Standing convective waves in poor conducting liquid

The paper investigates the impact of a steady electric field on convective wave regimes in the poorly conducting viscous incompressible liquid contained inside the infinite horizontal flat capacitor. The liquid is placed in the gravity field and heated from above. A constant electric charge is injected unipolar and autonomously on the top plate of the capacitor. Injection is the main mechanism responsible for the onset of convection instability, and the buoyancy forces directed against the Coulomb forces cause the appearance of non-stationary oscillations. A complete problem formulation is considered. This allows taking into account the electric field redistribution inside the capacitor due to the motion of charges in the electric field and their convective transport. The problem is solved numerically by applying the explicit finite difference schemes. Two sets of vertical boundary conditions are used: completely periodic conditions and limited periodic conditions, inhibiting the liquid motion along the horizontal direction. In both cases, convection occurs as result of forward bifurcation and is of oscillatory nature. In the case of the second set of boundary conditions, we obtain a solution that is unstable under other periodic conditions. When studying the standing wave regimes, the time series and the spatial distributions of the physical parameters of the system are investigated using the Fourier analysis of spatial harmonics. The analysis has revealed standing waves (SW) in which the horizontal motion of convective patterns is absent and modulated standing waves (MSW) where a decrease in wave intensity is accompanied by the appearance of the higher (second) spatial harmonic. The bifurcation diagram of the liquid flow is completed with the unstable standing wave regimes. The same properties are shown for both stable and unstable solutions.