Thermal convection in dielectric liquids in a cylindrical annulus*

Thermal convection is investigated in a dielectric liquid of thermal expansion coefficient $\alpha $, kinematic viscosity $\nu $, thermal diffusivity $\kappa $ and electric permittivity $\varepsilon $ in a cylindrical annulus of inner radius $a$ and outer radius $b $with a radial temperature gradient and a high-frequency electric tension. The coupling between the electric field and the gradient of the permittivity yields the dielectrophoretic force. The control parameters are $\eta =$ a/b, Pr $=$ $\nu $/$\kappa $, the classic Rayleigh number Ra $= \quad \alpha \Delta $T gd$^{\mathrm{3}}$/$\nu \kappa $, and the electric Rayleigh number L $=$ $\alpha \Delta $T g$_{\mathrm{e}}$d$^{\mathrm{3}}$/$\nu \kappa $ The electric gravity g$_{\mathrm{e}}$ is the gradient of the electric energy in the condenser. Linear stability analysis shows that for infinite annulus, depending on values of $\eta $, Ra and L, critical modes are either hydrodynamic or thermal modes, helical electric modes or columnar vortices. Experiments in an annulus of aspect ratio $\Gamma =$19.6 during parabolic flight campaigns indicate the existence of columns. Columnar vortices result from the competition between Archimedean buoyancy and dielectrophoretic force. Direct numerical simulations in the annulus of $\Gamma =$20 show that the columnar vortices occupy the central part of the annulus, while near the end-zones the flow is laminar and dominated by an azimuthal vorticity.