EQUILIBRIUM SHAPES AND STABILITY OF CHARGED AND CONDUCTING DROPS

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. 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. © 1990 American Institute of Physics.