EQUILIBRIUM SHAPES AND STABILITY OF NONCONDUCTING PENDANT DROPS SURROUNDED BY A CONDUCTING FLUID IN AN ELECTRIC-FIELD

The shapes and stability of pendant drops in the presence of an electr ic field is a classical problem in capillarity. This problem has been studied in great detail by numerous investigators when the drops are e ither perfect conductors or nonconductors and the surrounding fluid is a nondonductor. In this paper, the axisymmetric equilibrium shapes an d stability of a nonconducting drop hanging from a nonconducting nozzl e that is immersed in a perfectly conducting ambient fluid, a problem that has heretofore not been considered in the literature, are determi ned by solving the free boundary problem comprised of the Young-Laplac e equation for drop shape and an integral equation for the electric fi eld distribution. Here the free boundary problem is discretized by a h ybrid technique in which the Young-Laplace equation is solved by the f inite element method and the electrostatic problem is solved by the bo undary element method. An electrode in the form of a metal rod or an a nnulus is placed inside and coaxial with the tube whose tip is located a distance H-1 above or below the tube outlet. The electric field is generated by connecting the rod or the annulus electrode to a source o f high voltage at potential P and grounding the ambient conducting flu id that surrounds the drop and the tube. When the force due to surface (interfacial) tension is large compared to those due to gravity and e lectric field, equilibrium drop shapes are sections of spheres and can be parametrized by a parameter -1 less than or equal to D less than o r equal to 1: D = 0 corresponds to a hemisphere, as D --> 1 the drop a pproaches a sphere, and as D --> -1 the drop becomes vanishingly small . The results show that equilibrium families of fixed drop volume, or fixed D, lose stability at turning points with respect to the applied potential, where P = P. Detailed computations reveal the importance o f varying the drop size and geometric factors such as the location of the tip of the electrode H-1, electrode thickness W, and, in the case of the annulus electrode, the inner radius of the annulus R(A) on the value of P. Except for very skinny drops, i.e., D --> -1, the compute d profiles of drops at their limits of stability imply that nonconduct ing drops in a conducting ambient fluid should become unstable by pinc hing off near the contact line. This finding agrees with recent experi ments on such drops, but stands in marked contrast to previous studies on conducting and nonconducting drops immersed in a nonconducting amb ient fluid where the drops become unstable by developing conical tips from which a jet issues. Very skinny drops, however, take on a dog-bon ed appearance at their limits of stability regardless of whether the d rop is a conductor or a nonconductor immersed in a nonconducting ambie nt fluid or a nonconductor immersed in a conducting ambient fluid. (C) 1995 Academic Press, Inc.