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Modal and non-modal stability analysis of electrohydrodynamic flow with and without cross-flow

Published online by Cambridge University Press:  01 April 2015

Mengqi Zhang ,
Fulvio Martinelli ,
Jian Wu ,
Peter J. Schmid  and
Maurizio Quadrio
Mengqi Zhang*
Affiliation:
Département Fluides, Thermique, Combustion, Institut PPrime, CNRS-Université de Poitiers-ENSMA, UPR 3346, 43 Route de l’Aérodrome, Poitiers CEDEX F86036, France
Fulvio Martinelli
Affiliation:
Dipartimento di Scienze e Tecnologie Aerospaziali, Politecnico di Milano, via La Masa 34, 20156 Milano, Italy
Jian Wu
Affiliation:
Département Fluides, Thermique, Combustion, Institut PPrime, CNRS-Université de Poitiers-ENSMA, UPR 3346, 43 Route de l’Aérodrome, Poitiers CEDEX F86036, France
Peter J. Schmid
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
Maurizio Quadrio
Affiliation:
Dipartimento di Scienze e Tecnologie Aerospaziali, Politecnico di Milano, via La Masa 34, 20156 Milano, Italy
*
†Email address for correspondence: mengqi.zhang@univ-poitiers.fr
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Abstract

We report the results of a complete modal and non-modal linear stability analysis of the electrohydrodynamic flow for the problem of electroconvection in the strong-injection region. Convective cells are formed by the 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 electrohydrodynamics, 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 flow. 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- $\mathit{Re}$ Poiseuille flow yields a more unstable flow in both modal and non-modal 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 centreline is broken due to the additional electric field acting in the wall-normal direction. As a result, the centres 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.

JFM classification

Instability: Instability MHD and Electrohydrodynamics: MHD and Electrohydrodynamics

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Papers
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Journal of Fluid Mechanics , Volume 770 , 10 May 2015 , pp. 319 - 349
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© 2015 Cambridge University Press 

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