# Unsteady electrorotation of a drop in a constant electric field

## TL;DR

This paper presents a theoretical analysis of the unsteady electrorotation of a viscous drop in a uniform electric field, deriving solutions and stability conditions for the flow and electric field around the drop.

## Contribution

It introduces a series solution approach with asymptotic expansions to analyze unsteady electrorotation, considering large viscosity ratios and stability of the flow.

## Key findings

- Explicit stationary solutions for electric field and flow are obtained.
- Transitions to deformational oscillations are not observed up to second-order terms.
- Stability analysis confirms the robustness of the solutions under specified conditions.

## Abstract

The unsteady electrorotation of a drop of a viscous weakly conducting polarizable liquid suspended in another viscous weakly conducting polarizable liquid immiscible with the former in an applied constant uniform electric field is theoretically investigated. The surface tension of the drop is regarded as sufficiently large for deformation of the drop under the action of the applied electric field and the electrohydrodynamic flow can be considered small and the drop can be considered spherical in the calculation of the electric field and flow. The electric field intensity and the velocity and pressure in the electrohydrodynamic flow are sought for in the form of a series solution with time-dependent scalar, vector, and tensor coefficients for which relations are found allowing one to determine them. The coefficients are sought for in the form of asymptotic expansions over the parameter, the smallness of which corresponds to the sufficiently large viscosity of the drop with respect to that of the surrounding liquid. The differential equations for the terms of the dimensionless expansions up to the first and second orders with respect to this parameter are obtained. Their stationary solutions are found in the explicit form and their stability is investigated. Up to the terms of the second order, transitions to deformational oscillations of the drop are established not to occur.