The electro-hydrodynamic linear stability of a flat interface between two viscous, immiscible and incompressible liquids in plane Poiseuille flow has been shown to be useful in microfluidic devices. In some applications (e.g., material deposition) stability is desired, and in others (e.g., mixing or drop formation) instability needs to be induced. Depending on the direction of the electric field, i.e., parallel or normal to the flat interface, and in the case of fast electric times, it was shown analytically and without solving the complete set of equations that the electric field can either stabilize or destabilize the interface [1]. In this paper, we fully solve the equations and determine the maximum growth rates and the critical wavenumbers in the conductivity versus permittivity ratio space.

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