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Dynamics of sessile drops. Part 2. Experiment

Published online by Cambridge University Press:  10 March 2015

Chun-Ti Chang ,
J. B. Bostwick ,
Susan Daniel  and
P. H. Steen
Chun-Ti Chang*
Affiliation:
Theoretical and Applied Mechanics, Cornell University, Ithaca, NY 14853, USA
J. B. Bostwick
Affiliation:
Department of Engineering Science and Applied Mathematics, Northwestern University, Evanston, IL 60208, USA
Susan Daniel
Affiliation:
Theoretical and Applied Mechanics, Cornell University, Ithaca, NY 14853, USA School of Chemical and Biomolecular Engineering, Cornell University, Ithaca, NY 14853, USA
P. H. Steen
Affiliation:
Theoretical and Applied Mechanics, Cornell University, Ithaca, NY 14853, USA School of Chemical and Biomolecular Engineering, Cornell University, Ithaca, NY 14853, USA Center for Applied Mathematics, Cornell University, Ithaca, NY 14853, USA
*
†Email addresses for correspondence: [email protected], [email protected]
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Abstract

High-speed images of driven sessile water drops recorded under frequency scans are analysed for resonance peaks, resonance bands and hysteresis of characteristic modes. Visual mode recognition using back-lit surface distortion enables modes to be associated with frequencies, aided by the identifications in Part 1 (Bostwick & Steen, J. Fluid Mech., vol. 760, 2014, pp. 5–38). Part 1 is a linear stability analysis that predicts how inviscid drop spectra depend on base state geometry. Theoretically, the base states are spherical caps characterized by their ‘flatness’ or fraction of the full sphere. Experimentally, quiescent shapes are controlled by pinning the drop at a circular contact line on the flat substrate and varying the drop volume. The response frequencies of the resonating drop are compared with Part 1 predictions. Agreement with theory is generally good but does deteriorate for flatter drops and higher modes. The measured frequency bands agree better with an extended model, introduced here, that accounts for forcing and weak viscous effects using viscous potential flow. As the flatness varies, regions are predicted where modal frequencies cross and where the spectra crowd. Frequency crossings and spectral crowding favour interaction of modes. Modal interactions of two kinds are documented, called ‘mixing’ and ‘competing’. Mixed modes are two pure modes superposed with little evidence of hysteresis. In contrast, modal competition involves hysteresis whereby one or the other mode disappears depending on the scan direction. Perhaps surprisingly, a linear inviscid irrotational theory provides a useful framework for understanding observations of forced sessile drop oscillations.

JFM classification

Drops and Bubbles: Drops Drops and Bubbles: Drops and Bubbles
Type
Papers
Information
Journal of Fluid Mechanics , Volume 768 , 10 April 2015 , pp. 442 - 467
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Copyright
© 2015 Cambridge University Press 

