(499e) Linear Stability of the Belt-Constrained Rayleigh Drop

The eigenfrequencies and eigenmodes of inviscid vibrations of a drop held by surface tension and constrained by a spherical belt are solved for. The belt support extends between two latitudes. The full problem is solved with focus on eigenvalue multiplicities over the 2-parameter plane of geometrical constraints. Rayleigh (1879) and Strani and Sabetta (1984) results are recovered. An approximation as two coupled harmonic oscillators is derived and shown to be effective.  Constrained oscillations are of importance in a number of emerging applications.

Surface tension resists deformation of a liquid mass from rest configurations in much the same way as the linear-elastic spring resists excursions of a mass from a rest position.  Indeed, under broad conditions, the stability of a liquid mass held by surface tension is determined by an operator equation on small disturbances to the interface of the same form as that of the damped harmonic oscillator. Motions are assumed inviscid and irrotational.  The governing equations for the velocity potential within the drop are solved for general boundary conditions, following the boundary-integral approach. The resulting operator equation maps one function space to another, where the geometrical constraints of the belt are incorporated into the function space.  The linear operator equation is an integro-differential boundary-value problem and has the form of an eigenvalue problem.  The eigenvalue problem is reduced to a set of algebraic equations using a Rayleigh-Ritz procedure, in the standard way. Solutions are obtained by using basis functions expressed as series of Legendre polynomials, suitably orthonormalized.

The two spherical-cap interfaces are coupled through the underlying fluid and act as oscillators. Depending on the position and extent of the constraint, qualitatively different couplings can occur. In particular, there are regions of multiplicity and near-multiplicity where different mode shapes have the same or nearly the same frequencies. In a neighborhood of these geometries, the spectrum is particularly sensitive to geometry.  The regions of sensitivity may be of importance to application since there any coupling through nonlinearity (not considered in this paper) will be most dramatic.  Finally, for much of the plane of geometrical constraint, the spectral dependence can be thought of as arising from a superposition of single-interface constrained drops.