# Stability of a horizontal viscous fluid layer in a vertical time periodic electric field

## TL;DR

This paper analyzes the stability of a viscous fluid interface under a vertical time-periodic electric field, revealing how electric parameters and electrode spacing influence the onset of instabilities and mode behaviors.

## Contribution

It develops a Floquet theory-based method to determine marginal stability curves for arbitrary time-periodic electric signals, including multi-frequency cases, and compares results with Mathieu equation predictions.

## Key findings

- Electric field frequency and amplitude affect stability thresholds.
- Mode coupling leads to harmonic and subharmonic instabilities.
- Electrode spacing significantly influences system stability.

## Abstract

The stability of a horizontal interface between two viscous fluids, one of which is conducting and the other is dielectric, acted upon by a vertical time-periodic electric field is considered. The two fluids are bounded by electrodes separated by a finite distance. By means of Floquet theory, the marginal stability curves are obtained, thereby elucidating the dependency of the critical voltage and wavenumber upon the fluid viscosities. The limit of vanishing viscosities is shown to be in excellent agreement with the marginal stability curves predicted by means of a Mathieu equation. The methodology to obtain the marginal stability curves developed here is applicable to any arbitrary but time periodic-signal, as demonstrated for the case of a signal with two different frequencies. As a special case, the marginal stability curves for an applied ac voltage biased by a dc voltage are depicted. It is shown that the mode coupling caused by the normal stress at the interface due to the electric field leads to appearance of harmonic modes and subharmonic modes. This is in contrast to the application of a voltage with a single frequency which always leads to a harmonic mode. Whether a harmonic or subharmonic mode is the most unstable one depends on details of the excitation signal. It is also shown that the electrode spacing has a distinct effect on the stability bahavior of the system.