  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 $Ha=19.7$.
