Breakup of a leaky dielectric drop in a uniform electric field

Electrohydrodynamic (EHD) breakup phenomena for a leaky dielectric drop suspended in another immiscible viscous dielectric and subjected to a uniform electric field are examined using the leaky dielectric theory and the explicit forcing lattice Boltzmann method, by taking into account full nonlinear inertia effects. The breakup modes are first computed for varied conductivity of the drop fluid, as the viscosity ratio λ (=μ_{in}/μ_{out}) is momentarily set to unity, that is, for the slightly conducting (R=σ_{in}/σ_{out}<10), moderately conducting (10≤R≤20), and highly conducting (R>20) cases. For slightly conducting drops (R=5) only one breakup mode via two symmetrical necks persists for permittivity ratios 0.05Ca_{E,critical} (ratio of electric and surface tension forces), despite significant length-scale variation of mother and daughter drops. At higher Q (for increased drop permittivity) two necks move closer to the bulbous midpart of the extended droplet, which helps enlarge two daughter drops. However, in the case of moderately conducting drops (10≤R≤20) the number of necks increased to four for increased Ca_{E}. Accordingly two pairs of symmetrical daughter drops are created because of recurrent fluctuations of the electrical shear stress and centerline momentum flux. For highly conducting cases of R>20, depending on Ca_{E}, three distinctly elongated droplet states are formed prior to breakup, which results in the onset of three different breakup modes, namely, via formations of lobed ends (Ca_{E}≤0.264), pointed ends (Ca_{E}≤0.68), and nonpointed ends (Ca_{E}>0.83). While being consistent with past measurements, here we precisely characterize the associated breakup mechanisms and physics in terms of the interactive electric pressure, electric shear stress, and hydrodynamic pressure plus velocity gradients. Since the EHD drop breakup is a dynamic process, on an elongated slender drop the activated locally distinct driving forces, i.e., electric pressure at the end regions and tangential electric stress in the midsection, effectively lead to neck formations by virtue of the created high centerline velocity gradient. Accordingly, resulting variations of local extension rate and net mass flux toward drop ends or into intermediate bulbous regions facilitate the multiple-mode drop breakup via the inertia effect, whereas the developed negative curvature around a neck encourages capillary breakup. We also explicitly reveal the effect of the viscosity contrast λ, which particularly influences the breakup characteristics over a broader range of conductivity ratios.