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The Faraday threshold in small cylinders and the sidewall non-ideality

  • W. Batson (a1) (a2), F. Zoueshtiagh (a2) and R. Narayanan (a1)
Abstract

In this work we investigate, by way of experiments and theory, the Faraday instability threshold in cylinders at low frequencies. This implies large wavelengths where effects from mode discretization cannot be ignored. Careful selection of the working fluids has resulted in an immiscible interface whose apparent contact line with the sidewall can glide over a tiny film of the more wetting fluid, without detachment of its actual contact line. This unique behaviour has allowed for a system whose primary dissipation is defined by the bulk viscous effects, and in doing so, for the first time, close connection is seen with the viscous linear stability theory for which a stress-free condition is assumed at the sidewalls. As predicted, mode selection and co-dimension 2 points are observed in the experiment for a frequency range including subharmonic, harmonic, and superharmonic modes. While agreement with the predictions are generally excellent, there are deviations from the theory for certain modes and these are explained in the context of harmonic meniscus waves. A review of previous work on single-mode excitation in cylinders is given, along with comparison to the viscous model and analysis based upon the conclusions of the current experiments.

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Email address for correspondence: wbatson@gmail.com
References
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Abramowitz, M. & Stegun, I. A. 1964 Handbook of Mathematical Functions, 10th edn. Dover.
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, 325350.
Benjamin, T. B. & Ursell, F. 1954 The stability of a plane free surface of a liquid in vertical periodic motion. Proc. R. Soc. Lond. A 225, 505515.
Case, K. M. & Parkinson, W. C. 1957 Damping of surface waves in an incompressible fluid. J. Fluid Mech. 2, 172184.
Christiansen, B., Alstrøm, P. & Levinsen, M. T. 1994 Dissipation and ordering in capillary waves at high aspect ratios. J. Fluid Mech. 291, 323341.
Ciliberto, S. & Gollub, J. P. 1985 Chaotic mode competition in parametrically forced surface waves. J. Fluid Mech. 158, 381398.
Das, S. P. & Hopfinger, E. J. 2008 Parametrically forced gravity waves in a circular cylinder and finite-time singularity. J. Fluid Mech. 599, 205228.
Dodge, F. T., Kana, D. D. & Abramson, H. N. 1965 Liquid surface oscillations in longitudinally excited rigid cylindrical containers. AIAA J. 3, 685695.
Douady, S. 1990 Experimental study of the Faraday instability. J. Fluid Mech. 221, 383409.
Fauve, S., Kumar, K., Laroche, C., Beysens, D. & Garrabos, Y. 1992 Parametric instability of a liquid–vapor interface close to the critical point. Phys. Rev. Lett. 68 (21), 31603163.
Friend, J. & Yeo, L. Y. 2011 Microscale acoustofluidics: microfluidics driven via acoustics and ultrasonics. Rev. Mod. Phys. 83, 647704.
Henderson, D. & Miles, J. 1990 Single-mode Faraday waves in small cylinders. J. Fluid Mech. 213, 95109.
Ito, T. & Kukita, Y. 2008 Interface behavior between two fluids vertically oscillated in a circular cylinder under nonlinear contact line condition. J. Fluid Sci. Technol. 3, 690711.
Ito, T., Tsuji, Y. & Kukita, Y. 1999 Interface waves excited by vertical vibration of stratified fluids in a circular cylinder. J. Nucl. Sci. Technol. 36, 508521.
Keulegan, G. H. 1958 Energy dissipation in standing waves in rectangular basins. J. Fluid Mech. 6, 3350.
Kityk, A. V., Embs, J., Mekhonoshin, V. V. & Wagner, C. 2005 Spatiotemporal characterization of interfacial Faraday waves by means of a light absorption technique. Phys. Rev. E 72, 036209.
Kumar, K. 1996 Linear theory of Faraday instability in viscous fluids. Proc. R. Soc. Lond. A 452, 11131126.
Kumar, K. & Tuckerman, L. 1994 Parametric instability of the interface between two fluids. J. Fluid Mech. 279, 4967.
Landau, L. D. & Lifshitz, L. M. 1987 Fluid Mechanics, vol. 6. Course of Theoretical Physics, Butterworth-Heinemann.
Miles, J. W. 1967 Surface-wave damping in closed basins. Proc. R. Soc. Lond. A 297, 459475.
Miles, J. W. 1984 Nonlinear Faraday resonance. J. Fluid Mech. 146, 285302.
Milner, S. T. 1991 Square patterns and secondary instabilities in driven capillary waves. J. Fluid Mech. 225, 81100.
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), 23572360.
Nayfeh, A. H. & Mook, D. T. 1979 Nonlinear Oscillations. Wiley.
Skeldon, A. C. & Guidoboni, G. 2007 Pattern selection for Faraday waves in an incompressible fluid. SIAM J. Appl. Maths. 67 (4), 10641100.
Someya, S. & Munakata, T. 2005 Measurement of the interface tension of immiscible liquids interface. J. Cryst. Growth 275, 343348.
Tipton, C. 2003 Interfacial Faraday waves in a small cylindrical cell. PhD thesis, University of Manchester.
Tipton, C. R. & Mullin, T. 2004 An experimental study of Faraday waves formed on the interface between two immiscible liquids. Phys. Fluids 16, 23362341.
Virnig, J. C., Berman, A. S. & Sethna, P. R. 1988 On three-dimensional nonlinear subharmonic resonant surface waves in a fluid. Part 2. Experiment. Trans. ASME E: J. Appl. Mech. 55, 220224.
Wagner, C., Müller, H.-W. & Knorr, K. 2003 Pattern formation at the bicritical point of the Faraday instability. Phys. Rev. E 68, 066204.
Zoueshtiagh, F., Amiroudine, S. & Narayanan, R. 2009 Experimental and numerical study of miscible Faraday instability. J. Fluid Mech. 628, 4355.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
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