Existence, Uniqueness, and Asymptotic Behavior of Regular Time-Periodic Viscous Flow Around a Moving Body

Abstract

We show existence and uniqueness for small data of regular time-periodic solutions to the Navier-Stokes problem in the exterior of a rigid body, $\mathscr B$, that moves by time-periodic translational motion of the same period along a constant direction, $\bfe_1$, and spins with constant angular velocity $\bfomega$ parallel to $\bfe_1$. We also study the spatial asymptotic behavior of such solutions and show, in particular, that if $\mathscr B$ has a net motion characterized by a non-zero average translational velocity $\bar{\bfxi}$, then the solution exhibit a wake-like behavior in the direction $-\bar{\bfxi}$ entirely analogous to that of a steady-state flow around a body that moves with velocity $\bar{\bfxi}$ and angular velocity $\bfomega$.