J. zyxwvutsrqpon Fluid Mech. zyxwvutsrqp (1993). zyxwvutsrqp vol. 249, pp. 207-225 zyxwvut Copyright zyxwvutsrq 0 1993 Cambridge University Press 207 Bifurcation diagrams of axisymmetric liquid bridges of arbitrary volume in electric and gravitational axial fields By ANTONIO RAMOS AND ANTONIO CASTELLANOS Departamento de Electrhica y Electromagnetismo, Universidad de Sevilla, 41012 Seville, Spain Finite-amplitude bifurcation diagrams of axisymmetric liquid bridges anchored between two plane parallel electrodes subjected to a potential difference and in the presence of an axial gravity field are found by solving simultaneously the Laplace equation for the electric potential and the Young-Laplace equation for the interface by means of the Galerkinlfinite element method. Results show the strong stabilizing effect of the electric field, which plays a role somewhat similar to the inverse of the slenderness. It is also shown that the electric field may determine whether the breaking of the liquid bridge leads to two equal or unequal drops. Finally, the sensitivity of liquid bridges to an axial gravity in the presence of the electric field is studied. 1. Introduction The statics of liquid bridges as a function of the slenderness A zyx = L/2R (where L is the height and R the radius) and the non-dimensional volume r = V/nR2L is well established (Martinez 1983; Sanz & Martinez 1983; Martinez 1986). The effect of gravity upon its stability is also well known (Vega & Perales 1983; Meseguer, Sanz & Perales 1990). Recently, the application of an electric field has been considered with the aim of forming longer liquid bridges. Linear bifurcation studies have shown that the electric field always increases the value of the critical liquid bridge height thus increasing the stability region (Gonzalez et al. 1989; Ramos & Castellanos 1991). It turns out that for moderate values of electrical stresses this augmentation is approximately a linear function of the square of the applied electric field. The principal reason for the increase in liquid bridge stability is that the electric field always tends to suppress perturbations of the interface shape perpendicular to itself. This may be understood since there is a decrease in electrostatic energy stored in the system caused by these perturbations. In particular, a sinusoidal deformation of a plane interface subjected to a tangential electric field induces polarization charges that perturb the field in such a way that the total electrostatic energy decreases. An elementary estimate of this variation per unit of volume leads to &WE (e1-e2)ESE < 0 (see figure 1). Given that at equilibrium with fixed potentials the electrostatic energy has to be a maximum (Jackson 1975), the latter effect implies that dielectric forces favour an interface parallel to the electric field. In a companion paper to this one (Gonzalez & Castellanos 1993) a nonlinear bifurcation analysis is made in order to determine the way in which a cylindrical liquid bridge loses its stability in the presence of residual axial gravity. The method, based on the Lyapunov-Schmidt projection technique, is quite powerful but it is