Linear global stability of liquid metal mixed convection in a horizontal bottom-heating duct under strong transverse magnetic field

Abstract

Two-dimensional steady-state solutions of liquid metal mixed convection in a horizontal bottom-heating duct under a strong magnetic field are first computed numerically by the Newton iteration method along with the spatial discretization of the Taylor–Hood finite element. Two branches of steady solutions with symmetrical rolls and a pair of asymmetrical solutions with a single roll are identified and can be regarded as the base flow for linear global stability analysis. The symmetrical steady solution for the first branch has a nearly uniform distribution for the temperature field in the transverse direction, while the second branch occurs at much larger Grashof numbers and the temperature field becomes nonuniform transversely. The linear stability analysis is performed for a fixed Reynolds number and Prandtl number with Re = 5000 and Pr = 0.0321. For the symmetrical rolls of the first branch, with an increase in the Grashof number, two-dimensional stationary instabilities first occur at small Hartmann numbers, while three-dimensional oscillatory instabilities first appear at moderate or large Hartmann numbers. From the further study of the two-dimensional instabilities, it is revealed that the asymmetrical solution is actually bifurcated supercritically from the symmetrical solution at a two-dimensional critical Grashof number. In addition, the critical curve of the Grashof number with respect to the Hartmann number for the three-dimensional oscillatory mode shows that there exists a minimum critical Grashof number, which occurs at a moderate Hartmann number. The critical curves of the one-roll asymmetrical solution are also exhibited and determined by two three-dimensional oscillatory unstable modes. It is revealed that there exists a minimum Hartmann number below which the asymmetrical steady-state can always remain stable for all Grashof numbers (5 × 105–107). The energy analyses at the oscillatory critical thresholds with different Hartmann numbers are performed to exhibit that buoyancy is the dominant destabilizing term, and the magnetic force is always the main stabilization term for both symmetrical and asymmetrical solutions. In addition, both streamwise and cross-sectional shears of the basic flow are important for the determination of the linear stability boundary of the asymmetrical solution.