Modal and nonmodal stability analysis of electrohydrodynamic flow with and without cross-flow

TL;DR

This study conducts a comprehensive linear stability analysis of electrohydrodynamic flow with and without cross-flow, incorporating charge diffusion and transient growth, revealing how electric fields influence flow stability and structure.

Contribution

It provides the first combined modal and nonmodal stability analysis of EHD flow including charge diffusion and cross-flow effects, highlighting electric field impacts on flow stability and transient energy growth.

Findings

Charge diffusion lowers the linear stability threshold.

Electric fields enhance transient energy growth and lift-up mechanisms.

Cross-flow and electric fields shift flow structures and increase optimal wavenumber.

Abstract

We report the results of a complete modal and nonmodal linear stability analysis of the electrohydrodynamic flow (EHD) for the problem of electroconvection in the strong injection region. Convective cells are formed by Coulomb force in an insulating liquid residing between two plane electrodes subject to unipolar injection. Besides pure electroconvection, we also consider the case where a cross-flow is present, generated by a streamwise pressure gradient, in the form of a laminar Poiseuille flow. The effect of charge diffusion, often neglected in previous linear stability analyses, is included in the present study and a transient growth analysis, rarely considered in EHD, is carried out. In the case without cross-flow, a non-zero charge diffusion leads to a lower linear stability threshold and thus to a more unstable low. The transient growth, though enhanced by increasing charge diffusion, remains small and hence cannot fully account for the discrepancy of the linear stability threshold between theoretical and experimental results. When a cross-flow is present, increasing the strength of the electric field in the high-$Re$ Poiseuille flow yields a more unstable flow in both modal and nonmodal stability analyses. Even though the energy analysis and the input-output analysis both indicate that the energy growth directly related to the electric field is small, the electric effect enhances the lift-up mechanism. The symmetry of channel flow with respect to the centerline is broken due to the additional electric field acting in the wall-normal direction. As a result, the centers of the streamwise rolls are shifted towards the injector electrode, and the optimal spanwise wavenumber achieving maximum transient energy growth increases with the strength of the electric field.