Modulating droplet electrohydrodynamics via the interplay of extensional flow and an alternating current electric field

Electric fields can be used to exert controlled time-varying forces on a droplet under progressive stretching in an extensional flow, allowing for its precise manipulation in various industrial and scientific applications, including microfluidics, materials science, and biological studies. However, the interaction between the combined extensional flow field and electric field may trigger a complex electrohydrodynamic response, as determined primarily by the capillary and viscous forces and the convection of surface charge. Here, we theoretically and computationally analyze the deformation and breakup of a droplet subjected to an alternating current (AC) electric field and uniaxial extensional flow. Our asymptotic theory, applicable in the small-deformation limit, quantifies the contributions of each applied field to the shape oscillations. Numerical simulations are employed to explore the dynamical evolution of the droplet in the nonlinear regime of variation in the capillary number. Our theoretical and numerical results are in excellent agreement, demonstrating that an AC electric field can significantly alter transient deformation depending on its magnitude and frequency. We identify the threshold frequency, dependent on the ratios of electrical properties, which can induce periodic oblate-prolate shape transitions. The interaction between viscous and electric stresses driving these transients is discussed. Contrary to intuition, strong electric fields greatly suppress shape oscillations, leading instead to large continuous elongations that eventually result in an end-pinching breakup mode, forming elongated bulbous-ended droplets. The breakup state, characterized by droplet length and shape at the onset of breakup, is determined by the field parameters and the physical properties of the fluids. Notably, the breakup state length and total breakup time have a non-monotonic relationship with the applied electric field frequency. The insights gained into the physics of oscillatory stable deformation and non-oscillatory unstable deformation offer new means of droplet manipulation in droplet-based micro-mechano-electrical systems that remained unexplored thus far.