Numerical simulation of electrohydrodynamic convection induced by fast oscillating field emission

Abstract

The paper presents a mathematical model of electrohydrodynamic convection in a mixture of benzene and chlorobenzene subjected to highly inhomogeneous alternating electric field. The field inhomogeneity is determined by the shape of the electrodes, of which one is a flat electrode, and the other is a tip oriented perpendicular to the plane of the first electrode. Recent experiments carried out with a field oscillating at a frequency of 50 Hz have shown that the excitation of an averaged convective flow at a given maximum voltage between the electrodes (8 kV) is observed only at a mole fraction of chlorobenzene of at least 0.4. The flow arises in a threshold manner and is a narrow intense jet directed from the sharp electrode towards the plane closed by an axisymmetric return flow. We propose a model that explains this phenomenon by analyzing the interaction between two fields of charged particles. The sharp electrode injects electrons every half period due to cold field emission. We assume that a chlorobenzene molecule has an uncompensated dipole moment, i.e., in a strong electric field, a chlorobenzene molecule gains an electron and forms a negative ion. However, an electron can exist in a polar dielectric solution for a long time due to the effect of its salvation by the medium. Therefore, the ionization process occurs both at the injector electrode and in the bulk of the medium. The mathematical model includes the Navier–Stokes equation, the transport equations for electrons and ionized chlorobenzene molecules, and the equation of alternating electric field. The sharp electrode boundary conditions are formulated to take into account the cold emission of electrons, which is described by a piecewise linear approximation of the Fowler–Nordheim law. The study of the performance of the proposed model was carried out by the method of direct numerical simulations of non-stationary physical fields. It is shown that if the presence of electrons in the volume of the medium is ignored, the averaged electrohydrodynamic convection does not occur, even if the sharp electrode continues to inject chlorobenzene molecules. The proposed model explains most of the experimental facts. As in the experiment, the excitation of convection critically depends on the mole fraction of chlorobenzene in the binary mixture. The calculated and experimentally observed structures of the convective flow are in good qualitative and quantitative agreement. The results of numerical experiments provide a good basis for the development of an averaged theory of electrohydrodynamic convection.