The influence of thermal convection on the MHD flow in a toroidal duct

The non-steady, three-dimensional thermal convection in a laminar MHD flow that is confined in a toroidal duct with square cross-section, in the presence of a constant axial magnetic field is studied. The temperature difference between the liquid metal flow and the walls of the toroidal duct is generated by a homogeneous internal heat source. The transition from the steady state pure forced convection, at subcritical Rayleigh number, to time dependent mixed convection at supercritical Rayleigh number is investigated. The dimensionless equations governing the flow of the incompressible liquid metal under the Boussinesq and quasi-static (induction-less) approximations are solved by using the spectral-element method. The parameters governing the system are the Hartmann, the Reynolds, the Prandtl and the Rayleigh numbers. The walls of the toroidal duct are no-slip for velocity, while the side walls are electrically perfectly conducting and the top and bottom walls are electrically perfectly insulating. The four walls of the duct have the same constant dimensionless temperature. A three dimensional map where the Hartmann, Reynolds and Rayleigh numbers are involved (considering only one Prandtl number) is generated to identify the critical Rayleigh number at which the onset of thermal convection appears. The effect of the thermal convection on the flow patterns is presented. Additionally, the influence of the governing parameters on the temperature distribution and the heat transfer rate at the walls of the toroidal duct is reported.