Linear oscillations of constrained drops, bubbles, and plane liquid surfaces

The small-amplitude oscillations of constrained drops, bubbles, and plane liquid surfaces are studied theoretically. The constraints have the form of closed lines of zero thickness which prevent the motion of the liquid in the direction normal to the undisturbed free surface. It is shown that, by accounting explicitly for the singular nature of the curvature of the interface and the force exerted by the constraint, methods of analysis very close to the standard ones applicable to the unconstrained case can be followed. Weak viscous effects are accounted for by means of the dissipation function. Graphical and numerical results for the oscillations of constrained drops and bubbles are presented. Examples of two- and three-dimensional gravity-capillary waves are treated by the same method. A brief consideration of the Rayleigh-Taylor unstable configuration shows that the nature of the instability is not affected, although its growth rate is decreased