The influence of thermal convection on the liquid metal flow in a rectangular duct in the presence of a magnetic obstacle

The nonsteady, three dimensional thermal convection in a  pressure driven liquid metal flow, that is confined  in a rectangular duct heated from below, in the presence of heterogenous vertical magnetic field  is studied. The transition from steady state pure forced convection to time dependent  mixed convection is investigated. The four  walls of the duct confining the flow  are no slip and electrically insulating. The dimensionless equations  governing the flow of the incompressible fluid, under the Boussinesq and  quasi-static (induction-less) approximations, are solved by using the  spectral  element method. The dimensionless  parameters of the system are the Hartmann, Reynolds, Prandtl and Rayleigh numbers, and the constrainment factor (defining the spanwise distribution of the magnetic field). Three dimensional maps, where the Hartmann, Reynolds and Rayleigh numbers are involved, are generated. The results show the critical  Rayleigh number at which the thermal convection is dominant. The influence of the thermal convection on the  flow patterns generated in the vicinity the  magnetic obstacle is presented. The influence of the Hartmann and Reynolds numbers on the temperature distribution and heat transfer rate in the region close to the magnetic obstacle is reported.