# Simulations of a weakly conducting droplet under the influence of an alternating electric field

## TL;DR

This paper uses numerical simulations to explore how weakly conducting droplets deform under alternating electric fields, revealing differences from steady fields and effects of electrical properties on shape dynamics.

## Contribution

It provides new insights into droplet deformation behavior under alternating electric fields, especially for weakly conducting fluids, highlighting differences from steady fields and parameter effects.

## Key findings

- Deformation under alternating and direct fields is equivalent for leaky dielectrics.
- Weakly conducting media show different deformation behaviors between alternating and direct fields.
- Electrical conductivity ratio influences droplet deformation, especially when it exceeds permittivity ratio.

## Abstract

We investigate the electrohydrodynamics of an initially spherical droplet under the influence of an external alternating electric field by conducting axisymmetric numerical simulations using a charge-conservative volume-of-fluid based finite volume flow solver. The mean amplitude of shape oscillations of a droplet subjected to an alternating electric field for leaky dielectric fluids is the same as the steady-state deformation under an equivalent root mean squared direct electric field for all possible electrical conductivity ratio $(K_r)$ and permittivity ratio $(S)$ of the droplet to the surrounding fluid. In contrast, our simulations for weakly conducting media show that this equivalence between alternating and direct electric fields does not hold for $K_r \ne S$. Moreover, for a range of parameters, the deformation obtained using the alternating and direct electric fields is qualitatively different, i.e. for low $K_r$ and high $S$, the droplet becomes prolate under alternating electric field but deforms to an oblate shape in the case of the equivalent direct electric field. A parametric study is conducted by varying the time period of the applied alternating electric field, the permittivity and the electrical conductivity ratios. It is observed that while increasing $K_r$ has a negligible effect on the deformation dynamics of the droplet for $K_r<S$, it enhances the deformation of the droplet when $K_r>S$ for both alternating and direct electric fields. We believe that our results may be of immense consequence in explaining the morphological evolution of droplets in a plethora of scenarios ranging from nature to biology.