Skip to main content Accessibility help
×
×
Home

Two-frequency excitation of single-mode Faraday waves

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

Copyright
Corresponding author
Email address for correspondence: wbatson@gmail.com
References
Hide All
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.
Arbell, H. & Fineberg, J. 2002 Pattern formation in two-frequency forced parametric waves. Phys. Rev. E 65 (3), 036224.
Batson, W., Zoueshtiagh, F. & Narayanan, R. 2013 The Faraday threshold in small cylinders and the sidewall non-ideality. J. Fluid Mech. 729, 496523.
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. & Scott, J. C. 1979 Gravity–capillary waves with edge constraints. J. Fluid Mech. 92 (2), 241267.
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.
Besson, T., Edwards, W. S. & Tuckerman, L. S. 1996 Two-frequency parametric excitation of surface waves. Phys. Rev. E 54, 507514.
Ciliberto, S. & Gollub, J. P. 1985 Chaotic mode competition in parametrically forced surface waves. J. Fluid Mech. 158 (1), 381398.
Cobelli, P. J., Maurel, A., Pagneux, V. & Petitjeans, P. 2009 Global measurement of water waves by Fourier transform profilometry. Exp. Fluids 46 (6), 10371047.
Crawford, J. D., Knobloch, E. & Riecke, H. 1990 Period-doubling mode interactions with circular symmetry. Physica D 44 (3), 340396.
Das, S. P. & Hopfinger, E. J. 2008 Parametrically forced gravity waves in a circular cylinder and finite-time singularity. J. Fluid Mech. 599, 205228.
Douady, S. 1990 Experimental study of the Faraday instability. J. Fluid Mech. 221, 383409.
Edwards, W. S. & Fauve, S. 1993 Parametrically excited quasicrystalline surface waves. Phys. Rev. E 47 (2), R788R791.
Edwards, W. S. & Fauve, S. 1994 Patterns and quasi-patterns in the Faraday experiment. J. Fluid Mech. 278, 123148.
Faraday, M. 1831 On the forms and states of fluids on vibrating elastic surfaces. Phil. Trans. R. Soc. Lond. 52, 319340.
Grindrod, P. 1991 Patterns and Waves: The Theory and Applications of Reaction–Diffusion Equations. Clarendon.
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), 425439.
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.
Henderson, D. & Miles, J. 1990 Single-mode Faraday waves in small cylinders. J. Fluid Mech. 213, 95109.
Henderson, D. M. & Miles, J. W. 1991 Faraday waves in 2:1 internal resonance. J. Fluid Mech. 222, 449470.
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), 136.
Hocking, L. M. 1987 The damping of capillary–gravity waves at a rigid boundary. J. Fluid Mech. 179, 253266.
Kidambi, R. 2013 Inviscid Faraday waves in a brimful circular cylinder. J. Fluid Mech. 724, 671694.
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), 133154.
Kudrolli, A., Pier, B. & Gollub, J. P. 1998 Superlattice patterns in surface waves. Physica D 123 (1), 99111.
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.
McLachlan, N. W. 1947 Theory and Application of Mathieu Functions. Clarendon.
Melde, F. 1860 Über die Erregung stehender Wellen eines fadenförmigen Körpers. Ann. Phys. 187 (12), 513537.
Meron, E. & Procaccia, I. 1986 Low-dimensional chaos in surface waves: theoretical analysis of an experiment. Phys. Rev. A 34 (4), 32213237.
Miles, J. W. 1967 Surface-wave damping in closed basins. Proc. R. Soc. Lond. A 297, 459475.
Milner, S. T. 1991 Square patterns and secondary instabilities in driven capillary waves. J. Fluid Mech. 225, 81100.
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), 10211036.
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.
Müller, H. W. 1993 Periodic triangular patterns in the Faraday experiment. Phys. Rev. Lett. 71 (20), 32873290.
Nayfeh, A. H. & Mook, D. T. 1979 Nonlinear Oscillations. Wiley.
Perinet, N., Juric, D. & Tuckerman, L. S. 2009 Numerical simulation of Faraday waves. J. Fluid Mech. 635, 126.
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), 381390.
Prakash, G., Hu, S., Raman, A. & Reifenberger, R. 2009 Theoretical basis of parametric-resonance-based atomic force microscopy. Phys. Rev. B 79 (9), 094304.
Raman, C. V. 1912 Experimental investigations on the maintenance of vibrations. Bull. Indian Assoc. Cultiv. Sci. 6, 140.
Ruby, L. 1996 Applications of the Mathieu equation. Am. J. Phys. 64 (1), 3944.
Simonelli, F. & Gollub, J. P. 1989 Surface wave mode interactions: effects of symmetry and degeneracy. J. Fluid Mech. 199 (1), 471494.
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, e343e348.
Strogatz, S. H. 2001 Nonlinear Dynamics and Chaos: with Applications to Physics, Biology and Chemistry. Perseus.
Tipton, C. R. & Mullin, T. 2004 An experimental study of Faraday waves formed on the interface between two immiscible liquids. Phys. Fluids 16, 23362341.
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), 149152.
Umeki, M. & Kambe, T. 1989 Nonlinear dynamics and chaos in parametrically excited surface waves. J. Phys. Soc. Japan 58, 140154.
Xu, J. & Attinger, D. 2007 Control and ultrasonic actuation of a gas–liquid interface in a microfluidic chip. J. Micromech. Microengng 17 (3), 609616.
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), 191199.
Zhang, W. & Viñals, J. 1997a Pattern formation in weakly damped parametric surface waves. J. Fluid Mech. 336 (1), 301330.
Zhang, W. & Viñals, J. 1997b Pattern formation in weakly damped parametric surface waves driven by two frequency components. J. Fluid Mech. 341, 225244.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

JFM classification

Type Description Title
VIDEO
Movie

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◦].

 Video (8.8 MB)
8.8 MB

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed