The stability and small oscillations of an unduloidal fluid bridge between two coaxial discs under conditions of weightlessness

Abstract

A fluid bridge between two identical coaxial discs is considered which, in equilibrium, has the form of a convex unduloid (that is, a wave-like surface). It is shown that the stability of the equilibrium and the existence of small oscillations of the fluid depend on the coercivity of the bilinear form associated with the operator arising in the problem which is determined by the potential of the surface tension forces. The problem reduces to an operator equation in which one of the operators is associated, by virtue of Laplace's law, with the mean curvature of the perturbed free surface. The problem of coercivity reduces to an auxiliary eigenvalue problem. The conditions of stability are found to be satisfied if all of the eigenvalues of the problem are strictly greater than unity. Sufficient conditions for stability are obtained using arguments based on the theory of elliptic functions. The existence of natural frequencies is proved using functional analysis methods.