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Evolution and breaking of parametrically forced capillary waves in a circular cylinder

Published online by Cambridge University Press:  25 August 2009

BABURAJ A. PUTHENVEETTIL  and
E. J. HOPFINGER
BABURAJ A. PUTHENVEETTIL*
Affiliation:
Department of Applied Mechanics, Indian Institute of Technology Madras, Chennai-600 036, India
E. J. HOPFINGER
Affiliation:
LEGI-CNRS-UJF, BP 53, 38041 Grenoble Cedex 9, France
*
†Email address for correspondence: apbraj@iitm.ac.in
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Abstract

We present results on parametrically forced capillary waves in a circular cylinder, obtained in the limit of large fluid depth, using two low-viscosity liquids whose surface tensions differ by an order of magnitude. The evolution of the wave patterns from the instability to the wave-breaking threshold is investigated in a forcing frequency range (f = ω/2π = 25–100 Hz) that is around the crossover frequency (ωot) from gravity to capillary waves (ωot/2≤ω/2≤4ωot). As expected, near the instability threshold the wave pattern depends on the container geometry, but as the forcing amplitude is increased the wave pattern becomes random, and the wall effects are insignificant. Near breaking, the distribution of random wavelengths can be fitted by a Gaussian. A new gravity–capillary scaling is introduced that is more appropriate, than the usual viscous scaling, for low-viscosity fluids and forcing frequencies <103 Hz. In terms of these scales, a criterion is derived to predict the crossover from capillary- to gravity-dominated breaking. A wave-breaking model is developed that gives the relation between the container and the wave accelerations in agreement with experiments. The measured drop size distribution of the ejected drops above the breaking threshold is well approximated by a gamma distribution. The mean drop diameter is proportional to the wavelength determined from the dispersion relation; this wavelength is also close to the most likely wavelength of the random waves at drop ejection. The dimensionless drop ejection rate is shown to have a cubic power law dependence on the dimensionless excess acceleration ε′d an inertial–gravitational ligament formation model is consistent with such a power law.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 633 , 25 August 2009 , pp. 355 - 379
Copyright
Copyright © Cambridge University Press 2009

