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Experimental investigation of capillarity effects on surface gravity waves: non-wetting boundary conditions

Published online by Cambridge University Press:  26 April 2006

Bruno Cocciaro
Dipartimento di Fisica dell’ Universita’ di Pisa and Gruppo Nazionale di Struttura della Materia del CNR and Istituto Nazionale di Fisica della Materia, Piazza Torricelli 2, 56100 Pisa, Italy
Sandro Faetti
Dipartimento di Fisica dell’ Universita’ di Pisa and Gruppo Nazionale di Struttura della Materia del CNR and Istituto Nazionale di Fisica della Materia, Piazza Torricelli 2, 56100 Pisa, Italy
Crescenzo Festa
Dipartimento di Chimica e Chimica Industriale dell’ Universita’ di Pisa and Gruppo Nazionale di Struttura della Materia del CNR and Istituto Nazionale di Fisica della Materia. Via Risorgimento, 56100 Pisa, Italy


Damping and eigenfrequencies of surface capillary—gravity waves greatly depend on the boundary conditions. To the best of our knowledge, so far no direct measurement has been made of the dynamic behaviour of the contact angle at the three-phase interface (fluid—vapour—solid walls) in the presence of surface oscillation. Therefore, theoretical models of surface gravity–capillary waves involve ad hoc phenomenological assumptions as far as the behavior of the contact angle is concerned. In this paper we report a systematic experimental investigation of the static and dynamic properties of surface waves in a cylindrical container where the free surface makes a static contact angle $\theta_{\rm c} = 62^{\circ}$ with the vertical walls. The actual boundary condition relating the contact angle to the velocity of the contact line is obtained using a new stroboscopic optical method. The experimental results are compared with the theoretical expressions to be found in the literature. Two different regimes are observed: (i) a low-amplitude regime, where the contact line always remains at rest and the contact angle oscillates during the oscillation of the free surface; (ii) a higher-amplitude regime, where the contact line slides on the vertical walls. The profile, the eigenfrequency and the damping rate of the first non-axisymmetric mode of the surface gravity waves are investigated. The eigenfrequency and damping rate in regime (i) are in satisfactory agreement with the predictions of the Graham-Eagle theory (1983) of pinned-end edge conditions. The eigenfrequency and damping rate in regime (ii) show a strongly nonlinear dependence on the oscillation amplitude of the free surface. All the experimental results concerning regime (ii) can be explained in terms of the Hocking (1987 a) and Miles (1967, 1991) models of capillary damping by introducing an ‘effective’ capillary coefficient $\lambda_{\rm eft}$. This coefficient is directly obtained for the first time in our experiment from dynamic measurements on the contact line. A satisfactory agreement is found to exist between theory and experiment.

Research Article
© 1993 Cambridge University Press

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Ablett, R. 1923 An investigation of the angle of contact between paraffin wax and water. Phil. Mag. 46, 244256.Google Scholar
Benjamin, T. B. & Scott, J. C. 1979 Gravity-capillary waves with edge constraints. J. Fluid Mech. 92, 241267.Google Scholar
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, 505515.Google Scholar
Case, K. M. & Parkinson, W. C. 1957 Damping of surface waves in an incompressible liquid. J. Fluid Mech. 2, 172184.Google Scholar
Cocciaro, B., Faetti, S. & Nobili, M. 1991 Experimental investigation of capillarity effects on surface gravity waves in a cylindrical container: wetting boundary conditions. J. Fluid Mech. 231, 325343.Google Scholar
Davies, J. T. & Vose, R. B. 1965 On the damping of capillary waves by surface films. Proc. R. Soc. Lond. A 286, 218234.Google Scholar
Douady, S. 1988 Capillary–gravity surface wave modes in a closed vessel with edge constraint: eigen-frequency and dissipation. Woods Hole Ocean Inst. Tech. Rep. WHOI-88-(Sum. Stu. Prog, in GFD).Google Scholar
Douady, S. 1989 Experimental study of the Faraday instability. J. Fluid Mech. 221, 383409.Google Scholar
Dussan, V., E. B. 1979 On the spreading of liquids on solid surfaces: static and dynamic contact lines. Ann. Rev. Fluid Mech. 11. 371400.Google Scholar
Dussan, V., E. B., Ramé, E. & Garoff, S. 1991 On identifying the appropriate boundary conditions at a moving contact line: an experimental investigation. J. Fluid Mech. 230, 97116.Google Scholar
Graham-Eagle, J. 1983 A new method for calculating eigenvalues with application to gravity–capillary waves with edge constraints. Math. Proc. Camb. Phil. Soc. 94, 553564.Google Scholar
Graham-Eagle, J. 1984 Gravity-capillary waves with edge constraints. D. Phil thesis. University of Oxford.
Hocking, L. M. 1987 a The damping of capillary–gravity waves at a rigid boundary. J. Fluid Mech. 179, 253266.Google Scholar
Hocking, L. M. 1987b Waves produced by a vertically oscillating plate. J. Fluid Mech. 179. 267281.Google Scholar
Keulegan, G. H. 1959 Energy dissipation in standing waves in rectangular basins. J. Fluid Mech. 6, 33.Google Scholar
Kitchener, J. A. & Cooper, C. F. 1959 Current concepts in the theory of foaming Q. Rev. Chem. Soc. 13, 7197.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.
Mei, C. C. & Liu, L. F. 1973 The damping of surface gravity waves in a bounded liquid. J. Fluid Mech. 59, 239256.Google Scholar
Miles, J. W. 1967 Surface-wave damping in closed basins. Proc. R. Soc. Lond. A 297, 459475.Google Scholar
Miles, J. W. 1984 Resonantly forced gravity waves in a circular cylinder. J. Fluid Mech. 149, 1545.Google Scholar
Miles, J. W. 1990 Capillary–viscous forcing of surface waves. J. Fluid Mech. 219, 635646.Google Scholar
Miles, J. W. 1991 The capillary boundary layer for standing waves. J. Fluid Mech. 222, 197205.Google Scholar
Miles, J. W. & Henderson, D. 1990 Parametrically forced surface waves. Ann. Rev. Fluid Mech. 22, 143165.Google Scholar
Nobili, M., Ciliberto, S., Cocciaro, B., Faetti, S. & Fronzonl, L. 1988 Time-dependent surface waves in a horizontally oscillating container. Europhys. Lett. 7, 587592.Google Scholar
Scott, J. C. 1981 The propagation of capillary–gravity waves on a clean water surface. J. Fluid Mech. 108, 127131.Google Scholar
Ursell, F. 1952 Edge waves on a sloping beach. Proc. R. Soc. bond. A 214, 7997.Google Scholar
Van Dorn, W. G. 1966 Boundary dissipation of oscillatory waves. J. Fluid Mech. 24, 769779.Google Scholar