On the asymptotic structure of steady Stokes and Navier–Stokes flows around a rotating two-dimensional body

Toshiaki Hishida and Mads Kyed

Vol. 315 (2021), No. 1, 89–109

DOI: 10.2140/pjm.2021.315.89

Abstract

We establish pointwise decay estimates for the velocity field of a steady two-dimensional Stokes flow around a rotating body via a new approach rather than analysis adopted in the previous literature. The novelty is to analyze the singular behavior of the constants in these estimates with respect to the angular velocity of the body, where such singularity is reasonable since they cannot hold in the absence of rotation. We then employ the estimates to identify the asymptotic structure at infinity of a steady scale-critical Navier–Stokes flow, assumed to be small, around a rotating body. It is proved that the leading term is given by a self-similar Navier–Stokes flow which exhibits a circular profile $x^{\perp }∕|x{|}^2$ and whose coefficient is the torque acting on the body.

Keywords

Navier–Stokes, Stokes, rotating body, asymptotic expansion

Mathematical Subject Classification 2010

Primary: 35B40, 35C20, 35Q30, 76D05, 76D07

Milestones

Received: 10 September 2018

Revised: 24 November 2020

Accepted: 6 September 2021

Published: 13 December 2021