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References

Arscott, F. M., Sneddon, I. N. & Ulam, S. 2014 Periodic Differential Equations: An Introduction to Mathieu, Lamé, and Allied Functions. Elsevier Science.Google Scholar
Basaran, O. A. 2002 Small-scale free surface flows with breakup: drop formation and emerging applications. AIChE J. 48 (9), 1842–1848.CrossRefGoogle Scholar
Batson, B., Zoueshtiagh, F. & Narayanan, R. 2013 The Faraday threshold in small cylinders and the sidewall non-ideality. J. Fluid Mech. 729, 496–523.CrossRefGoogle Scholar
Bauer, H. F. 1992 Response of axially excited spherical and conical liquid systems with anchored edges. Forsch. Ing. Wes. 58 (4), 96–103.Google Scholar
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 (1163), 505–515.Google Scholar
Bostwick, J. & Steen, P. 2013 Coupled oscillations of deformable spherical-cap droplets. Part 2. Viscous motions. J. Fluid Mech. 714, 336–360.CrossRefGoogle Scholar
Bostwick, J. & Steen, P. 2014 Dynamics of sessile drops. Part 1. Inviscid theory. J. Fluid Mech. 760, 5–38.Google Scholar
Bouwhuis, W., Winkels, K., Peters, I., Brunet, P., van der Meer, D. & Snoeijer, J. 2013 Oscillating and star-shaped drops levitated by an airflow. Phys. Rev. E 88, 023017.CrossRefGoogle ScholarPubMed
Brunet, P. & Snoeijer, J. 2011 Star-drops formed by periodic excitation and on an air cushion a short review. Eur. Phys. J. B 192 (1), 207–226.Google Scholar
Busse, F. H. 1984 Oscillations of a rotating liquid drop. J. Fluid Mech. 142, 1–8.Google Scholar
Castrejon-Pita, J. R., Baxter, W. R. S., Morgan, J., Temple, S., Martin, G. D. & Hutchings, I. M. 2013 Future, opportunities and challenges of inkjet technologies. Atomiz. Sprays 23 (6), 541–565.Google Scholar
Chang, C.-T., Bostwick, J. B., Steen, P. H. & Daniel, S. 2013 Substrate constraint modifies the Rayleigh spectrum of vibrating sessile drops. Phys. Rev. E 88, 023015.Google Scholar
Chebel, N. A., Risso, F. & Masbernat, O. 2011 Inertial modes of a periodically forced buoyant drop attached to a capillary. Phys. Fluids 23 (10), 102104.Google Scholar
Ciliberto, S. & Gollub, J. P. 1985 Chaotic mode competition in parametrically forced surface waves. J. Fluid Mech. 158, 381–398.Google Scholar
Cui, Y., Paxson, A. T., Smyth, K. M. & Varanasi, K. K. 2012 Hierarchical polymeric textures via solvent-induced phase transformation: a single-step production of large-area superhydrophobic surfaces. Colloids Surf. A 394 (2012), 8–13.Google Scholar
Daniel, S., Chaudhury, M. K. & Chen, J. C. 2001 Fast drop movements resulting from the phase change on a gradient surface. Science 291 (5504), 633–636.Google Scholar
Daniel, S., Chaudhury, M. K. & de Gennes, P.-G. 2005 Vibration-actuated drop motion on surfaces for batch microfluidic processes. Langmuir 21 (9), 4240–4248; pMID: 15836001.Google Scholar
Dong, L., Chaudhury, A. & Chaudhury, M. 2006 Lateral vibration of a water drop and its motion on a vibrating surface. Eur. Phys. J. E 21 (3), 231–242.CrossRefGoogle ScholarPubMed
Dorbolo, S., Terwagne, D., Vandewalle, N. & Gilet, T. 2008 Resonant and rolling droplet. New J. Phys. 10 (11), 113021.Google Scholar
Feng, J. Q. & Beard, K. V. 1991 Resonances of a conducting drop in an alternating electric field. J. Fluid Mech. 222, 417–435.Google Scholar
Hartog, J. P. D. 1956 Mechanical Vibrations. McGraw-Hill.Google Scholar
Hill, R. J. A. & Eaves, L. 2010 Vibrations of a diamagnetically levitated water droplet. Phys. Rev. E 81, 056312.Google Scholar
Hill, R. J. A. & Eaves, L. 2012 Shape oscillations of an electrically charged diamagnetically levitated droplet. Appl. Phys. Lett. 100 (11), 114106.Google Scholar
