Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2025-01-01T09:11:48.230Z Has data issue: false hasContentIssue false
  • English
  • Français

Experimental study on travelling and standing pattern formation and capillary waves in a pinned liquid film: effects of multi-axis lateral (horizontal) vibrations and substrate geometry

Published online by Cambridge University Press:  12 August 2020

Talha Khan Open the ORCID record for Talha Khan [Opens in a new window]  and
Morteza Eslamian Open the ORCID record for Morteza Eslamian [Opens in a new window]
Talha Khan
Affiliation:
University of Michigan–Shanghai Jiao Tong University Joint Institute, Shanghai200240, PR China Department of Mechanical Engineering, University of Engineering and Technology (Main Campus), Lahore54890, Pakistan
Morteza Eslamian*
Affiliation:
University of Michigan–Shanghai Jiao Tong University Joint Institute, Shanghai200240, PR China
*
†Email address for correspondence: morteza.eslamian@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

Pattern-forming instability in various fields is an interesting research topic because of its complex physical nature and numerous applications. In this paper, we experimentally study capillary surface waves and patterns formed on a liquid film, cast on a plane substrate without physical walls, but pinned to the substrate edges, and subjected to multi-axis horizontal (lateral) oscillations (55–333 Hz). The effect of single-axis ultrasonic horizontal vibrations (20–170 kHz) was also investigated. We show that using substrates with different geometrical shapes and various travelling paths created by multi-axis vibrations with a phase angle difference between the axes produce a plethora of standing and travelling wave patterns on the liquid film surface. We report perfect standing square and spiral-like patterns for low-frequency multi-axis horizontal vibrations, which are commonly observed for vertical vibrations, while the mechanisms of momentum transfer to the liquid film from the vibrating substrate are different in vertical and horizontal vibrations. Other patterns forming on the liquid film surface in our experiments include lines/stripes, circles, swirls, pentagons, triangles, etc. It is also reported that low-frequency excitations create harmonic travelling waves and standing patterns, while the frequency of response waves generated by the application of ultrasonic horizontal vibrations is several orders of magnitude less than the excitation frequency. No subharmonic cross-waves are observed in this study, which strengthens the idea that plane substrates (without walls) are a good approximation for the theoretical case of a horizontally vibrated liquid film with infinite lateral length.

