  We study the one-dimensional, longitudinally boost-invariant motion of an
ideal fluid with infinite conductivity in the presence of a transverse magnetic
field, i.e., in the ideal transverse magnetohydrodynamical limit. In an
extension of our previous work Roy et al., [Phys. Lett. B 750, 45 (2015)], we
consider the fluid to have a non-zero magnetization. First, we assume a
constant magnetic susceptibility $\chi_{m}$ and consider an ultrarelativistic
ideal gas equation of state. For a paramagnetic fluid (i.e., with
$\chi_{m}>0$), the decay of the energy density slows down since the fluid gains
energy from the magnetic field. For a diamagnetic fluid (i.e., with
$\chi_{m}<0$), the energy density decays faster because it feeds energy into
the magnetic field. Furthermore, when the magnetic field is taken to be
external and to decay in proper time $\tau$ with a power law $\sim\tau^{-a}$,
two distinct solutions can be found depending on the values of $a$ and
$\chi_m$. Finally, we also solve the ideal magnetohydrodynamical equations for
one-dimensional Bjorken flow with a temperature-dependent magnetic
susceptibility and a realistic equation of state given by lattice-QCD data. We
find that the temperature and energy density decay more slowly because of the
non-vanishing magnetization. For values of the magnetic field typical for
heavy-ion collisions, this effect is, however, rather small. Only for magnetic
fields which are about an order of magnitude larger than expected for heavy-ion
collisions, the system is substantially reheated and the lifetime of the quark
phase might be extended.
