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Liquid-metal flow in a horizontal pipe with an electrically insulating wall, subject to a transverse magnetic field and bottom heating, can undergo a buoyancy dominated instability in the turbulence-suppressing magnetic-field regime. In this paper, we explore whether this instability persists under a stronger magnetic field and study the effects of small thin-wall electric conductance on the instability as well. The unstable region in the Hartmann number (Ha) –Grashof number (Gr) plane is determined for Reynolds number Re =9046 and Prandtl number Pr=0. 022. First, we extended Ha=O (10²) of former studies up to Ha=2 10⁴ and found that, in the Gr range investigated (up to 1 10⁹), there does exist a marginal stability boundary in the large- Ha regime beyond which the linear instability will be fully suppressed so that the flow will be stabilised again by the magnetic field. Overall, the marginal stability boundary has a moderate- Ha branch and a large- Ha branch, between which the flow is linearly unstable. The results suggest a power-law scaling of Gr Ha^4/3 as Ha increases on the large- Ha branch. Second, as the wall conductivity increases, the unstable region shifts towards smaller Ha and Gr, and shrinks in extent with respect to Ha, exhibiting a destabilising effect at moderate Ha and a stabilising effect at large Ha. In contrast to the insulating wall case where relatively short waves are always the leading mode on the marginal stability boundary, long waves can be destabilised and become the leading modes on both the moderate- Ha and large- <jats: inline-graphic xmlns: xlink="http: //www. w3. org/1999/xlink"
Xing et al. (Thu,) studied this question.