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Two-frequency excitation of single-mode Faraday waves

Published online by Cambridge University Press:  09 January 2015

W. Batson ,
F. Zoueshtiagh  and
R. Narayanan
W. Batson*
Affiliation:
Department of Chemical Engineering, University of Florida, Gainesville, FL 32601, USA
F. Zoueshtiagh
Affiliation:
Institut d’Electronique, de Microélectronique et de Nanotechnologie (IEMN), UMR CNRS 8520, University of Lille 1, Avenue Poincaré, 59652 Villeneuve d’Ascq, France
R. Narayanan
Affiliation:
Department of Chemical Engineering, University of Florida, Gainesville, FL 32601, USA
*
†Email address for correspondence: wbatson@gmail.com
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Abstract

The purpose of this work is to investigate, for the first time, excitation of Faraday waves in small containers using two commensurate frequencies. This spatial restriction, which is encountered at low frequencies, leads to a wave composed primarily of one spatial eigenmode of the container. When two frequencies are used, the mode resonates primarily with one frequency, while the role of the second is to alter the instability threshold and the resulting nonlinear dynamics. As the parameter space expands greatly as a result of the introduction of three new degrees of freedom, viz. the frequency, amplitude and phase of the new component, the linear theory is first used as a guide to highlight basic two-frequency phenomena. These predictions and nonlinear phenomena are then studied experimentally with the system of Batson, Zoueshtiagh & Narayanan (J. Fluid Mech., vol. 729, 2013, pp. 496–523), who studied single-frequency excitation of different modes in a cylindrical cell. The two-frequency experiments of this work focus on excitation of the fundamental axisymmetric mode, and are quantitatively compared to the model via a posteriori Fourier decomposition of the parametric input. In doing so, experimental dependence of the instability on the new degrees of freedom is demonstrated, in accordance with the model predictions. This is done for a variety of frequency ratios, and overall agreement between the observed and predicted onset conditions is identical to that already reported for the single-frequency experiment. For each frequency ratio, the nonlinear behaviour is experimentally characterized by bifurcation and time series data, which is shown to differ significantly from comparable single-frequency excitations. Finally, we present and discuss a wave in which both temporal frequencies are used to simultaneously excite different spatial modes.

JFM classification

Instability: Instability Instability: Nonlinear instability Instability: Parametric instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 764 , 10 February 2015 , pp. 538 - 571
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Copyright
© 2015 Cambridge University Press 

