On Bifurcating Time-Periodic Flow of a Navier-Stokes Liquid past a Cylinder

TL;DR

This paper establishes conditions under which time-periodic flow solutions bifurcate from steady states in 2D Navier-Stokes equations around a cylinder, using a novel approach involving spectral analysis of the linearized operator.

Contribution

It provides a new, more direct method to identify bifurcation points for time-periodic solutions in exterior flow problems, focusing on eigenvalue crossings of the linearized operator.

Findings

Bifurcation occurs when a simple eigenvalue crosses the imaginary axis.

The approach applies to supercritical and subcritical bifurcations.

The problem is formulated as a coupled elliptic-parabolic system.

Abstract

We provide general sufficient conditions for branching out of a time-periodic family of solutions from steady-state solutions to the two-dimensional Navier-Stokes equations in the exterior of a cylinder. To this end, we first show that the problem can be formulated as a coupled elliptic-parabolic nonlinear system in appropriate function spaces. This is obtained by separating the time-independent averaged component of the velocity field from its "purely periodic" one. We then prove that time-periodic bifurcation occurs, provided the linearized time-independent operator of the parabolic problem possess a simple eigenvalue that crosses the imaginary axis when the Reynolds number passes through a (suitably defined) critical value. We also show that only supercritical or subcritical bifurcation may occur. Our approach is different and, we believe, more direct than those used by previous authors in similar, but distinct, context.