Boundary driven instabilities of Couette flows

TL;DR

This paper proves that boundary effects cause instabilities in Couette flows at a specific threshold related to viscosity, revealing the formation of unstable boundary layers and secondary instabilities.

Contribution

It establishes the boundary-driven instability threshold in Couette flows within high regularity spaces and describes the resulting boundary layer dynamics.

Findings

Instability threshold is proportional to ^{1/2} for large s.

Boundary layers of width ^{1/2} are linearly unstable.

Secondary instabilities lead to complex sub-layer formations.

Abstract

In this article, we prove that the threshold of instability of the classical Couette flow in $H^s$ for large $s$ is $\nu^{1/2}$. The instability is completely driven by the boundary. The dynamic of the flow creates a Prandtl type boundary layer of width $\nu^{1/2}$ which is itself linearly unstable. This leads to a secondary instability which in turn creates a sub-layer.