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 cn . DELTA-zeta = zeta-t (zeta is the free-surface displacement and c is a complex constant) are determined through a boundary-layer approximation for l/a << 1, where a is a characteristic length of the cross-section and l is the capillary length. The primary result is omega-2 = omega-n(2)[1 + (l/a) F (zeta-n;c/omega-nl)], where omega is the frequency of a free oscillation, omega-n is the natural frequency for a particular normal mode, zeta = zeta-n, in the limit l/a --> 0, and F (zeta-n;c/omega-nl) is a corresponding form factor. The imaginary part of F is positive (for the complex time dependence exp (i-omega-t)) 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.