On the flow instability under thermal and electric fields: A linear analysis

With increasing application of technologies related to electric field being used to enhance heat transfer, it is important to understand better the effects of electric field and lateral advection on the thermal field. In this work, we present a detailed study of the linear dynamics of the flows subjected to simultaneously a thermal field and an electric field (which we call electro–thermo–hydrodynamic flow, ETHD) with a pressure-driven cross-flow from the perspective of modal and non-modal linear stability analyses. The flow takes place in a planar layer of a dielectric fluid heated from below and is subjected to unipolar space-charge-limited injection from above. The effects of various parameters are studied and energy analyses are performed to gain more physical insights into these effects. Our results of the linear modal stability analysis indicate that at a small Reynolds number (Re, quantifying inertia to viscosity), the critical Rayleigh number (Ra, measuring the strength of thermal gradient) increases monotonically with increasing Re when the Coulomb force is weak, while the trend becomes non-monotonic in a stronger electric field. The critical electric Rayleigh number (T, quantifying the strength of electric field) first decreases and then increases with increasing Re. Besides, the linear system becomes more stable with increasing Prandtl number (Pr, a ratio of kinematic viscosity to thermal diffusivity) and mobility ratio (M, a ratio of hydrodynamic mobility to ion mobility) at the Re investigated. In the non-modal results, we demonstrate that the large-Re scaling law holds in the ETHD–Poiseuille flow when the flow is inertia-dominant. The results at large Re indicate that at sufficiently large Pr, Ra exerts a negligible influence on the transient growth. Furthermore, an input–output analysis is adopted to demonstrate that electric field perturbations can boost the transient growth of total perturbation energy compared to the case without the electric field. This result may be helpful for subsequent investigations of heat transfer enhancement using an electric field.