Subcritical bifurcations of shear flows

TL;DR

This paper investigates the nature of bifurcations in shear flows governed by the Navier-Stokes equations, providing numerical evidence that these bifurcations are subcritical near the marginal stability curves.

Contribution

It offers the first numerical evidence suggesting that the Hopf bifurcation in shear flows is subcritical, enhancing understanding of flow stability and transition.

Findings

Bifurcations are likely subcritical near the upper marginal stability curve.

Shear flows exhibit instability for small viscosity and specific wave numbers.

Numerical evidence supports the subcritical nature of bifurcations in these flows.

Abstract

It is well-known that shear flows in a strip or in the half plane are unstable for the incompressible Navier-Stokes equations if the viscosity $\nu$ is small enough, provided the horizontal wave number $\alpha$ lies in a small interval, between the so called lower and upper marginal stability curves. Moreover, a Hopf bifurcation occurs at the upper marginal stability curve. In this article, for various shear flows, we give numerical evidences that this bifurcation is subcritical.