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

TL;DR

This paper proves existence, uniqueness, and describes the asymptotic behavior of time-periodic solutions to Navier-Stokes equations around a moving rigid body with general periodic motion, extending previous results.

Contribution

It allows both translational and angular velocities to depend on time without restrictions on period or average velocity, and characterizes the spatial decay and wake behavior of solutions.

Findings

Existence and uniqueness of solutions under small data conditions.

Spatial asymptotic behavior includes wake-like flow at large distances.

No restrictions on period T or average velocity of the body.

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, $\mathscr B$, 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 $\bar{\bfxi}$.