# Viscous Flow past a Body Translating by Time-Periodic Motion with Zero Average

## TL;DR

This paper rigorously analyzes the Navier-Stokes flow around a body with time-periodic, zero-average motion, demonstrating the existence of a steady streaming phenomenon where oscillatory effects decay with distance, revealing a steady-state behavior far from the body.

## Contribution

It provides a rigorous mathematical proof of the steady streaming phenomenon for viscous flow past a body with time-periodic motion and zero average.

## Key findings

- Flow exhibits steady-state characteristics far from the body.
- Oscillatory components decay faster than the steady component.
- Rigorous proof of steady streaming phenomenon.

## Abstract

We study existence, uniqueness, regularity and asymptotic spatial behavior of a Navier-Stokes flow past a body moving by a time-periodic translational motion of period $T$, and with zero average. For example, $\mathscr B$ moves in an oscillating fashion. The flow is also time-periodic with same period $T$. However, sufficiently ``far" from the body, the oscillatory component decays faster than the averaged component, so that the flow shows there a distinctive steady-state character. This provides a rigorous proof of the ``steady streaming" phenomenon.