Hostname: page-component-848d4c4894-r5zm4 Total loading time: 0 Render date: 2024-06-23T07:06:17.688Z Has data issue: false hasContentIssue false
  • English
  • Français

The capillary boundary layer for standing waves

Published online by Cambridge University Press:  26 April 2006

John Miles
John Miles
Affiliation:
Institute of Geophysics and Planetary Physics, University of California, San Diego, La Jolla, CA 92093, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The linear, free-surface oscillations of an inviscid fluid in a cylindrical basin subject to the contact-line condition en∇ζ = ζt (ζ is the free-surface displacement and c is a complex constant) are determined through a boundary-layer approximation for l/a [Lt ] 1, where a is a characteristic length of the cross-section and l is the capillary length. The primary result is ω2 = ω2n [1 + (l/a) [Fscr ] (ζn; c/ωnl)] where ω is the frequency of a free oscillation, ωn is the natural frequency for a particular normal mode, ζ = ζn in the limit l/a → 0, and l/a→0, and [Fscr ](ζn;c/ωnl)] is a corresponding form factor. The imaginary part of [Fscr ] is positive (for the complex time dependence exp (iωt) if Re (c) > 0, which implies positive dissipation through contact-line motion. Explicit results are derived for circular and rectangular cylinders and compared with Graham-Eagle's (1983) results for the circular cylinder for c = 0 and Hocking's (1987) results for the two-dimensional problem. The exact eigenvalue equation for the circular cylinder and a variational approximation for an arbitrary cross-section are derived on the assumption that the static meniscus is negligible.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 222 , January 1991 , pp. 197 - 205
Copyright
© 1991 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramowitz, M. & Stegun, I. A., 1964 Handbook of Mathematical Functions. National Bureau of Standards, Washington.
Benjamin, T. B. & Scott, J. C., 1979 Gravity-capillary waves with edge constraints. J. Fluid Mech. 92, 241–267.Google Scholar
Graham-Eagle, J.: 1983 A new method for calculating eigenvalues with applications to gravity-capillary waves with edge constraints. Math. Proc. Camb. Phil. Soc. 94, 553–564.Google Scholar
Hocking, L. M.: 1987 The damping of capillary-gravity waves at a rigid boundary. J. Fluid Mech. 179, 253–266.Google Scholar
Lamb, H.: 1932 Hydrodynamics. Cambridge University Press.
Miles, J.: 1967 Surface-wave damping in closed basins. Proc. R. Soc. Lond. A 297, 459–475.Google Scholar
Miles, J.: 1990 Capillary—viscous forcing of surface waves. J. Fluid Mech. 219, 635–646.Google Scholar