Holter, N. J. & Glasscock, W. R. 1952 Vibrations of evaporating liquid drops. J. Acoust. Soc. Am. 24 (6), 682–686.Google Scholar
Joseph, D. 2006 Helmholtz decomposition coupling rotational to irrotational flow of a viscous fluid. Proc. Natl Acad. Sci. USA 103, 14272–14277.Google Scholar
Joseph, D. D. 2003 Viscous potential flow. J. Fluid Mech. 479, 191–197.Google Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
MacRobert, T. 1967 Spherical Harmonics. Pergamon.Google Scholar
Marston, P. L. & Apfel, R. E. 1980 Quadrupole resonance of drops driven by modulated acoustic radiation pressure – experimental properties. J. Acoust. Soc. Am. 67 (1), 27–37.Google Scholar
Noblin, X., Kofman, R. & Celestini, F. 2009 Ratchetlike motion of a shaken drop. Phys. Rev. Lett. 102, 194504.Google Scholar
Padrino, J., Funada, T. & Joseph, D. 2007 Purely irrotational theories for the viscous effects on the oscillations of drops and bubbles. Intl J. Multiphase Flow 34, 61–75.Google Scholar
Perez, M., Brechet, Y., Salvo, L., Papoular, M. & Suery, M. 1999 Oscillation of liquid drops under gravity: influence of shape on the resonance frequency. Eur. Phys. Lett. 47 (2), 189–195.Google Scholar
Qi, A., Yeo, L., Friend, J. & Ho, J. 2010 The extraction of liquid, protein molecules and yeast cells from paper through surface acoustic wave atomization. Lab on a Chip 10, 470–476.Google Scholar
Rajchenbach, J., Clamond, D. & Leroux, A. 2013 Observation of star-shaped surface gravity waves. Phys. Rev. Lett. 110, 094502.Google Scholar
Rayleigh, L. 1879 On the capillary phenomena of jets. Proc. R. Soc. Lond. 29 (196–199), 71–97.Google Scholar
Rodot, H., Bisch, C. & Lasek, A. 1979 Zero-gravity simulation of liquids in contact with a solid surface. Acta Astronaut. 6, 1083–1092.Google Scholar
Sharp, J., Farmer, D. & Kelly, J. 2011 Contact angle dependence of the resonant frequency of sessile water droplets. Langmuir 27 (15), 9367–9371.Google Scholar
Shen, C. L., Xie, W. J. & Wei, B. 2010 Parametrically excited sectorial oscillation of liquid drops floating in ultrasound. Phys. Rev. E 81, 046305.Google Scholar
Shi, T. & Apfel, R. E. 1995 Oscillations of a deformed liquid drop in an acoustic field. Phys. Fluids 7 (7), 1545–1552.CrossRefGoogle Scholar
Shilton, R., Tan, M. K., Yeo, L. Y. & Friend, J. R. 2008 Particle concentration and mixing in microdrops driven by focused surface acoustic waves. J. Appl. Phys. 104 (1), 014910.Google Scholar
Trinh, E., Zwern, A. & Wang, T. G. 1982 An experimental study of small-amplitude drop oscillations in immiscible liquid systems. J. Fluid Mech. 115, 453–474.CrossRefGoogle Scholar
Trinh, E. H., Holt, R. G. & Thiessen, D. B. 1996 The dynamics of ultrasonically levitated drops in an electric field. Phys. Fluids 8 (1), 43–61.Google Scholar
Vukasinovic, B., Smith, M. & Glezer, A. 2007 Dynamics of a sessile drop in forced vibration. J. Fluid Mech. 587, 395–423.Google Scholar
Wang, T. G., Anilkumar, A. V. & Lee, C. P. 1996 Oscillations of liquid drops: results from USML-1 experiments in space. J. Fluid Mech. 308, 1–14.Google Scholar
Whitehill, J., Neild, A., Ng, T. W. & Stokes, M. 2010 Collection of suspended particles in a drop using low frequency vibration. Appl. Phys. Lett. 96 (5), 053501.CrossRefGoogle Scholar
Wright, P. H. & Saylor, J. R. 2003 Patterning of particulate films using Faraday waves. Rev. Sci. Instrum. 74 (9), 4063–4070.Google Scholar
Yoshiyasu, N., Matsuda, K. & Takaki, R. 1996 Self-induced vibration of a water drop placed on an oscillating plate. J. Phys. Soc. Japan 65 (7), 2068–2071.Google Scholar