Linear stability of quasi-two-dimensional liquid metal duct flow subjected to a transverse magnetic field and vertical temperature stratification

The linear stability of a magnetohydrodynamic duct flow with heating from below is investigated and serves as an extension to the class of Poiseuille–Rayleigh–Bénard flows. Such flows can be found in the blankets of nuclear fusion reactors and have been of great interest in recent times due to the endeavour of demonstrating the viability of nuclear fusion as a future energy source. The flow is described by the quasi-two-dimensional model proposed by Sommeria and Moreau [12] coupled with the energy transport equation through the Boussinesq approximation. The onset of several instability modes in this system is studied as functions of the non-dimensional governing parameters: Reynolds number Re, Rayleigh number Ra, and modified Hartmann number H. As H → 0, the classical plane plane Poiseuille and Rayleigh–Bénard flows are recovered with critical values that are in agreement with previous literature. As H → ∞, the relationships Re_c ∞ H^1/2 and Ra_c ∞ H are obtained. Neutral stability curves are obtained for fixed values of low Re and low Ra conditions and are mapped onto Ra_c–H and Re_c–H regimes, respectively. The critical eigenmodes feature characteristics that are consistent with Tollmien–Schlichting waves and Rayleigh–Bénard convection cells.