Deformation and breakup of a leaky dielectric drop in a quadrupole electric field

The deformation and breakup of a leaky dielectric drop suspended in a leaky dielectric medium subjected to a quadrupole electric field are studied. Analytical (linear and nonlinear asymptotic expansions in the electric capillary number, Ca-Q, a ratio of electric to capillary stress) and numerical (boundary element) methods are used. A complete phase diagram for the drop deformation in the R-Q plane is presented, where R and Q are the non-dimensional ratios of the resistivities and dielectric constants, respectively, of the drop and the medium phase. The prolate and oblate deformations are mapped in the phase diagram, and the flow contours are also shown. The large deformation and breakup of a drop at higher Ca-Q are analysed using the boundary element method. Several non-trivial shapes are observed at the onset of breakup of a drop. A prolate drop always breaks above a certain critical value of Ca-Q. In the oblate deformation cases, breakup as well as steady shapes are observed at a higher value of Ca-Q. A detailed study of prolate and oblate deformation tendencies due to the normal and tangential electric stresses and the countervailing role of viscous stresses is presented. The circulation inside a drop is found to be more intense for a quadrupole field as compared with a uniform electric field. More intense internal circulations can lead to enhanced mixing characteristics and will have implications in microfluidic devices.