  This work studies the effects of a through-flow on two-dimensional
electrohydrodynamic (EHD) flows of a dielectric liquid confined between two
plane plates, as a model problem to further our understanding of the fluid
mechanics in the presence of an electric field. The liquid is subjected to a
strong unipolar charge injection from the bottom plate and a pressure gradient
along the streamwise direction. Highly-accurate numerical simulations and
weakly nonlinear stability analyses based on multiple-scale expansion and
amplitude expansion methods are used to unravel the nonlinear spatiotemporal
instability mechanisms in this combined flow. We found that the through-flow
makes the hysteresis loop in the EHD flow narrower. In the numerical simulation
of an impulse response, the leading and trailing edges of the wavepacket within
the nonlinear regime are consistent with the linear ones, a result which we
also verified against that in natural convection. In addition, as the
bifurcation in EHD-Poiseuille flows is of a subcritical nature, nonlinear
finite-amplitude solutions exist in the subcritical regime, and our calculation
indicates that they are convectively unstable. The validity of the
Ginzburg-Landau equation (GLE), derived from the weakly nonlinear expansion of
Navier-Stokes equations and the Maxwell's equations in the quasi-electrostatic
limit, serving as a physical reduced-order model for probing the spatiotemporal
dynamics in this flow, has also been investigated. We found that the
coefficients in the GLE calculated using amplitude expansion method can predict
the absolute growth rates even when the parameters are away from the linear
critical conditions, compared favourably with the local dispersion relation,
whereas the validity range of the GLE derived from the multiple-scale expansion
method is confined to the vicinity of the linear critical conditions.
