Linear stability of horizontal, laminar fully developed, quasi-two-dimensional liquid metal duct flow under a transverse magnetic field and heated from below

摘要

This study considers the linear stability of Poiseuille-Rayleigh-B\'enardflows, subjected to a transverse magnetic field to understand the instabilitiesthat arise from the complex interaction between the effects of shear, thermalstratification and magnetic damping. This fundamental study is motivated inpart by the desire to enhance heat transfer in the blanket ducts of nuclearfusion reactors. In pure MHD flows, the imposed transverse magnetic fieldcauses the flow to become quasi-2D and exhibit disturbances that are localisedto the horizontal walls. However, the vertical temperature stratification inRayleigh-B\'enard flows feature convection cells that occupy the interiorregion and therefore the addition of this aspect provides an interesting pointfor investigation. The linearised governing equations are described by the \qtwod\ modelproposed by Sommeria and Moreau (1982) which incorporates a Hartmann frictionterm, and the base flows are considered fully developed and 1D. The neutralstability curves for critical Reynolds and Rayleigh numbers, $Re_c$ and $Ra_c$,respectively, as functions of Hartmann friction parameter $H$ have beenobtained over $10^{-2}\leq H\leq10^4$. Asymptotic trends are observed as$H\rightarrow\infty$ following $Re_c\propto H^{\,1/2}$ and $Ra_c\propto H$. Thelinear stability analysis reveals multiple instabilities which alter the flowboth within the Shercliff boundary layers and the interior flow, withstructures consistent with features from plane Poiseuille and Rayleigh-B\'enardflows.