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

TL;DR

This paper investigates the linear stability of quasi-two-dimensional liquid metal duct flows under a transverse magnetic field with heating from below, revealing multiple instability mechanisms influenced by magnetic damping and thermal stratification.

Contribution

It provides a comprehensive linear stability analysis of MHD Rayleigh-Bénard flows with magnetic damping, highlighting the effects of Hartmann friction on critical stability parameters.

Findings

Critical Reynolds number scales as H^{1/2} for large H

Critical Rayleigh number scales linearly with H

Multiple instability modes affect boundary layers and interior flow

Abstract

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