The influence of inertia and charge relaxation on electrohydrodynamic drop deformation

Abstract

We quantify the transient deformation of a droplet immersed in a weakly conductive (leaky dielectric) fluid upon exposure to a uniform DC electric field. Capillary forces are assumed to be sufficiently large that the drop only slightly deviates from its equilibrium spherical shape. In particular, we account for transient (or linear) fluid inertia via the unsteady Stokes equations, and also account for a finite electrical relaxation time over which the drop interface charges. The temporal droplet deformation is governed by two dimensionless groups: (i) the ratio of capillary to momentum diffusion time scales: an Ohnesorge number Oh and (ii) the ratio of charge relaxation to momentum diffusion time scales, which we denote by Sa. If charge and momentum relaxation occur quickly compared to interface deformation, Sa ≪ 1 and Oh ≫ 1 for the droplet and medium, a monotonic deformation is acquired. In contrast, Sa > 1 and Oh < 1 for either phase can lead to a non-monotonic development in the deformation. Numerical values for the deformation are calculated by inverting an analytical expression obtained in the Laplace domain, and are corroborated by asymptotic expansions at early and late times. The droplet and medium behave as perfect dielectrics at early times, which always favors an initial prolate (parallel to the applied field) deformation. As a consequence, for a final oblate (normal to the applied field) deformation, there is a shape transition from prolate to oblate at intermediate times. This transition is caused by the accumulation of sufficient charge at the interface to generate electrical and viscous shear stresses. Notably, after the transition, there may be an “overshoot” in the deformation, i.e., the magnitude exceeds its steady-state value, which is proceeded by an algebraic tail describing the arrival towards the final, steady deformation. Our work demonstrates that transient inertia or a non-zero electrical relaxation time can yield non-monotonic electrohydrodynamic drop deformation.

A nonlinear small-deformation theory for transient droplet electrohydrodynamics

“…We summarize here the results of the theory and outline the solution procedure at first and second order. We also compare and contrast our predictions with the existing theories of Taylor (1966), Ajayi (1978), Esmaeeli & Sharifi (2011) and Lanauze et al (2013).…”

“…Accordingly, it satisfies the conservation equation oblate shapes, suggesting an inconsistency in the model. This discrepancy was recently resolved by Lanauze et al (2013), who showed using a small-deformation theory that either transient charge relaxation or fluid acceleration, combined with transient shape deformations, needs to be included in the model to capture the correct behavior.…”

“…We first summarize the first-order theory, which allows us to compare our results with those of Taylor (1966), Esmaeeli & Sharifi (2011) and Lanauze et al (2013). The zerothorder stress balance equations (3.50b) and (3.51a), together with the dipole relaxation equation (3.25), provide three coupled equations for the three unknowns B 03 , f 12 and P 01 .…”

Nonlinear electrohydrodynamics of slightly deformed oblate drops

“…The transient deformation of a leaky dielectric drop has also received theoretical (Esmaeeli & Sharifi 2011;Lanauze, Walker & Khair 2013;Zhang, Zahn & Lin 2013) and computational (Supeene et al 2008;Halim & Esmaeeli 2013) treatment. For such low-conductivity systems, the finite amount of time taken for a quantity of charge q * to arrive at the interface, characterized by the charge relaxation time scales τ e(i,o) , implies that another form of charge transport must be accounted for: the interfacial accumulation of charge.…”

“…Supeene et al (2008) implemented the finite-element method to predict the transient deformation of a leaky dielectric drop via a general electrokinetic model and a surface charge conservation equation. Lanauze et al (2013) developed an O(Ca) perturbation theory that accounts for charge relaxation and linear inertia. These computational and theoretical efforts have predicted prolate-oblate shape transitions at intermediate times for ultimately steady oblate conformations due to the dominance of the charge relaxation time scales τ e(i,o) (i.e.…”