Transient dynamics of confined liquid drops in a uniform electric field

We analyze the effect of confinement on the transient dynamics of liquid drops, suspended in another immiscible liquid medium, under the influence of an externally applied uniform dc electric field. For our analysis, we adhere to an analytical framework conforming to a Newtonian-leaky-dielectric liquid model in the Stokes flow regime, under the small deformation approximation. We characterize the transient relaxation of the drop shape towards its asymptotic configuration, attributed by the combined confluence of the charge-relaxation time scale and the intrinsic shape-relaxation time scale. While the former appears due to the charge accumulation process on the drop surface over a finite interval of time, the genesis of the latter is found to be intrinsic to the hydrodynamic situation under consideration. In an unbounded condition, the intrinsic shape-relaxation time scale is strongly governed by the viscosity ratio, defined as the ratio of dynamic viscosities of the droplet and the background liquid. However, when the wall effects are brought into consideration, the combined influence of the relative extent of the confinement and the intrinsic viscosity effects, acting in tandem, alter this time scale in a rather complicated and nontrivial manner. We reveal that the presence of confinement may dramatically increase the effective viscosity ratio that could have otherwise been required in an unconfined domain to realize identical time-relaxation characteristics. We also bring out the alterations in the streamline patterns because of the combinations of transient and confinement effects. Thus, our results reveal that the extent of fluidic confinement may provide an elegant alternative towards manipulating the transient dynamics of liquid drops in the presence of an externally applied electric field, bearing far-ranging consequences towards the design and functionalities of several modern-day microfluidic applications.