# On the instability of the magnetohydrodynamic pipe flow subject to a transverse magnetic field

## TL;DR

This study investigates the linear stability of liquid-metal MHD pipe flow under a transverse magnetic field, revealing instability at high Hartmann numbers and wall conductance ratios, with a critical Reynolds number of 45230.

## Contribution

We developed a BiGlobal stability analysis method for non-axisymmetric MHD pipe flows and identified the conditions leading to flow instability due to magnetic effects.

## Key findings

- Flow becomes unstable at high Hartmann numbers and wall conductance ratios.
- The critical Reynolds number is 45230 for a perfectly conducting pipe wall.
- Most unstable modes depend on the wall conductance ratio.

## Abstract

The linear stability of a fully-developed liquid-metal MHD pipe flow subject to a transverse magnetic field is studied numerically. Because of the lack of axial symmetry in the mean velocity profile, we need to perform a BiGlobal stability analysis. For that purpose, we develop a two-dimensional complex eigenvalue solver relying on a Chebyshev-Fourier collocation method in physical space. By performing an extensive parametric study, we show that in contrast to Hagen-Poiseuille flow known to be linearly stable for all Reynolds numbers, the MHD pipe flow with transverse magnetic field is unstable to three-dimensional disturbances at sufficiently high values of the Hartmann number and wall conductance ratio. The instability observed in this regime is attributed to the presence of velocity overspeeds in the so-called Roberts layers and the corresponding inflection points in the mean velocity profile. The nature and characteristics of the most unstable modes are investigated, and we show that they vary significantly depending on the wall conductance ratio. A major result of this paper is that the global critical Reynolds number for the MHD pipe flow with transverse magnetic field is $Re=45230$, and it occurs for a perfectly conducting pipe wall and the Hartmann number $Ha=19.7$.