Navier-Stokes Flow past a Rigid Body that Moves by Time-Periodic Motion

Abstract

We study existence, uniqueness and asymptotic spatial behavior of time-periodic strong solutions to the Navier-Stokes equations in the exterior of a rigid body, B, moving by timeperiodic motion of given period T , when the data are sufficiently regular and small. Our contribution improves all previous ones in several directions. For example, we allow both translational, ξ, and angular, ω, velocities of B to depend on time, and do not impose any restriction on the period T nor on the averaged velocity, ξ, of B. If ξ ≡ 0 we assume that ξ and ω are both parallel to a constant direction, while no further assumption is needed if ξ ≡ 0. We also furnish the spatial asymptotic behavior of the velocity field, u, associated to such solutions. In particular, if B has a net motion characterized by ξ = 0, we then show that, at large distances from B, u manifests a wake-like behavior in the direction −ξ, entirely similar to that of the velocity field of the steady-state flow occurring when B moves with velocity ξ.