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The capillary boundary layer for standing waves

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/ωn l)] 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.