On periodic motions of a harmonic oscillator interacting with incompressible fluids

We consider a mass-spring system immersed in an incompressible fluid flow governed by the Navier-Stokes equations subject to a prescribed time-periodic flow rate (and possibly external time-periodic body forces on the fluid and the mass). We show that, with no restriction on the period of the flow rate (and of the external forces), when the flow rate is “small”, there exits a weak time-periodic solution to the coupled system. Under some more regularity and “smallness” conditions on the flow rate (and the external forces) we also show that these solutions are, indeed, strong solutions.