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Modulated surface waves in large-aspect-ratio horizontally vibrated containers

Published online by Cambridge University Press:  02 May 2007

FERNANDO VARAS  and
JOSÉ M. VEGA
FERNANDO VARAS
Affiliation:
ETS Ingenieros de Telecomunicación, Universidad de Vigo, Campus Marcosende, 36280-Vigo, Pontevedra, Spain
JOSÉ M. VEGA
Affiliation:
ETS Ingenieros Aeronáuticos, Universidad Politécnica de Madrid, Plaza Cardenal Cisneros, 3, 28040-Madrid, Spain
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Abstract

We consider the harmonic and subharmonic modulated surface waves that appear upon horizontal vibration along the surface of the liquid in a two-dimensional large-aspect-ratio (length large compared to depth) container, whose depth is large compared to the wavelength of the surface waves. The analysis requires us also to consider an oscillatory bulk flow and a viscous mean flow. A weakly nonlinear description of the harmonic waves is made which provides the threshold forcing amplitude to trigger harmonic instabilities, which are of various qualitatively different kinds. A linear analysis provides the threshold amplitude for the appearance of subharmonic waves through a subharmonic instability. The results obtained are used to make several specific qualitative and quantitative predictions.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 579 , 25 May 2007 , pp. 271 - 304
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Aranson, I. S. & Tsimring, L. S. 2006 Patterns and collective behavior in granular media: theoretical concepts. Rev. Mod. Phys. 78, 641–692.CrossRefGoogle Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Mechanics. 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 viscous fluids. J. Fluid Mech. 288, 325–350.CrossRefGoogle Scholar
Billingham, J. 2002 Nonlinear sloshing in zero gravity. J. Fluid Mech. 464, 365–391.CrossRefGoogle Scholar
Cross, M. C. & Hohenberg, P. C. 1993 Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112.CrossRefGoogle Scholar
Daniels, P. G. 1978 Finite amplitude two-dimensional convection in a finite rotating system. Proc. R. Soc. Lond. A 363, 195–215.Google Scholar
Davey, A. & Stewartson, S. 1974 On three-dimensional packets of surface waves. Proc. R. Soc. Lond. A 338, 101–110.Google Scholar
Faltinsen, O. M. & Timokha, A. N. 2002 Asymptotic modal approximation of nonlinear resonant sloshing in a rectangular tank with small fluid depth. J. Fluid Mech. 470, 319–357.CrossRefGoogle Scholar
Faltinsen, O. M., Rognebakke, O. F. & Timokha, A. N. 2006 Transient and steady-state amplitudes of resonant three-dimensional sloshing in a square-based tank with a finite fluid depth. Phys. Fluids 18, 012103.CrossRefGoogle Scholar
Faraday, M. 1831 On the forms and states assumed by fluids in contact with vibrating elastic surfaces. Phil. Trans. R. Soc. Lond. 121, 319–340.Google Scholar
Fauve, S. 1995 Parametric instabilities. In Dynamics of Nonlinear and Disordered Systems (ed. Martínez-Mekler, G., Seligman, T. H.), pp. 67–115. World Scientific.CrossRefGoogle Scholar
Feng, Z. C. 1997 Transition to travelling waves from standing waves in a rectangular container subjected to horizontal excitations. Phys. Rev. Lett. 79, 415–418.CrossRefGoogle Scholar
Funakoshi, M. & Inoue, S. 1988 Surface waves due to resonant horizontal oscillation. J. Fluid Mech. 192, 219–247.CrossRefGoogle Scholar
Gavrilyuk, I., Lobovsky, I. & Tomokha, A. 2004 Two-dimensional variational vibroequilibria and Faraday's drops. Z. Angew. Math. Phys. 55, 1015–1033.CrossRefGoogle Scholar
González-Viñas, W. & Salam, J. 1994 Surface waves periodically excited in a CO2 tube. Europhys. Lett. 26, 665–670.CrossRefGoogle Scholar
Henderson, D. M. & Miles, J. W. 1994 Surface wave damping in a circular cylinder with a fixed contact line. J. Fluid Mech. 213, 285–299.CrossRefGoogle Scholar
