Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-07T20:23:50.359Z Has data issue: false hasContentIssue false
  • English
  • Français

Surface wave pattern formation in a cylindrical container

Published online by Cambridge University Press:  09 March 2021

X. Shao ,
P. Wilson ,
J.R. Saylor  and
J.B. Bostwick Open the ORCID record for J.B. Bostwick [Opens in a new window]
X. Shao
Affiliation:
Department of Mechanical Engineering, Clemson University, Clemson, SC29634, USA
P. Wilson
Affiliation:
Department of Mechanical Engineering, Clemson University, Clemson, SC29634, USA
J.R. Saylor
Affiliation:
Department of Mechanical Engineering, Clemson University, Clemson, SC29634, USA
J.B. Bostwick*
Affiliation:
Department of Mechanical Engineering, Clemson University, Clemson, SC29634, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Surface waves are excited by mechanical vibration of a cylindrical container having an air/water interface pinned at the rim, and the dynamics of pattern formation is analysed from both an experimental and theoretical perspective. The wave conforms to the geometry of the container and its spatial structure is described by the mode number pair ($n,\ell$) that is identified by long exposure time white light imaging. A laser light system is used to detect the surface wave frequency, which exhibits either a (i) harmonic response for low driving amplitude edge waves or (ii) sub-harmonic response for driving amplitude above the Faraday wave threshold. The first 50 resonant modes are discovered. Control of the meniscus geometry is used to great effect. Specifically, when flat, edge waves are suppressed and only Faraday waves are observed. For a concave meniscus, edge waves are observed and, at higher amplitudes, Faraday waves appear as well, leading to complicated mode mixing. Theoretical predictions for the natural frequency of surface oscillations for an inviscid liquid in a cylindrical container with a pinned contact line are made using the Rayleigh–Ritz procedure and are in excellent agreement with experimental results.

JFM classification

Waves/Free-surface Flows: Faraday waves Instability: Parametric instability Nonlinear Dynamical Systems: Pattern formation
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 915 , 25 May 2021 , A19
Check for updates
Copyright
© The Author(s), 2021. 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