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References

Agrawal, P., Gandhi, P. S. & Neild, A. 2014 Quantification and comparison of low frequency microparticle collection mechanism in an open rectangular chamber. J. Appl. Phys. 115 (17), 174505.Google Scholar
Arbell, H. & Fineberg, J. 2002 Pattern formation in two-frequency forced parametric waves. Phys. Rev. E 65 (3), 036224.Google Scholar
Batson, W., Zoueshtiagh, F. & Narayanan, R. 2013 The Faraday threshold in small cylinders and the sidewall non-ideality. J. Fluid Mech. 729, 496–523.Google 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.Google 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. 1954 The stability of a plane free surface of a liquid in vertical periodic motion. Proc. R. Soc. Lond. A 225, 505–515.Google Scholar
Besson, T., Edwards, W. S. & Tuckerman, L. S. 1996 Two-frequency parametric excitation of surface waves. Phys. Rev. E 54, 507–514.Google Scholar
Ciliberto, S. & Gollub, J. P. 1985 Chaotic mode competition in parametrically forced surface waves. J. Fluid Mech. 158 (1), 381–398.Google Scholar
Cobelli, P. J., Maurel, A., Pagneux, V. & Petitjeans, P. 2009 Global measurement of water waves by Fourier transform profilometry. Exp. Fluids 46 (6), 1037–1047.Google Scholar
Crawford, J. D., Knobloch, E. & Riecke, H. 1990 Period-doubling mode interactions with circular symmetry. Physica D 44 (3), 340–396.Google 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.Google Scholar
Douady, S. 1990 Experimental study of the Faraday instability. J. Fluid Mech. 221, 383–409.Google Scholar
Edwards, W. S. & Fauve, S. 1993 Parametrically excited quasicrystalline surface waves. Phys. Rev. E 47 (2), R788–R791.Google Scholar
Edwards, W. S. & Fauve, S. 1994 Patterns and quasi-patterns in the Faraday experiment. J. Fluid Mech. 278, 123–148.Google Scholar
Faraday, M. 1831 On the forms and states of fluids on vibrating elastic surfaces. Phil. Trans. R. Soc. Lond. 52, 319–340.Google Scholar
Grindrod, P. 1991 Patterns and Waves: The Theory and Applications of Reaction–Diffusion Equations. Clarendon.Google Scholar
HaQuang, N., Mook, D. T. & Plaut, R. H. 1987 A non-linear analysis of the interactions between parametric and external excitations. J. Sound Vib. 118 (3), 425–439.Google Scholar
Harris, D. M., Moukhtar, J., Fort, E., Couder, Y. & Bush, J. 2013 Wavelike statistics from pilot-wave dynamics in a circular corral. Phys. Rev. E 88 (1), 011001.Google Scholar
Henderson, D. & Miles, J. 1990 Single-mode Faraday waves in small cylinders. J. Fluid Mech. 213, 95–109.Google Scholar
Henderson, D. M. & Miles, J. W. 1991 Faraday waves in 2:1 internal resonance. J. Fluid Mech. 222, 449–470.Google Scholar
Hill, G. W. 1886 On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon. Acta Mathematica 8 (1), 1–36.Google Scholar
Hocking, L. M. 1987 The damping of capillary–gravity waves at a rigid boundary. J. Fluid Mech. 179, 253–266.Google Scholar
Kidambi, R. 2013 Inviscid Faraday waves in a brimful circular cylinder. J. Fluid Mech. 724, 671–694.Google 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 (1), 133–154.Google Scholar
Kudrolli, A., Pier, B. & Gollub, J. P. 1998 Superlattice patterns in surface waves. Physica D 123 (1), 99–111.Google Scholar
Kumar, K. 1996 Linear theory of Faraday instability in viscous fluids. Proc. R. Soc. Lond. A 452, 1113–1126.Google Scholar
Kumar, K. & Tuckerman, L. 1994 Parametric instability of the interface between two fluids. J. Fluid Mech. 279, 49–67.CrossRefGoogle Scholar
McLachlan, N. W. 1947 Theory and Application of Mathieu Functions. Clarendon.Google Scholar
Melde, F. 1860 Über die Erregung stehender Wellen eines fadenförmigen Körpers. Ann. Phys. 187 (12), 513–537.Google Scholar
Meron, E. & Procaccia, I. 1986 Low-dimensional chaos in surface waves: theoretical analysis of an experiment. Phys. Rev. A 34 (4), 3221–3237.Google Scholar
Miles, J. W. 1967 Surface-wave damping in closed basins. Proc. R. Soc. Lond. A 297, 459–475.Google Scholar
Milner, S. T. 1991 Square patterns and secondary instabilities in driven capillary waves. J. Fluid Mech. 225, 81–100.Google Scholar
Moisy, F., Rabaud, M. & Salsac, K. 2009 A synthetic Schlieren method for the measurement of the topography of a liquid interface. Exp. Fluids 46 (6), 1021–1036.Google Scholar
Monti, R. & Savino, R.1996. Influence of $g$ -jitter on fluid physics experimentation on-board the International Space Station. ESA Rep. SP-385, pp. 215–224. European Space Agency.Google Scholar
Müller, H. W. 1993 Periodic triangular patterns in the Faraday experiment. Phys. Rev. Lett. 71 (20), 3287–3290.Google Scholar
Nayfeh, A. H. & Mook, D. T. 1979 Nonlinear Oscillations. Wiley.Google Scholar
Perinet, N., Juric, D. & Tuckerman, L. S. 2009 Numerical simulation of Faraday waves. J. Fluid Mech. 635, 1–26.Google Scholar
Plaut, R. H., Gentry, J. J. & Mook, D. T. 1990 Non-linear structural vibrations under combined multi-frequency parametric and external excitations. J. Sound Vib. 140 (3), 381–390.Google Scholar
Prakash, G., Hu, S., Raman, A. & Reifenberger, R. 2009 Theoretical basis of parametric-resonance-based atomic force microscopy. Phys. Rev. B 79 (9), 094304.Google Scholar
Raman, C. V. 1912 Experimental investigations on the maintenance of vibrations. Bull. Indian Assoc. Cultiv. Sci. 6, 1–40.Google Scholar
Ruby, L. 1996 Applications of the Mathieu equation. Am. J. Phys. 64 (1), 39–44.Google Scholar
Simonelli, F. & Gollub, J. P. 1989 Surface wave mode interactions: effects of symmetry and degeneracy. J. Fluid Mech. 199 (1), 471–494.Google Scholar
Skeldon, A. C. & Guidoboni, G. 2007 Pattern selection for Faraday waves in an incompressible fluid. SIAM J. Appl. Maths 67 (4), 1064–1100.Google Scholar
Someya, S. & Munakata, T. 2005 Measurement of the interface tension of immiscible liquids interface. J. Cryst. Growth 275, e343–e348.Google Scholar
Strogatz, S. H. 2001 Nonlinear Dynamics and Chaos: with Applications to Physics, Biology and Chemistry. Perseus.Google Scholar
Tipton, C. R. & Mullin, T. 2004 An experimental study of Faraday waves formed on the interface between two immiscible liquids. Phys. Fluids 16, 2336–2341.Google Scholar
Turner, K. L., Miller, S. A., Hartwell, P. G., MacDonald, N. C., Strogatz, S. H. & Adams, S. G. 1998 Five parametric resonances in a microelectromechanical system. Nature 396 (6707), 149–152.Google Scholar
Umeki, M. & Kambe, T. 1989 Nonlinear dynamics and chaos in parametrically excited surface waves. J. Phys. Soc. Japan 58, 140–154.CrossRefGoogle Scholar
Xu, J. & Attinger, D. 2007 Control and ultrasonic actuation of a gas–liquid interface in a microfluidic chip. J. Micromech. Microengng 17 (3), 609–616.Google Scholar
Yoshikawa, H. N., Zoueshtiagh, F., Caps, H., Kurowski, P. & Petitjeans, P. 2010 Bubble splitting in oscillatory flows on ground and in reduced gravity. Eur. Phys. J. E 31 (2), 191–199.Google Scholar
Zhang, W. & Viñals, J. 1997a Pattern formation in weakly damped parametric surface waves. J. Fluid Mech. 336 (1), 301–330.Google Scholar
Zhang, W. & Viñals, J. 1997b Pattern formation in weakly damped parametric surface waves driven by two frequency components. J. Fluid Mech. 341, 225–244.Google Scholar

Batson et al. supplementary movie

Simultaneous excitation of a (0,1) and a (1,1) mode with [l/m, f, A, χ]=[3/2, 2.55 Hz, 1.40 mm, 48.0◦].

Download Batson et al. supplementary movie(Video)
Video 8.8 MB