Abstract
We find a quantitative approximation which explains the appearance and amplification of surface waves in a highly viscous fluids when it is submitted to vertical accelerations (Faraday’s instability). Although stationary surface waves with frequency equal to half of the frequency of the excitation are observed in fluids of different kinematical viscosities we show here that the mechanism which produces the instability is very different for a highly viscous fluid as compared with a weakly viscous fluid. This can be shown by-deriving an exact equation for the linear evolution of the surface which is non-local in time. For a highly viscous fluid, this equation becomes local and of second order and is a Mathieu equation which is different from the one found for weak viscosity. Analyzing the new equation, an intimate relation with the Rayleigh-Taylor instability can be found.
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References
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.
Benjamin, T. B. & Ursell, F. 1954 The stability of the plane free surface of a liquid in vertical periodic motion. Proc. R. Soc. Lond. A 225, 505–515.
Bergé, P., Pomeau, Y. & Vidal, C. 1984 Oscillateur libre-oscillateur amorti. In L’ordre dans le chaos. Hermman.
Besson, T., Edwards, W. S., & Tuckerman, L. S. 1996 Two-frequency parametric excitation of surface waves. Phys. Rev. E 54, 507–513.
Cerda, E. & Tirapegui, E. 1997 (a) Faraday’s Instability for Viscous Fluids. Phys. Rev. Lett. 78, 859–863.
Cerda, E. & Tirapegui, E. 1997 (b) On the linear evolution of the surface of a viscous incompressible fluid. Bulletin de l’Académie Royale des Sciences de Belgique (Classe des Sciences 7/12/1996).
Cerda, E. & Tirapegui, E. 1997 (c) Faraday’s Instability for Viscous Fluids. Submitted to J. Fluid Mech.
Chandrasekar 1981 The Rayleigh-Taylor instability. In Hydrodinamic and Hydromagnetic Stability. Dover.
Edwards, W. S. & Fauve, S. 1992 Parametric Instability of a Liquid-Vapor Interface Close to the Critical Point. Phys. Rev. Lett. 68, 3160–3164.
Eisenmenger, W. 1959 Dynamics properties of the surface tension of water and aqueous solutions of surface active agents with standing capillary waves in the frequency range 10 kc/s to 1.5 Mc/s. Acustica 9, 328–340.
De Gennes, P. G. 1985 Wetting: statics and dynamics. Reviews of Modern Physics 57, 827–863.
Kumar, K. & Tuckerman, L. S. 1994 Parametric instability of the interface between two fluids. J. Fluid Mech. 279, 49–68.
Kumar, K. 1996 Linear theory of Faraday instability in viscous liquids. Proc. R. Soc. Lond. A 452, 1113–1126.
Landau, L. D. & Lifshitz, E. M. 1970 Parametric resonance. In Mechanica. Reverté.
Landau, L. D. & Lifshitz, E. M. 1987 Fluid Mechanics. Pergamon.
Lioubashevsky, O., Flneberg, J. & Tuckerman, L. S. 1997 Scaling of the transition to parametrically driven surface waves in highly dissipative systems. Phys. Rev. E 55, 3832–3835.
Melo, F., Umbanhowar, P. & Swinney, H. L. 1994 Transition to Parametric Wave Patterns in a Vertically Oscillated Granular Layer. Phys. Rev. Lett. 72, 172–176.
Melo, F., Umbanhowar, P. & Swinney, H. L. 1995 Hexagons, Kinks, and Disorder in Oscillated Granular Layers. Phys. Rev. Lett. 75, 3838.
Sorokin, V. I. 1957 The effects of fountain formation at the surface of a vertically oscillating liquid. Soviet Physics Acoustic 3, 281–295.
Umbanhowar, P., Melo, F. & Swinney, H. L. 1996 Localized excitations in a vertically vibrated granular layer. Nature 382, 793.
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Cerda, E.A. (2000). A parametric oscillator in a highly viscous fluid. In: Tirapegui, E., Martínez, J., Tiemann, R. (eds) Instabilities and Nonequilibrium Structures VI. Nonlinear Phenomena and Complex Systems, vol 5. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-4247-2_7
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DOI: https://doi.org/10.1007/978-94-011-4247-2_7
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