REFERENCES

Batson, W., Zoueshtiagh, F. & Narayanan, R. 2013 The Faraday threshold in small cylinders and the sidewall non-ideality. J. Fluid Mech. 729, 496–523.CrossRefGoogle Scholar
Batson, W., Zoueshtiagh, F. & Narayanan, R. 2015 Two-frequency excitation of single-mode Faraday waves. J. Fluid Mech. 764, 538–571.CrossRefGoogle Scholar
Bechhoefer, J., Ego, V., Manneville, S. & Johnson, B. 1995 An experimental study of the onset of parametrically pumped surface waves in viscous fluids. J. Fluid Mech. 288, 325–350.CrossRefGoogle Scholar
Benjamin, T.B. & Scott, J.C. 1979 Gravity-capillary waves with edge constraints. J. Fluid Mech. 92 (2), 241–267.CrossRefGoogle Scholar
Benjamin, T.B. & Ursell, F.J. 1954 The stability of the plane free surface of a liquid in vertical periodic motion. Proc. R. Soc. Lond. A 225 (1163), 505–515.Google Scholar
Bostwick, J.B. & Steen, P.H. 2009 Capillary oscillations of a constrained liquid drop. Phys. Fluids 21, 032108.CrossRefGoogle Scholar
Bostwick, J.B. & Steen, P.H. 2013 a Coupled oscillations of deformable spherical-cap droplets. Part 1. Inviscid motions. J. Fluid Mech. 714, 312–335.CrossRefGoogle Scholar
Bostwick, J.B. & Steen, P.H. 2013 b Coupled oscillations of deformable spherical-cap droplets. Part 2. Viscous motions. J. Fluid Mech. 714, 336–360.CrossRefGoogle Scholar
Bostwick, J.B. & Steen, P.H. 2014 Dynamics of sessile drops. Part 1. Inviscid theory. J. Fluid Mech. 760, 5–38.CrossRefGoogle Scholar
Bostwick, J.B. & Steen, P.H. 2015 Stability of constrained capillary surfaces. Annu. Rev. Fluid Mech. 47, 539–568.CrossRefGoogle Scholar
Bostwick, J.B. & Steen, P.H. 2016 Response of driven sessile drops with contact-line dissipation. Soft Matt. 12 (43), 8919–8926.CrossRefGoogle ScholarPubMed
Briard, A., Gostiaux, L. & Gréa, B.-J. 2020 The turbulent Faraday instability in miscible fluids. J. Fluid Mech. 883, A57.CrossRefGoogle Scholar
Chang, C.-T., Bostwick, J.B., Daniel, S. & Steen, P.H. 2015 Dynamics of sessile drops. Part 2. Experiment. J. Fluid Mech. 768, 442–467.CrossRefGoogle Scholar
Chen, P., Güven, S., Usta, O.B., Yarmush, M.L. & Demirci, U. 2015 Biotunable acoustic node assembly of organoids. Adv. Healthc. Mater. 4 (13), 1937–1943.CrossRefGoogle ScholarPubMed
Chen, P., Luo, Z., Güven, S., Tasoglu, S., Ganesan, A.V., Weng, A. & Demirci, U. 2014 Microscale assembly directed by liquid-based template. Adv. Mater. 26 (34), 5936–5941.CrossRefGoogle ScholarPubMed
Christiansen, B., Alstrøm, P. & Levinsen, M.T. 1995 Dissipation and ordering in capillary waves at high aspect ratios. J. Fluid Mech. 291, 323–341.CrossRefGoogle Scholar
Ciliberto, S. & Gollub, J.P. 1985 Chaotic mode competition in parametrically forced surface waves. J. Fluid Mech. 158, 381–398.CrossRefGoogle Scholar
Ciliberto, S. & Gollub, J.P. 1984 Pattern competition leads to chaos. Phys. Rev. Lett. 52 (11), 922.CrossRefGoogle Scholar
Davis, S.H. 1980 Moving contact lines and rivulet instabilities. Part 1. The static rivulet. J. Fluid Mech. 98, 225–242.CrossRefGoogle Scholar
Douady, S. 1990 Experimental study of the Faraday instability. J. Fluid Mech. 221, 383–409.CrossRefGoogle Scholar
Douady, S. & Fauve, S. 1988 Pattern selection in Faraday instability. Europhys. Lett. 6 (3), 221.CrossRefGoogle Scholar
Edwards, W.S. & Fauve, S. 1994 Patterns and quasi-patterns in the Faraday experiment. J. Fluid Mech. 278, 123–148.CrossRefGoogle Scholar
Faraday, M. 1831 XVII. On a peculiar class of acoustical figures; and on certain forms assumed by groups of particles upon vibrating elastic surfaces. Phil. Trans. R. Soc. Lond. 121, 299–340.Google Scholar
Ghadiri, M. & Krechetnikov, R. 2019 Pattern formation on time-dependent domains. J. Fluid Mech. 880, 136–179.CrossRefGoogle Scholar
Gluckman, B.J., Marcq, P., Bridger, J. & Gollub, J.P. 1993 Time averaging of chaotic spatiotemporal wave patterns. Phys. Rev. Lett. 71 (13), 2034–2037.CrossRefGoogle ScholarPubMed
