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Hydrodynamics of compressible superfluids in confined geometries

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.