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

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, $\mathscr B$, moving by time-periodic 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, $\bfxi$, and angular, $\bfomega$, velocities of $\mathscr B$ to depend on time, and do not impose any restriction on the period $T$ nor on the averaged velocity, $\bar{\bfxi}$, of $\mathscr B$. If $\bfxi\not\equiv\0$ we assume that $\bfxi$ and $\bfomega$ are both parallel to a constant direction, while no further assumption is needed if $\bfxi\equiv\0$. We also furnish the spatial asymptotic behavior of the velocity field, $\bfu$, associated to such solutions. In particular, if $\mathscr B$ has a net motion characterized by $\bar{\bfxi}\neq\0$, we then show that, at large distances from $\mathscr B$, $\bfu$ manifests a wake-like behavior in the direction $-\bar{\bfxi}$, entirely similar to that of the velocity field of the steady-state flow occurring when $\mathscr B$ moves with velocity $\bar{\bfxi}$.

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