  We analyse near-extremal black brane configurations in asymptotically
$\mathrm{AdS}_4$ spacetime with the temperature $T$, chemical potential $\mu$,
and three-velocity $u^\nu$, varying slowly. We consider a low-temperature limit
where the rate of variation is much slower than $\mu$, but much bigger than
$T$. This limit is different from the one considered for conventional
fluid-mechanics in which the rate of variation is much smaller than both $T$,
$\mu$. We find that in our limit, as well, the Einstein-Maxwell equations can
be solved in a systematic perturbative expansion. At first order, in the rate
of variation, the resulting constitutive relations for the stress tensor and
charge current are local in the boundary theory and can be easily calculated.
At higher orders, we show that these relations become non-local in time but the
perturbative expansion is still valid. We find that there are four linearised
modes in this limit; these are similar to the hydrodynamic modes found in
conventional fluid mechanics with the same dispersion relations. We also study
some linearised time independent perturbations exhibiting attractor behaviour
at the horizon - these arise in the presence of external driving forces in the
boundary theory.
