Absolute and convective instabilities in electrohydrodynamic flow subjected to a Poiseuille flow: a linear analysis

We present a study of absolute and convective instabilities in electrohydrodynamic flow subjected to a Poiseuille flow (EHD-Poiseuille). The electric field is imposed on two infinite flat plates filled with a non-conducting dielectric fluid with unipolar ion injection. Mathematically, the dispersion relation of the linearised problem is studied based on the asymptotic response of an impulse disturbance imposed on the base EHD-Poiseuille flow. Transverse, longitudinal and oblique rolls are investigated to identify the saddle point satisfying the pinching condition in the corresponding complex wavenumber space. It is found that when the ratio of Coulomb force to viscous force increases, the transverse rolls can transit from convective instability to absolute instability. The ratio of hydrodynamic mobility to electric mobility, which exerts negligible effect on the linear stability criterion when the cross-flow is small, has significant influence on the convective-absolute instability transition, especially when the ratio is small. As we change the value of the mobility ratio, a saddle point shift phenomenon occurs in the case of transverse rolls. The unstable longitudinal rolls are convectively unstable as long as there is a cross-flow, a result which is deduced from a one-mode Galerkin approximation. Longitudinal rolls have a larger growth rate than transverse rolls except for a small cross-flow. Finally, regarding the oblique rolls, a numerical search for the saddle point simultaneously in the complex streamwise and transverse wavenumber spaces always yields an absolute transverse wavenumber of zero, implying that oblique rolls give way to transverse rolls when the flow is unstable.