ALBERT

Notes: It was shown by Rayleigh [Philos. Mag. 14, 184 (1882)] that a conducting spherical drop becomes unstable when the net dimensionless charge on its surface, Qc, exceeds the value of 4(π)1/2 . More recently, Tsamopoulos et al. [Proc. R. Soc. London Ser. A 401, 67 (1985)] have shown, both analytically and numerically, that at this point a transcritical bifurcation occurs. The finite element methodology that they employed is limited to cases where the drop shapes are not very deformed because of truncation problems with mesh representing the infinitely extending surrounding medium. This situation has now been rectified by employing the integral form of Laplace's equation, which only requires discretization and solution on the surface of the drop. Thus a hybrid method results with the integral equations solved via boundary element techniques, while finite elements are still used for the remaining governing equations. Using this hybrid method, previous results have been reproduced much more accurately and efficiently. In addition, new solution families have been discovered. In particular, several shape families that are not symmetric about the equatorial plane were found to bifurcate from the families of two- and four-lobed shapes. A disjoint family with saddle point shapes was found to extend to small values of charge. It corresponds to the Frankel–Metropolis family that is well known in nuclear physics (Cohen et al. [Ann. Phys. 82, 557 (1974)]). All newly discovered solution families are linearly unstable.

Notes: Small amplitude oscillations of viscous, capillary bridges are studied in the presence of an electric dc field. The electric field is proposed as a means to maintain bridges longer than their perimeter and of uniform cylindrical shape. This is desired in the fabrication of semiconductor crystals. The material of the bridge and the surrounding medium is modeled either as a perfect or as a leaky dielectric. The frequency and the damping rate of the oscillations are calculated numerically by solving a generalized eigenvalue problem. It is shown that they depend on the ratios of the dielectric constants, ε=εin/εout, and conductivities, S=σin/σout, of the two materials, the aspect ratio of the bridge, Λ=πR˜/L˜, the ratio of viscous to the capillary force, Oh=Re−1, which can also be viewed as the inverse Reynolds number of the flow, and, finally, the electrical Bond number, Cel, which is the ratio of the electric stresses to the capillary force. The stability limit of an initially cylindrical bridge is determined with respect to varicose disturbances. In agreement with previous studies it is shown that, if both materials are perfect dielectrics, application of an electric field has a stabilizing effect on the bridge, in the sense that the minimum value, Λmin, of the aspect ratio for the bridge to remain stable drops below 0.5, irrespective of the specific value of the ratio ε. If both materials are leaky dielectrics, bridge stability is determined by the sign of (S−ε) and (S−1)(ε−1), with the positive sign indicating bridge stabilization. The factor (S−ε) arises due to the appearance of a tangential electric stress in the perturbed state for leaky dielectrics. For both cases of leaky and perfect dielectrics, the most unstable mode is the one leading to amphora shaped bridges. It was also found that, when application of an electric field stabilizes the bridge, leaky dielectrics require a lower field than perfect dielectrics and that a large enough field tends to stabilize the bridge for almost the entire range of values of the aspect ratio Λ. These findings concur with earlier analytical results for the stability of jets in longitudinal electric fields and, in conjunction with certain experimental observations, point to the usefulness of the leaky dielectric model pertaining to the stability of bridges. © 2001 American Institute of Physics.