Capillary Surfaces: Stability from Families of Equilibria with Application to the Liquid Bridge

A method for determining the stability of general static capillary surfaces is illustrated by application to the liquid bridge. Axisymmetric bridges with fixed contact lines under gravity are parametrized by three quantities: bridge length L, bridge volume V, and Bond number B. The method delivers: (i) stability envelopes in the {L, V, B} parameter space for constant-pressure and constant-volume disturbances (generating new and recovering classical results), (ii) the number of unstable modes for any equilibrium (state of instability) once the stability of one equilibrium state is known (e.g. that of the sphere) based on (iii) a demonstration that all known families of equilibria are connected. The method uses `preferred' bifurcation diagrams, a plot of volume V against pressure p. The state of instability of an equilibrium shape relative to its neighbours is immediate from this plot. In addition, an invariant wavenumber classification is introduced and used to label the numerous families of liquid bridge equilibria. The preferred diagram method, which is based on properties of the Jacobi equation, gives stronger results than classical bifurcation theory. Application to other capillary surfaces, including drops and non-axisymmetric shapes, is discussed.