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References

REFERENCES

3M 2000 Fluorinert™ electronic liquid FC72 product information sheet. Tech Rep. 3M Speciality Materials.Google Scholar
Batchelor, G. K. 1969 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Bechhoefer, J., Ego, V., Manneville, S. & Johnson, B. 1995 An experimental study of the onset of parametrically pumped surface waves in a viscous fluid. J. Fluid. Mech. 288, 325–350.CrossRefGoogle Scholar
Benjamin, T. & Ursell, F. 1954 The stability of plane free surface of a liquid in periodic vertical motion. Phil. Trans. R. Soc. Lond. A. 225, 505–515.Google Scholar
Binks, D., Westra, M. T. & Van de Water, W. 1997 Effect of depth on the pattern formation of Faraday waves. Phys. Rev. Lett. 79, 5010–5013.CrossRefGoogle Scholar
Chen, P. & Vinals, J. 1997 Pattern selection in Faraday waves. Phys. Rev. Lett. 79 (14), 2670–2673.CrossRefGoogle Scholar
Ciliberto, S. & Gollub, J. P. 1985 Chaotic mode competition in parametrically forced surface waves. J. Fluid. Mech. 158, 381–398.CrossRefGoogle Scholar
Das, S. P. & Hopfinger, E. J. 2009 Mass transfer enhancement by gravity waves at a liquid-vapour interface. Intl J. Heat Mass Transfer. 52, 1400–1411.CrossRefGoogle Scholar
Das, S. P. & Hopfinger, E. J. 2008 Parametrically forced gravity waves in a circular cylinder and finite-time singularity. J. Fluid. Mech. 599, 205–228.CrossRefGoogle Scholar
Donnelly, T. D., Hogan, J., Mugler, A., Schommer, N. & Schubmehl, M. 2004 An experimental study of micron-scale droplet aerosols produced via ultrasonic atomization. Phys. Fluids 16 (8), 2843–2851.CrossRefGoogle Scholar
Edwards, W. S. & Fauve, S. 1995 Patterns and quasi-patterns in the Faraday experiment. J. Fluid. Mech. 278, 123–148.CrossRefGoogle Scholar
Falcon, E., Laroche, C. & Fauve, S. 2007 Observations of gravity–capillary wave turbulence. Phys. Rev. Lett. 98, 094503.CrossRefGoogle ScholarPubMed
Frank, H. & Althoen, S. C. 2002 Statistics, Concepts and Applications. Cambridge University Press.Google Scholar
Gollub, J. P. & Meyer, W. 1983 Symmetry-breaking instabilities on a fluid surface. Physica D 6, 337–346.CrossRefGoogle Scholar
Goodridge, C. L., Hentschel, H. G. E. & Lathrop, D. P. 1999 Breaking Faraday waves: critical slowing of droplet ejection rates. Phys. Rev. Lett. 82 (15), 3062–3065.CrossRefGoogle Scholar
Goodridge, C. L., Shi, T. & Lathrop, D. P. 1996 Threshold dynamics of singular gravity–capillary waves. Phys. Rev. Lett. 76 (11), 1824–1827.CrossRefGoogle ScholarPubMed
Goodridge, C. L., Tao Shi, W., Hentschel, H. G. E. & Lathrop, D. P. 1997 Viscous effects in droplet-ejecting capillary waves. Phys. Rev. E 56 (1), 472–475.Google Scholar
Hopfinger, E. J. & Das, S. P. 2009 Mass transfer enhancement by capillary waves at a liquid-vapour interface. Exp. Fluids 46, 597–605.CrossRefGoogle Scholar
James, A. J., Smith, M. K. & Glezer, A. 2003 a Vibration-induced drop atomisation and the numerical simulation of low frequency single-droplet ejection. J. Fluid. Mech. 476, 29–62.CrossRefGoogle Scholar
James, A. J., Vukasinovic, B., Smith, M. K. & Glezer, A. 2003 b Vibration induced drop atomisation and bursting. J. Fluid. Mech. 476, 1.CrossRefGoogle Scholar
Jiang, L., Perlin, M. & Schultz, W. W. 1998 Period tripling and energy dissipation of breaking standing waves. J. Fluid. Mech. 369, 273.CrossRefGoogle Scholar
Jiang, L., Ting, C. L., Perlin, M. & Schultz, W. W. 1996 Moderate and steep Faraday waves:instabilities, modulation and temporal asymmetries. J. Fluid. Mech. 329, 275.CrossRefGoogle Scholar
Kudrolli, A. & Gollub, J. P. 1996 Patterns and spatiotemporal chaos in parametrically forced surface waves: a systematic survey at large aspect ratio. Physica D 97, 133–154.CrossRefGoogle Scholar
Kumar, K. & Tuckerman, L. S. 1994 Parametric instability of the interface between two fluids. J. Fluid. Mech. 279, 49–68.CrossRefGoogle Scholar
Lang, R. 1962 Ultrasonic atomization of liquids. J. Acoust. Soc. Am. 34, 6–9.CrossRefGoogle Scholar
Miles, J. & Henderson, D. 1990 Parametrically forced surface waves. Annu. Rev. Fluid Mech. 22, 143–165.CrossRefGoogle Scholar
Saylor, J. R. & Handler, R. A. 1997 Gas transport across an air/water interface populated with capillary waves. Phys. Fluids 9, 2529.CrossRefGoogle Scholar
Taylor, G. I. 1953 An experimental study of standing waves. Proc. R. Soc. A 218, 44–59.Google Scholar
Topp, M. 1973 Ultrasonic atomisation-a photographic study of the mechanism of disintegration. Aerosol Sci. 4, 17–25.CrossRefGoogle Scholar
Villermaux, E. 2007 Fragmentation. Annu. Rev. Fluid Mech. 39, 419–446.CrossRefGoogle Scholar
Vukasinovic, B., Smith, M. K. & Glezer, A. 2004 Spray characterization during vibration-induced drop atomization. Phys. Fluids 16, 306–316.CrossRefGoogle Scholar
Vukasinovic, B., Smith, M. K. & Glezer, A. 2007 a Dynamics of a sessile drop in forced vibration. J. Fluid. Mech. 587, 395–423.CrossRefGoogle Scholar
Vukasinovic, B., Smith, M. K. & Glezer, A. 2007 b Mechanisms of free-surface breakup in vibration-induced liquid atomiation. Phys. Fluids 19, 012104.CrossRefGoogle Scholar
Wright, W., Budakian, R. & Putterman, S. 1996 Diffusing light photography of fully developed isotropic ripple turbulence. Phys. Rev. Lett. 76 (24), 4528–4531.CrossRefGoogle ScholarPubMed
Yule, A. J. & Al-Suleimani, Y. 2000 On droplet formation from capillary waves on a vibrating surface. Proc. R. Soc. Lond. A 456, 1069–1085.CrossRefGoogle Scholar
Zhang, W. & Vinals, J. 1997 Pattern formation in weakly damped parametric surface waves. J. Fluid. Mech. 336, 310–330.CrossRefGoogle Scholar