A leading term for the velocity of stationary viscous incompressible flow around a rigid body performing a rotation and a translation

Paul Deuring,Šárka Nečasová,Stanislav Kračmar

doi:10.3934/dcds.2017057

Abstract

We consider the Navier-Stokes system with Oseen and rotational terms describing the stationary flow of a viscous incompressible fluid around a rigid body moving at a constant velocity and rotating at a constant angular velocity. In a previous paper, we proved a representation formula for Leray solutions of this system. Here the representation formula is used as starting point for splitting the velocity into a leading term and a remainder, and for establishing pointwise decay estimates of the remainder and its gradient.

Highlights

Consider a rigid body moving with constant velocity and rotating with constant angular velocity in a viscous incompressible fluid
The usual mathematical model of this flow, with respect to a reference frame in which the body is at rest, is given by the Navier-Stokes system with an Oseen term and rotational terms
The function Γ may be continuously extended to a function from C∞ R3 × R3 × (0, ∞)

Summary

Introduction

Consider a rigid body moving with constant velocity and rotating with constant angular velocity in a viscous incompressible fluid. In a series of papers due to Farwig, Hishida [20], [21], [22] and Farwig, Galdi, Kyed [17], a satisfactory theory could be developed for the case that the Oseen term τ ∂1u is not present in (1) but the parameter does not vanish (flow around a body that rotates but does not translate; “purely rotational case”) In this situation, if is small, it turned out that a leading term is again given by the Landau solution. The function Γ may be continuously extended to a function from C∞ R3 × R3 × (0, ∞)

Thus we may define

So we have