Supplementary Movie

Generated by a micro-pipette at one tip of the drop, a liquid jet induces a strong deformation of the drop. Waves propagate ahead of a solitary depression wave and the latter travels at a speed lower than the propagation speed of the long-wavelength waves. This depression wave corresponds to a negative-amplitude Korteweg--de Vries soliton.

Download Le Doudic at al. Supplementary Movie(Video)
Video 12.5 MB