JFM classification

Interfacial Flows (free surface): Thin films Nonlinear Dynamical Systems: Pattern formation Waves/Free-surface Flows: Capillary waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 900 , 10 October 2020 , A30
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arbell, H. & Fineberg, J. 2002 Pattern formation in two-frequency forced parametric waves. Phys. Rev. E 65 (3), 036224.Google ScholarPubMed
Benjamin, T. B. & Ursell, F. 1954 The stability of the plane free surface of a liquid in vertical periodic motion. Proc. R. Soc. Lond. A 225, 505–515.Google Scholar
Bestehorn, M. 2013 Laterally extended thin liquid films with inertia under external vibrations. Phys. Fluids 25 (11), 114106.CrossRefGoogle Scholar
Bestehorn, M., Han, Q. & Oron, A. 2013 Nonlinear pattern formation in thin liquid films under external vibrations. Phys. Rev. E 88 (2), 023025.Google ScholarPubMed
Beyer, J. & Friedrich, R. 1995 Faraday instability: linear analysis for viscous fluids. Phys. Rev. E 51 (2), 1162–1168.Google ScholarPubMed
Binks, D. & van de Water, W. 1997 Nonlinear pattern formation of Faraday waves. Phys. Rev. Lett. 78, 4043.CrossRefGoogle Scholar
Bush, J. W. M. 2015 Pilot-wave hydrodynamics. Annu. Rev. Fluid Mech. 47 (1), 269–292.CrossRefGoogle Scholar
Chen, P. & Viñals, J. 1999 Amplitude equation and pattern selection in Faraday waves. Phys. Rev. E 60 (1), 559–570.Google ScholarPubMed
Christian, B., Alstrom, P. & Levinsen, M. T. 1992 Ordered capillary-wave states: quasicrystals, hexagons and radial waves. Phys. Rev. Lett. 68, 2157.CrossRefGoogle Scholar
Couchman, M., Turton, S. E. & Bush, J. W. M. 2019 Bouncing phase variations in pilot-wave hydrodynamics and the stability of droplet pairs. J. Fluid Mech. 871, 212–243.CrossRefGoogle Scholar
Currie, I. G. 2013 Fundamental Mechanics of Fluids. CRC.Google Scholar
Ding, Y. & Umbanhowar, P. 2006 Enhanced Faraday pattern stability with three-frequency driving. Phys. Rev. E 73, 046305.Google ScholarPubMed
Douady, S. 1990 Experimental study of the Faraday instability. J. Fluid Mech. 221 (1), 383.CrossRefGoogle Scholar
Douady, S. & Fauve, S. 1988 Pattern selection in Faraday instability. Europhys. Lett. 6 (3), 221–226.CrossRefGoogle Scholar
Edwards, W. S. & Fauve, S. 1993 Parametrically excited quasicrystalline surface waves. Phys. Rev. E 47, R788–R791.Google ScholarPubMed
Edwards, W. S. & Fauve, S. 1994 Patterns and quasi-patterns in the Faraday experiment. J. Fluid Mech. 278 (1), 123.CrossRefGoogle Scholar
Faraday, M. 1831 On forms and states assumed by fluids in contact with vibrating elastic surfaces. Phil. Trans. R. Soc. Lond. 121, 319–340.Google Scholar
Francois, N., Xia, H., Punzmann, H., Fontana, P. W. & Shats, M. 2017 Wave-based liquid-interface metamaterials. Nat. Commun. 8 (1), 14325.CrossRefGoogle ScholarPubMed
Gholampour, N., Brian, D. & Eslamian, M. 2018 Tailoring characteristics of PEDOT: PSS coated on glass and plastics by ultrasonic substrate vibration post treatment. Coatings 8 (10), 337.CrossRefGoogle Scholar
Hutton, R. E. 1963 An investigation of resonant, nonlinear, nonplanar, free surface oscillations of a fluid. NASA Tech. Note D-1870.Google Scholar
Kahouadji, L., Périnet, N., Tuckerman, L. S., Shin, S., Chergui, J. & Juric, D. 2015 Numerical simulation of supersquare patterns in Faraday waves. J. Fluid Mech. 772, R2.CrossRefGoogle Scholar
Kentaro, T. & Takeshi, M. 2015 Numerical simulation of Faraday waves oscillated by two-frequency forcing. Phys. Fluids 27 (3), 032108.Google Scholar
Khan, T. & Eslamian, M. 2019 Experimental analysis of one-dimensional Faraday waves on a liquid layer subjected to horizontal vibrations. Phys. Fluids 31 (8), 082106.CrossRefGoogle Scholar
Kumar, K. & Tuckerman, L. S. 1994 Parametric instability of the interface between two fluids. J. Fluid Mech. 279 (1), 49.CrossRefGoogle Scholar
Li, X., Yu, Z. & Liao, S. 2015 Observation of two-dimensional Faraday waves in extremely shallow depth. Phys. Rev. E 92, 033014.Google ScholarPubMed
Miles, J. 1976 Nonlinear surface waves in closed basins. J. Fluid Mech. 75, 419–448.CrossRefGoogle Scholar
Miles, J. 1984 Resonantly forced surface waves in a circular cylinder. J. Fluid Mech. 149, 15–31.CrossRefGoogle Scholar
Miles, J. 1993 On Faraday waves. J. Fluid Mech. 248, 671.CrossRefGoogle Scholar
Milner, S. T. 1991 Square patterns and secondary instabilities in driven capillary waves. J. Fluid Mech. 225, 81–100.CrossRefGoogle Scholar
Müller, H. W. 1993 Periodic triangular patterns in the Faraday experiment. Phys. Rev. Lett. 71 (20), 3287–3290.CrossRefGoogle ScholarPubMed
Or, A. C. 1997 Finite-wavelength instability in a horizontal liquid layer on an oscillating plane. J. Fluid Mech. 335, 213–232.CrossRefGoogle Scholar
Oza, A. U., Siéfert, E., Harris, D. M., Moláček, J. & Bush, J. W. M. 2017 Orbiting pairs of walking droplets: dynamics and stability. Phys. Rev. Fluids 2 (5), 053601.CrossRefGoogle Scholar
Pérez-Gracia, J. M., Jeff, P., Fernando, V. & José, M. V. 2014 Oblique cross-waves in horizontally vibrated containers. Fluid Dyn. Res. 46 (4), 041410.CrossRefGoogle Scholar
Pérez-Gracia, J. M., Porter, J., Varas, F. & Vega, J. M. 2013 Subharmonic capillary–gravity waves in large containers subject to horizontal vibrations. J. Fluid Mech. 739, 196–228.CrossRefGoogle Scholar
Perinet, N., Juric, D. & Tuckerman, L. S. 2009 Numerical simulation of Faraday waves. J. Fluid Mech. 635, 1.CrossRefGoogle Scholar
Porter, J., Tinao, I., Laverón-Simavilla, A. & Lopez, C. A. 2012 Pattern selection in a horizontally vibrated container. Fluid Dyn. Res. 44 (6), 065501.CrossRefGoogle Scholar
Qi, A., Yeo, L. Y. & Friend, J. R. 2008 Interfacial destabilization and atomization driven by surface acoustic waves. Phys. Fluids 20 (7), 074103.CrossRefGoogle Scholar
Rahimzadeh, A., Ahmadian-Yazdi, M. R. & Eslamian, M. 2018 Experimental study on the characteristics of capillary surface waves on a liquid film on an ultrasonically vibrated substrate. Fluid Dyn. Res. 50 (6), 065510.CrossRefGoogle Scholar
Rajchenbach, J., Clamond, D. & Leroux, A. 2013 Observation of star-shaped surface gravity waves. Phys. Rev. Lett. 110 (9), 094502.CrossRefGoogle ScholarPubMed
Richter, S. & Bestehorn, M. 2019 Direct numerical simulations of liquid films in two dimensions under horizontal and vertical external vibrations. Phys. Rev. Fluids 4 (4), 044004.CrossRefGoogle Scholar
Sanlı, C., Lohse, D. & van der Meer, D. 2014 From antinode clusters to node clusters: the concentration-dependent transition of floaters on a standing Faraday wave. Phys. Rev. E 89 (5), 053011.Google ScholarPubMed
Shats, M., Xia, H. & Punzmann, H. 2012 Parametrically excited water surface ripples as ensembles of oscillons. Phys. Rev. Lett. 108 (3), 034502.CrossRefGoogle ScholarPubMed
Shklyaev, S., Alabuzhev, A. A. & Khenner, M. 2009 Influence of a longitudinal and tilted vibration on stability and dewetting of a liquid film. Phys. Rev. E 79 (5), 051603.Google ScholarPubMed
Skeldon, A. C. & Guidoboni, A. G. 2007 Pattern selection for Faraday waves in an incompressible viscous fluid. SIAM J. Appl. Maths 67 (4), 1064–1100.CrossRefGoogle Scholar
Tan, M. K., Friend, J. R., Matar, O. K. & Yeo, L. Y. 2010 Capillary wave motion excited by high frequency surface acoustic waves. Phys. Fluids 22 (11), 112112.CrossRefGoogle Scholar
Varas, F. & Vega, J. M. 2007 Modulated surface waves in large-aspect-ratio horizontally vibrated containers. J. Fluid Mech. 579, 271.CrossRefGoogle Scholar
Yeo, L. Y., Lastochkin, D., Wang, S.-C. & Chang, H.-C. 2004 A new AC electrospray mechanism by Maxwell-Wagner polarization and capillary resonance. Phys. Rev. Lett. 92 (13), 133902.CrossRefGoogle ScholarPubMed
Yih, C.-S. 1968 Instability of unsteady flows or configurations. Part 1. Instability of a horizontal liquid layer on an oscillating plane. J. Fluid Mech. 31 (4), 737.CrossRefGoogle Scholar

