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Steep capillary-gravity waves in oscillatory shear-driven flows

Published online by Cambridge University Press:  10 November 2009

SHREYAS V. JALIKOP  and
ANNE JUEL
SHREYAS V. JALIKOP*
Affiliation:
School of Mathematics and Manchester Centre for Nonlinear Dynamics, The University of Manchester, Manchester M13 9PL, UK
ANNE JUEL
Affiliation:
School of Mathematics and Manchester Centre for Nonlinear Dynamics, The University of Manchester, Manchester M13 9PL, UK
*
†Email address for correspondence: sjalikop@ma.man.ac.uk
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Abstract

We study steep capillary-gravity waves that form at the interface between two stably stratified layers of immiscible liquids in a horizontally oscillating vessel. The oscillatory nature of the external forcing prevents the waves from overturning, and thus enables the development of steep waves at large forcing. They arise through a supercritical pitchfork bifurcation, characterized by the square root dependence of the height of the wave on the excess vibrational Froude number (W, square root of the ratio of vibrational to gravitational forces). At a critical value Wc, a transition to a linear variation in W is observed. It is accompanied by sharp qualitative changes in the harmonic content of the wave shape, so that trochoidal waves characterize the weakly nonlinear regime, but ‘finger’-like waves form for W ≥ Wc. In this strongly nonlinear regime, the wavelength is a function of the product of amplitude and frequency of forcing, whereas for W < Wc, the wavelength exhibits an explicit dependence on the frequency of forcing that is due to the effect of viscosity. Most significantly, the radius of curvature of the wave crests decreases monotonically with W to reach the capillary length for W = Wc, i.e. the lengthscale for which surface tension forces balance gravitational forces. For W < Wc, gravitational restoring forces dominate, but for W ≥ Wc, the wave development is increasingly defined by localized surface tension effects.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 640 , 10 December 2009 , pp. 131 - 150
Copyright
Copyright © Cambridge University Press 2009

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References

REFERENCES

Chandrasekhar, S. 1981 Hydrodynamic and hydromagnetic stability. Dover.Google Scholar
Cox, C. S. 1958 Measurements of slopes of high-frequency wind waves. J. Mar. Res. 16, 199–225.Google Scholar
Craik, A. D. D. 1985 Wave interactions and fluid flows. Cambridge University Press.Google Scholar
Crapper, G. D. 1957 An exact solution for progressive capillary waves of arbitrary amplitude. J. Fluid Mech. 2, 532–540.Google Scholar
Drazin, P. G. 1970 Kelvin–Helmholtz instability of finite amplitude. J. Fluid Mech. 42, 321–335.Google Scholar
Dussan, E. B. 1979 On the spreading of liquids on solid surfaces: static and dynamic contact lines. Annu. Rev. Fluid Mech. 11, 371–400.CrossRefGoogle Scholar
González-Viñas, W. & Salán, G. 1994 Surface waves periodically excited in a CO2 tube. Europhys. Lett. 26, 665–670.Google Scholar
Helfrich, K. R. & Melville, W. K. 2006 Long nonlinear internal waves. Annu. Rev. Fluid Mech. 38, 395–425.Google Scholar
Holyer, J. Y. 1979 Large amplitude progressive interfacial waves. J. Fluid Mech. 93, 433–448.Google Scholar
Hou, T. Y., Lowengrub, J. S. & Shelley, M. J. 1997 The long-time motion of vortex sheets with surface tension. Phys. Fluids 9, 1933–1954.Google Scholar
Ivanova, A. A., Kozlov, V. G. & Evesque, P. 2001 Interface dynamics of immiscible fluids under horizontal vibration. Fluid Dyn. 36, 362–368.Google Scholar
Jiang, L., Perlin, M. & Schultz, W. W. 1998 Period tripling and energy dissipation of breaking standing waves. J. Fluid Mech. 369, 273–299.Google Scholar
Khenner, M. V., Lyubimov, D. V., Belozerova, T. S. & Roux, B. 1999 Stability of plane-parallel vibrational flow in a two-layer system. Eur. J. Mech. – B/Fluids 18, 1085–1101.Google Scholar
Kraskovskii, Y. P. 1960 On the theory of steady waves of not small amplitude (in Russian). Dokl. Akad. Nauk SSSR 130, 1237.Google Scholar
LaRocca, M. Rocca, M., Sciortino, G. & Boniforti, M. A. 2002 Interfacial gravity waves in a two-fluid system. Fluid Dyn. Res. 30, 31–66.Google Scholar
Longuet-Higgins, M. S. 1963 The generation of capillary waves by steep gravity waves. J. Fluid Mech. 16, 138–159.CrossRefGoogle Scholar
Lyubimov, D. V. & Cherepanov, A. A. 1987 Development of a steady relief at the interface of fluids in a vibrational field. Fluid Dyn. 86, 849–854.Google Scholar
McGoldrick, L. F. 1970 a An experiment on second-order capillary gravity resonant wave interactions. J. Fluid Mech. 40, 251–271.Google Scholar
McGoldrick, L. F. 1970 b On Wilton's ripples: a special case of resonant interactions. J. Fluid Mech. 42, 193–200.Google Scholar
Meiron, D. I. & Saffman, P. G. 1983 Overhanging interfacial gravity waves of large amplitude. J. Fluid Mech. 129, 213–218.Google Scholar
Phillips, O. M. 1988 Remote sensing of the sea surface. Annu. Rev. Fluid Mech. 20, 89–109.Google Scholar
Schultz, W. W., Vanden-Broeck, J. M., Jiang, L. & Perlin, M. 1998 Highly nonlinear standing waves with small capillary effect. J. Fluid Mech. 369, 273–299.Google Scholar
Shyh, C. K. & Munson, B. R. 1986 Interfacial instability of an oscillating shear layer. J. Fluid Eng. 108, 89–92.Google Scholar
Stokes, G. G. 1847 On the theory of oscillatory waves. Camb. Phil. Soc. Trans. 8, 441–455.Google Scholar
Talib, E., Jalikop, S. V. & Juel, A. 2007 The influence of viscosity on the frozen wave instability: theory and experiment. J. Fluid Mech. 584, 45–68.CrossRefGoogle Scholar
Talib, E. & Juel, A. 2007 Instability of a viscous interface under horizontal oscillation. Phys. Fluids 19, 092102.Google Scholar
Taylor, G. I. 1953 An experimental study of standing waves. Proc. R. Soc. Lond. A 218, 44–59.Google Scholar
Thorpe, S. A. 1978 On the shape and breaking of finite amplitude internal gravity waves in a shear flow. J. Fluid Mech. 85, 7–31.Google Scholar
Wolf, G. H. 1969 The dynamic stabilization of the Rayleigh–Taylor instability and the corresponding dynamic equilibrium. Z. Physik 227, 291–300.Google Scholar
Wunenburger, R., Evesque, P., Chabot, C., Garrabos, Y., Fauve, S. & Beysens, D. 1999 Frozen wave instability by high frequency horizontal vibrations on a CO2 liquid–gas interface near the critical point. Phys. Rev. E 59, 5440–5445.Google Scholar
Yoshikawa, H. N. 2006 Instabilitiés des interfaces, sous oscillations. PhD Thesis, Université Paris 6.Google Scholar