  In the analogue gravity framework, the acoustic disturbances in a moving
fluid can be described by an equation of motion identical to a relativistic
scalar massless field propagating in a curved spacetime. This description is
possible only when the fluid under consideration is barotropic, inviscid and
irrotational. In this case, the propagation of the perturbations is governed by
an acoustic metric which depends algebrically on the local speed of sound,
density and the background flow velocity, the latter assumed to be vorticity
free. In this work we provide an straightforward extension in order to go
beyond the irrotational constraint. Using a charged --- relativistic and
non-relativistic --- Bose--Einstein condensate as a physical system, we show
that in the low momentum limit and performing the eikonal approximation we can
derive a d'Alembertian equation of motion for the charged phonons where the
emergent acoustic metric depends on a flow velocity in the presence of
vorticity.
