  We present a study of the hydrodynamics of compressible superfluids in
confined geometries. We use a perturbative procedure in terms of the
dimensionless expansion parameter $(v/v_s)^2$ where $v$ is the typical speed of
the flow and $v_s$ the speed of sound. A zero value of this parameter
corresponds to the incompressible limit. We apply the procedure to two specific
problems: the case of a trapped superfluid with a gaussian profile of the local
density, and that of a superfluid confined in a rotating obstructed cylinder.
We find that the corrections due to finite compressibility which are, as
expected, negligible for liquid He, are important but amenable to the
perturbative treatment for typical ultracold atomic systems.
