Deformation and stability of a viscous electrolyte drop in a uniform electric field

We study the deformation and breakup of an axisymmetric electrolyte drop which is freely suspended in an infinite dielectric medium and subjected to an imposed electric field. The electric potential in the drop phase is assumed small, so that its governing equation is approximated by a linearized Poisson-Boltzmann or modified Helmholtz equation (the Debye-H{u}ckel regime). An accurate and efficient boundary integral method is developed to solve the low-Reynolds-number flow problem for the time-dependent drop deformation, in the case of arbitrary Debye layer thickness. Extensive numerical results are presented for the case when the viscosity of the drop and surrounding medium are comparable. Qualitative similarities are found between the evolution of a drop with a thick Debye layer (characterized by the parameter $chill 1$, which is an inverse dimensionless Debye layer thickness) and a perfect dielectric drop in an insulating medium. In this limit, a highly elongated steady state is obtained for sufficiently large imposed electric field, and the field inside the drop is found to be well approximated using slender body theory. In the opposite limit $chigg 1$, when the Debye layer is thin, the drop behaves as a highly conducting drop, even for moderate permittivity ratio $Q=epsilon_1/epsilon_2$, where $epsilon_1, epsilon_2$ is the dielectric permittivity of drop interior and exterior, respectively. For parameter values at which steady solutions no longer exist, we find three distinct types of unsteady solution or breakup modes. These are termed conical end formation, end splashing, and open end stretching. The second breakup mode, end splashing, resembles the breakup solution presented in a recent paper [R. B. Karyappa et al., J. Fluid Mech. 754, 550-589 (2014)]. We compute a phase diagram which illustrates the regions in parameter space in which the different breakup modes occur.