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

TL;DR

This paper derives pointwise decay estimates for steady 2D Stokes flows around rotating bodies, analyzes the singular behavior related to angular velocity, and characterizes the asymptotic structure of small scale-critical Navier-Stokes flows, revealing a self-similar circular profile influenced by torque.

Contribution

It introduces a new approach to estimate decay in 2D Stokes flows around rotating bodies and characterizes the asymptotic structure of small steady Navier-Stokes flows, highlighting the role of torque.

Findings

Decay estimates depend on angular velocity, showing singularity due to Stokes paradox.

Asymptotic structure of Navier-Stokes flow is a self-similar circular profile.

Leading term in flow at infinity is determined by the torque on the body.

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 on account of the Stokes paradox. We then employ the estimates to identify the asymptotic structure at infinity of a steady scale-critical Navier-Stokes flow, being 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 and whose coefficient is the torque acting on the body.