Stability and bifurcation of annular electro-thermo-convection

We numerically investigated the global linear instability and bifurcations in electro-thermo-convection (ETC) of a dielectric liquid confined in a two-dimensional (2-D) concentric annulus subjected to a strong unipolar injection. Seven kinds of solutions exist in this ETC system due to the complex bifurcations, i.e. saddle-node, subcritical and supercritical Hopf bifurcations. These bifurcation routes constitute at most four solution branches. Global linear instability analysis and energy analysis were conducted to explain the instability mechanism and transition of different solutions and to predict the local instability regions. The linearized lattice Boltzmann method (LLBM) for global linear instability analysis, first proposed by Pérez et al. (Theor. Comput. Fluid Dyn., vol. 31, 2017, pp. 643-664) to analyse incompressible flows, was extended here to solve the whole set of coupled linear equations, including the linear Navier-Stokes equations, the linear energy equation, Poisson's equation and the linear charge conservation equation. A multiscale analysis was also performed to recover the macroscopic linearized Navier-Stokes equations from the four different discrete lattice Boltzmann equations (LBEs). The LLBM was validated by calculating the linear critical value of 2-D natural convection; it has an error of 1.39% compared with the spectral method. Instability with global travelling wave behaviour is a unique behaviour in the annulus configuration electrothermohydrodynamic system, which may be caused by the baroclinity. Finally, the chaotic behaviour was quantitatively analysed through calculation of the fractal dimension and Lyapunov exponent.