Lattice Boltzmann modelling of electro-thermo-convection in a planar layer of dielectric liquid subjected to unipolar injection and thermal gradient

In this paper, the electro-thermo-convective phenomena induced by the simultaneous action of a unipolar injection of ions and a thermal gradient in a dielectric liquid lying between two parallel plates are studied. We develop a lattice Boltzmann model (LBM) to solve the whole set of coupled governing equations, including the Navier-Stokes equations, the conservation equation of charge density, the Poisson's equation for electric potential and the energy equation. In this method, four different particle distribution functions are used to calculate the flow field, electric potential, charge density distribution and temperature field, respectively. A multi-scale analysis is also performed to recover the macroscopic equations from the discrete lattice Boltzmann equations (LBEs). The present method is validated with several carefully chosen test cases, and all LBM results are found to be highly consistent with available analytical solutions or other numerical works. Besides, the typical subcritical bifurcations and the hysteresis loops in electro-thermo-convective are clearly presented and analyzed.