Inertial effects on the electrohydrodynamic of liquid drops and the urnaround from oblate to prolate phenomenon

The effect of a uniform DC electric field on the deformation of liquid drops and the induced fluid flow circulation in the absence of fluid inertia is well understood. This behavior is most clearly represented by the so-called deformation-circulation map (in the context of leaky-dielectric theory), which illustrates the sense of drop deformation (oblate vs. prolate) and the direction of fluid flow circulation (pole-to-equator or equator-to-pole) as a function of electric permittivity and conductivity ratios. The map is divided into three distinct regions: in Region (1), drops are oblate with pole-to-equator circulation; in Region (2), drops are prolate with equator-to-pole circulation; and in Region (3), drops are prolate with pole-to-equator circulation.

Fluid inertia becomes relevant in many practically important applications, and while a few studies have considered its role, they have been mostly limited to observational findings. The goal of this study is to develop a fundamental understanding of the effect of inertia. This is accomplished through direct numerical simulations in conjunction with analysis of the jump in the normal and tangential electrohydrodynamic stresses at the interface. The study demonstrates how modifications in the hydrodynamic stresses influence the interfacial force balance. Thus, depending on the region within the deformation-circulation map, inertia can either enhance or reduce deformation. Notably, within a small subregion of Region (1), inertia can even reverse the deformation from oblate to prolate, leading to the turnaround phenomenon.