The stability of electrohydrodynamic flow between two horizontal plates with a vertical electrical conductivity gradient has been investigated in the presence of an imposed weak shear flow. The weak shear flow is driven by the horizontal pressure gradient, and the electrical conductivity gradient is generated by the concentration variation of the charge-carrying solute. An external electric field is applied across the fluid layer, and then the interaction between the unstable stratification of electrohydrodynamic flow and the shear arising from the plane Poiseuille flow is studied. A linear stability analysis has been implemented by considering both the longitudinal and transverse modes. Unlike the thermally stratified plane Poiseuille flow in which the longitudinal mode always dominates the onset of instability and is virtually unaffected by the superimposed shear flow, the instability of this mixed electrohydrodynamic–Poiseuille flow system is found to depend heavily on the shear flow, and the transverse mode may prevail over the longitudinal mode when the momentum of shear flow is sufficiently small. Particularly, an oscillatory longitudinal mode is found to exist, and it may become the critical mode when the conductivity gradient is small enough. The present results verify that an imposed weak shear flow may enhance the electrohydrodynamic instability in a fluid layer with electrical conductivity gradient.

Thermal convection in a two-layer system comprised by a fluid layer overlying a layer of porous media saturated with the same fluid has been investigated. The system is heated from below and subjected to a horizontally plane Poiseuille flow. The interaction between both instability mechanisms, the unstably stratification and the shear arising from the plane Poiseuille flow, is studied and a completely linear stability analysis has been carried out by considering both longitudinal rolls ͑LRs͒ and transverse rolls ͑TRs͒. It is found that the neutral curves of both modes may be bimodal, which is dependent on the depth ratio, the ratio of the depth of fluid layer to that of the porous layer. The stability characteristics of LRs are found to be invariant with the Reynolds number based on the horizontal Poiseuille velocity and the Prandtl number of the fluid, and the instability is always dominated by this mode if the Reynolds number is within low or moderate strength. The superimposed Poiseuille flow seems to exert a stabilizing effect on the traveling TRs and the critical transverse mode tends to be within the porous layer with low propagating speed. However, if the Reynolds number is sufficiently large, a shear instability mode appears abruptly with higher propagating speed and then dominates the system instability.

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The electrohydrodynamic instability of a horizontal rotating fluid layer with a vertical electrical conductivity gradient is considered. An external electric field is applied across the fluid layer to induce an unstably stratified electrical body force. A linear stability analysis has been performed to study the effect of rotation on the onset of electrohydrodynamic instability in the fluid layer. Results show that the instability behaviors depend heavily on the boundary condition of bottom surface. In the case of stress-free condition, rotation enhances the stability and the onset of instability will be dominated by the oscillatory mode once the speed of rotation ͑or Taylor number͒ exceeds a critical value. In contrast, in the case of rigid bottom surface, rotation also tends to stabilize the fluid layer and the stationary mode will prevail eventually with increasing Taylor number. However rotation becomes destabilizing as the critical mode shifts from oscillatory to stationary. Moreover, under the same electrical conductivity gradient, the case with stress-free bottom surface is always more unstable than that with rigid bottom surface in the small Taylor number domain. However this situation is reversed at high Taylor number region since the stability of the stress-free case will be enhanced more rapidly than the rigid case.

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Thermal convection in a two-layer system with the presence of a plane Couette flow is investigated. The system comprises a fluid layer overlying a porous layer saturated with the same fluid and the lower plate is maintained warmer than the upper. A shear is generated by moving the upper wall horizontally at a constant speed. Both convection modes including longitudinal and transverse rolls are considered. Numerical results show that the neutral curves of both modes may be bimodal which depends on the depth ratio ͑i.e., the ratio of the depth of fluid layer to that of porous layer͒. The longitudinal rolls are always preferred at the onset of convection and the stability of this mode is independent of Reynolds and Prandtl numbers. But the transverse rolls may have the same stability as the longitudinal rolls when Reynolds number or Prandtl number approaches zero. Either increase of Reynolds number or Prandtl number raises its stability, and both produce the same effects on the stability behaviors of transverse rolls and the critical convection mode of transverse rolls tends to be within the porous layer with long wavelength.