Higuera, M., Vega, J. M. & Knobloch, E. 2002 Coupled amplitude-mean flow equations for nearly-inviscid Faraday waves in moderate aspect ratio containers. J. Nonlin. Sci. 12, 505–551.CrossRefGoogle Scholar
Hill, D. F. 2003 Transient and steady state amplitudes of forced waves in rectangular basins. Phys. Fluids 15, 1576–1587.CrossRefGoogle Scholar
Hocking, L. M. 1987 Reflection of capillary-gravity waves. Wave Motion 9, 217–226.CrossRefGoogle Scholar
Ivanova, A., Kozlov, V. & Evesque, P. 1996 Patterning of liquified sand surface in a cylinder filled with liquid and subject to horizontal vibrations. Europhys. Lett. 35, 159–164.CrossRefGoogle Scholar
Jaeger, H. M. & Nagel, S. R. 1996 Granular solids, liquids, and gases. Rev. Mod. Phys. 68, 1259–1273.CrossRefGoogle Scholar
Kozlov, V. G. 1991 Experimental investigation of vibrational convection in pseudoliquid layer. In Hydrodynamics and Heat Transfer in Microgravity (ed. Avduevsky, G.), pp. 57–61. Gordon and Breach.Google Scholar
Kudrolli, A. & Gollub, J. P. 1997 Patterns and spatio-temporal chaos in parametrically forced surface waves: a systematic survey at large aspect ratio. Physica D 97, 133–154.Google Scholar
Lapuerta, V., Mancebo, F. J. & Vega, J. M. 2001 Control of Rayleigh–Taylor instability by vertical vibration in large aspect ratio containers. Phys. Rev. E 64, 016318-1-17.Google ScholarPubMed
Lapuerta, V., Martel, C. & Vega, J. M. 2002 Interaction of nearly-inviscid Faraday waves and mean flows in 2-D containers of quite large aspect ratio. Physica D 173, 178–203.Google Scholar
Martel, C. & Vega, J. M. 1996 Finite size effects near the onset of the oscillatory instability. Nonlinearity 9, 1129–1171.CrossRefGoogle Scholar
Martel, C. & Vega, J. M. 1998 Dynamics of a hyperbolic system that applies at the onset of the oscillatory instability. Nonlinearity 11, 105–142.CrossRefGoogle Scholar
Martel, C., Vega, J. M. & Knobloch, K. 2003 Dynamics of counterpropagating waves in parametrically driven systems: dispersion vs. advection. Physica D 174, 198–217.Google Scholar
Miles, J. W. 1984 Resonantly forced surface waves in a circular cylinder. J. Fluid Mech. 149, 15–31.CrossRefGoogle Scholar
Miles, J. & Henderson, D. 1990 Parametrically forced surface waves. Annu. Rev. Fluid Mech. 22, 143–165.CrossRefGoogle Scholar
Newell, A. C. 1985 Solitons in Mathematics and Physics. Society for Industrial and Applied Mathematics.CrossRefGoogle Scholar
Nicolás, J. A., Rivas, D. & Vega, J. M. 1998 On the steady streaming flow due to high frequency vibration in nearly-inviscid liquid bridges. J. Fluid Mech. 354, 147–174.CrossRefGoogle Scholar
Nobili, M., Ciliberto, S., Cocciaro, B., Faetti, S. & Fronzoni, L. 1988 Time dependent surface waves in a horizontally oscillating container. Euro. Phys. Lett. 7, 587–592.CrossRefGoogle Scholar
Ockendon, J. R. & Ockendon, H. 1973 Resonant surface waves. J. Fluid Mech. 59, 397–413.CrossRefGoogle Scholar
Rayleigh, Lord 1883 On the crispations of fluid resting upon a vibrating support. Phil. Mag. 16, 50–58.CrossRefGoogle Scholar
Riley, M. 2001 Steady streaming. Annu. Rev. Fluid Mech. 33, 43–65.CrossRefGoogle Scholar
Ristow, G. H. 2000 Pattern Formation in Granular Materials. Springer.Google Scholar
Schlichting, H. 1968 Boundary Layer Theory. McGraw–Hill.Google Scholar
Vega, J. M., Knobloch, E. & Martel, C. 2001 Nearly inviscid Faraday waves in annular containers of moderately large aspect ratio. Physica D 154, 147–171.Google Scholar
Vukasinovic, B., Smith, B. & Glezer, A. 2006 Dynamics of a sessile drop in forced vibration. J. Fluid Mech. (sumitted).CrossRefGoogle Scholar
Wolf, G. H. 1969 The dynamic stabilization of the Rayleigh–Taylor instability and the corresponding dynamic equilibrium. Z. Phys. 227, 291–300.CrossRefGoogle Scholar
Wunenburger, R., Evesque, P., Chabot, C., Garrabos, Y., Fauve, S. & Beysens, D. 1999 Frozen wave induced by high frequency horizontal vibrations on a CO2 liquid–gas interface near the critical point. Phys. Rev. E 59, 5440–5445.Google ScholarPubMed