Influence of Prandtl number on three-dimensional instability of magnetohydrodynamic natural convection in an annular enclosure under a toroidal magnetic field

This study investigates the linear stability of natural convection of liquid metals in a vertically mounted annular enclosure subjected to an azimuthal static magnetic field. While previous studies have demonstrated the stabilizing influence of magnetic fields for specific fluid properties, the role of the Prandtl number (Pr) in the transition from an axisymmetric steady state to three-dimensional flow structures has not yet been fully clarified. To address this issue, a systematic linear stability analysis is conducted over a wide range of Prandtl numbers (Pr = 0.005–0.05), with the Rayleigh number (Ra) and the Hartmann number (Ha) as governing parameters. The results reveal that the critical Hartmann number (Hac), which defines the stability threshold, exhibits a strong negative correlation with Pr. To generalize this stability threshold, a unified correlation equation for Hac was formulated using the Churchill–Usagi method, successfully blending the distinct asymptotic scaling laws identified for the low- and high-Pr regimes. The proposed correlation demonstrates excellent agreement with the numerical data, confirmed by a parity plot yielding a coefficient of determination of 0.9592. These findings provide new insights into mode selection and instability mechanisms in low-Prandtl-number magnetohydrodynamic convection, with direct relevance to applications such as crystal growth and fusion reactor engineering.