The stability margin for stable weightless liquid bridges

Abstract

The stability of weightless axisymmetric liquid bridge equilibrium configurations to “large” disturbances is examined by calculating the stability margin. For bridges held between coaxial equidimensional circular disks (radius R0) separated by a distance H, the stability to infinitesimal perturbations (linear stability) has been thoroughly investigated and the stability region is constructed in the (Λ,V) plane. Here, the slenderness Λ (=H/2R0) and the relative volume V (ratio of the actual liquid volume to that of a cylinder with radius R0 and height H) are the parameters that define the system. To assess stability with respect to finite amplitude disturbances we use a potential energy analysis based on the concepts of a potential energy well and the equilibrium stability margin introduced by Myshkis [USSR Comput. Math. Math. Phys. 5, 193 (1965); Math. Notes Acad. Sci. USSR 33, 131 (1983); Introduction to the Dynamics of a Body Containing a Liquid Under Zero-Gravity Conditions (Vychisl. Tsentr Akad. Nauk SSSR, Moscow, 1968)]. The stability margin represents the height of a local potential energy barrier adjacent to the well of a given stable equilibrium. Wherever a linearly stable equilibrium is nonunique equilibrium, the stability margin corresponds to the smallest among the heights of saddle points on the potential energy surface that are adjacent to the well. The saddle point that determines the stability margin is the point of emergence from the well and leads to the energy wells corresponding to other equilibria or to infinity. Unless the total energy of perturbations exceeds the stability margin for a given stable equilibrium, the liquid bridge will return to that equilibrium state. In this work we determined the stability margin in part of the stability region where axisymmetric bridges that are already unstable to small axisymmetric perturbations coexist with stable ones. The domains of existence of a variety of unstable axisymmetric bridges are constructed using previous results concerning the bifurcation structure. This enabled us to construct contours of the dimensionless stability margin within the linear stability region not only in the vicinity of the stability boundary, but also far from it. The stability margins for bridges with fixed values of the slenderness, as well as for cylindrical and catenoidal bridges, are also calculated.