Electroconvection of a weakly conducting liquid subjected to unipolar injection and heated from above

Abstract

The problem of a weakly conducting viscous incompressible liquid placed in an infinite horizontal capacitor is solved numerically by the two-field method using finite difference schemes. It is assumed that a free charge arises in the liquid (heated from above) due to a homogeneous unipolar autonomous injection from the cathode. In mechanical equilibrium, the Coulomb force acting on the charge injected from the cathode and the buoyancy force are oppositely directed, and the oscillatory instability and wave supercritical modes of convection occur. The problem is investigated in full statement, i.e., the electric field redistribution created by the charge redistribution is taken into account. Periodic boundary conditions are used on the lateral boundaries of the studied area, which makes it possible to detect and analyze not only the mode of stationary electroconvection, but also the solutions in the form of traveling waves. A mixed convection regime that arises as a result of a direct Hopf bifurcation from the state of mechanical equilibrium was found and investigated. In this regime, the phases of standing and traveling waves happen one after another. Traveling waves, modulated traveling waves and stationary convection successively replace each other with an increase in the control parameter (the electric Rayleigh number) proportional to the voltage on the capacitor plates. A bifurcation diagram that characterizes the intensity and phase velocity of supercritical fluid flow modes is plotted. The intensity of the stationary electroconvective flow is an order of magnitude higher than the intensity of the flow in the traveling wave mode. The influence of the grid scale on the observed regimes is studied.