Three-dimensional (3D) numerical simulations were performed to investigate the natural convection of a paramagnetic fluid (air) under a magnetic quadrupole field in the absence of gravity, extending a previously studied two-dimensional model. A customized OpenFOAM solver incorporating the magnetic force term into the governing equations was developed to evaluate heat transfer and flow structures. As the magnetic Rayleigh number Rm increased, the average Nusselt number Nu followed a power-law scaling, and increasingly intense convection led to the pronounced development of 3D vortex structures. Entropy generation was decomposed into contributions due to heat conduction and viscous dissipation, both of which exhibited power-law dependence on Rm. The relationship of the form Be−1 − 1 = a Nub was established between Nu and the average Bejan number Be, directly linking a first-law-based metric (heat transfer) with a second-law-based metric (irreversibility). In the high-Rm regime, entropy generation was shown to be governed predominantly by fluid shear rather than heat conduction, clarifying the thermodynamic characteristics of convection-dominated regimes. The Nu–Be correlation remained nearly unchanged even when the enclosure aspect ratio was varied, suggesting a degree of universality in the macroscopic thermodynamic behavior despite differences in flow topology. Furthermore, this correlation is theoretically supported by a scaling analysis.
Masuda et al. (2026) studied this question.