M.Sc Thesis
M.Sc Student Rhodes Dov Slender-Body Analysis of Drop Deformation by a Strong Electric Field Department of Applied Mathematics Professor Ehud Yariv

Abstract

A dielectric drop is suspended within a dielectric fluid and exposed to a uniform electric field. There is a jump in the electric field across the interface, proportional to the permittivitiy ratio of the drop and the suspending fluid, creating Maxwell stresses on the surface. The drop, which in the absence of the electric field is held spherical by pressure and surface tension, thus deforms to become prolate in the field direction. As larger electric fields are considered, the drop is found to elongate significantly, becoming long and slender, up to eventual breakup.

The problem of determining the equilibrium shape of the dielectric drop, which depends upon the permittivity ratio and the electric field strength, has been thoroughly analyzed for near-spherical shapes (small deformations), but becomes more challenging when a strong electric field leads to large deformation. The linear near-sphere approximations break down at some point. The key in this case is to make use of the problem geometry by applying slender-body analysis.

The slender limit was originally analyzed by Sherwood in 1991, using a singularity (line of charges) representation of the electric field. This analysis, however, contained errors in the handling of significant logarithmic terms, and did not agree with Sherwood's own previous numerical results from 1988. Here, we revisit the problem using matched asymptotic expansions. The electric field within the drop is continued into a comparable solution in the `inner' region, at the drop cross-sectional scale, which is itself matched with the singularity representation in the `outer' region, at the drop longitudinal scale.

The expansion parameter of the problem is chosen as the elongated drop slenderness. In contrast to familiar slender-body analysis, the value of this parameter is not provided by the problem formulation, and must be found within the course of the solution. The resulting drop aspect-ratio scaling, which is found proportional to the 6/7-power of the electric field magnitude, is identical to that found by Sherwood (1991).

The predicted drop shape is compared with the boundary-integral solutions of Sherwood (1988). While the first-order agreement is better than that found by Sherwood's slender-body analysis, the weak logarithmic decay of the error terms still hinders an accurate calculation. We obtain the leading-order correction to the drop shape, improving the asymptotic approximation.