Khan et al. supplementary movie 1

Movie 1 shows surface waves on an elliptical substrate (a = 20 mm, b = 10 mm) under 1D horizontal vibration (f = 200 Hz).
Download Khan et al. supplementary movie 1(Video)
Video 4.6 MB

Khan et al. supplementary movie 2

Movie 2 shows the spiral pattern in surface waves on a circular substrate (d = 50 mm, f = 165 Hz, φ ≈ 90 deg).
Download Khan et al. supplementary movie 2(Video)
Video 8.6 MB

Khan et al. supplementary movie 3

Movie 3 shows the standing square pattern in a square substrate (s = 25 mm, f = 55 Hz, φ ≈ 0 deg).

Download Khan et al. supplementary movie 3(Video)
Video 8.8 MB

Khan et al. supplementary movie 4

Movie 4 shows the line switching pattern in a square substrate (s = 25 mm, f = 55 Hz, φ ≈ 90 deg).

Download Khan et al. supplementary movie 4(Video)
Video 4.7 MB

Khan et al. supplementary movie 5

Movie 5 shows surface waves on a stadium-shaped substrate showing a swirl pattern (f = 167 Hz, and φ ≈ 0 deg).

Download Khan et al. supplementary movie 5(Video)
Video 2.9 MB

Khan et al. supplementary movie 6

Movie 6 shows surface waves for the ultrasonic vibrations at 20 kHz.

Download Khan et al. supplementary movie 6(Video)
Video 7.2 MB