  The study focuses on the 3D electro-hydrodynamic (EHD) instability for flow
between to parallel electrodes with unipolar charge injection with and without
cross-flow. Lattice Boltzmann Method (LBM) with two-relaxation time (TRT) model
is used to study flow pattern. In the absence of cross-flow, the base-state
solution is hydrostatic, and the electric field is one-dimensional. With strong
charge injection and high electrical Rayleigh number, the system exhibits
electro-convective vortices. Disturbed by different perturbation patterns, such
as rolling pattern, square pattern, and hexagon pattern, the flow patterns
develop according to the most unstable modes. The growth rate and the unstable
modes are examined using dynamic mode decomposition (DMD) of the transient
numerical solutions. The interactions between the applied Couette and
Poiseuille cross-flows and electroconvective vortices lead to the flow patterns
change. When the cross-flow velocity is greater than a threshold value, the
spanwise structures are suppressed; however, the cross-flow does not affect the
streamwise patterns. The dynamics of the transition is analyzed by DMD.
Hysteresis in the 3D to 2D transition is characterized by the non-dimensional
parameter Y, a ratio of the coulombic force to viscous term in the momentum
equation. The change from 3D to 2D structures enhances the convection marked by
a significant increase in the electric Nusselt number.