Graham-Eagle, J. 1983 A new method for calculating eigenvalues with applications to gravity-capillary waves with edge constraints. In Mathematical Proceedings of the Cambridge Philosophical Society, vol. 94, pp. 553–564. Cambridge University Press.CrossRefGoogle Scholar
Guven, S., Chen, P., Inci, F., Tasoglu, S., Erkmen, B. & Demirci, U. 2015 Multiscale assembly for tissue engineering and regenerative medicine. Trends Biotechnol. 33 (5), 269–279.CrossRefGoogle ScholarPubMed
Henderson, D.M. & Miles, J.W. 1990 Single-mode Faraday waves in small cylinders. J. Fluid Mech. 213, 95–109.CrossRefGoogle Scholar
Henderson, D.M. & Miles, J.W. 1991 Faraday waves in 2: 1 internal resonance. J. Fluid Mech. 222, 449–470.CrossRefGoogle Scholar
Henderson, D.M. & Miles, J.W. 1994 Surface-wave damping in a circular cylinder with a fixed contact line. J. Fluid Mech. 275, 285–299.CrossRefGoogle Scholar
Hocking, L.M. 1987 The damping of capillary-gravity waves at a rigid boundary. J. Fluid Mech. 179, 253–266.CrossRefGoogle Scholar
James, A., Vukasinovic, B., Smith, M.K. & Glezer, A. 2003 Vibration-induced drop atomization and bursting. J. Fluid Mech. 476, 1–28.CrossRefGoogle Scholar
Kim, J. 2007 Spray cooling heat transfer: the state of the art. Intl J. Heat Fluid Flow 28 (4), 753–767.CrossRefGoogle Scholar
Kumar, K. 1996 Linear theory of Faraday instability in viscous liquids. Proc. R. Soc. Lond. A 452 (1948), 1113–1126.Google Scholar
Kumar, K. & Tuckerman, L.S. 1994 Parametric instability of the interface between two fluids. J. Fluid Mech. 279, 49–68.CrossRefGoogle Scholar
Martel, C., Nicolas, J.A. & Vega, J.M. 1998 Surface-wave damping in a brimful circular cylinder. J. Fluid Mech. 360, 213–228.CrossRefGoogle Scholar
Melo, F., Umbanhowar, P. & Swinney, H.L. 1994 Transition to parametric wave patterns in a vertically oscillated granular layer. Phys. Rev. Lett. 72 (1), 172–175.CrossRefGoogle Scholar
Miles, J. & Henderson, D. 1990 Parametrically forced surface waves. Annu. Rev. Fluid Mech. 22 (1), 143–165.CrossRefGoogle Scholar
Miles, J.W. & Henderson, D.M. 1998 A note on interior vs. boundary-layer damping of surface waves in a circular cylinder. J. Fluid Mech. 364, 319–323.CrossRefGoogle Scholar
Müller, H.W., Wittmer, H., Wagner, C., Albers, J. & Knorr, K. 1997 Analytic stability theory for Faraday waves and the observation of the harmonic surface response. Phys. Rev. Lett. 78 (12), 2357.CrossRefGoogle Scholar
Perlin, M. & Schultz, W.W. 2000 Capillary effects on surface waves. Annu. Rev. Fluid Mech. 32 (1), 241–274.CrossRefGoogle Scholar
Prosperetti, A. 2012 Linear oscillations of constrained drops, bubbles, and plane liquid surfaces. Phys. Fluids 24 (3), 032109.CrossRefGoogle Scholar
Saylor, J.R. & Kinard, A.L. 2005 Simulation of particle deposition beneath Faraday waves in thin liquid films. Phys. Fluids 17 (4), 047106.CrossRefGoogle Scholar
Segel, L.A. 1987 Mathematics Applied to Continuum Mechanics. Dover Publications.Google Scholar
Steen, P.H., Chang, C.-T. & Bostwick, J.B. 2019 Droplet motions fill a periodic table. Proc. Natl Acad. Sci. 116 (11), 4849–4854.CrossRefGoogle ScholarPubMed
Strickland, S.L., Shearer, M. & Daniels, K.E. 2015 Spatiotemporal measurement of surfactant distribution on gravity–capillary waves. J. Fluid Mech. 777, 523–543.CrossRefGoogle Scholar
Tsai, C.S., Mao, R.W., Lin, S.K., Zhu, Y. & Tsai, S.C. 2014 Faraday instability-based micro droplet ejection for inhalation drug delivery. Technology 2 (01), 75–81.CrossRefGoogle ScholarPubMed
Vukasinovic, B., Smith, M.K. & Glezer, A. 2007 Dynamics of a sessile drop in forced vibration. J. Fluid Mech. 587, 395–423.CrossRefGoogle Scholar
Ward, K., Zoueshtiagh, F. & Narayanan, R. 2019 Faraday instability in double-interface fluid layers. Phys. Rev. Fluids 4 (4), 043903.CrossRefGoogle Scholar
Wright, P.H. & Saylor, J.R. 2003 Patterning of particulate films using Faraday waves. Rev. Sci. Instrum. 74 (9), 4063–4070.CrossRefGoogle Scholar