On the stability of symmetric flows in a two-dimensional channel

TL;DR

This paper investigates the stability of symmetric flows in a 2D channel, proving stability outside certain parameter regions where previous work indicated instability, by analyzing the boundedness of the Orr-Sommerfeld operator.

Contribution

It establishes stability conditions for symmetric flows in a 2D channel outside known unstable parameter regions, extending prior instability results.

Findings

Flows are stable outside the identified instability region.

The Orr-Sommerfeld operator is bounded on the half-plane for specific parameter ranges.

Provides rigorous mathematical proof of stability conditions.

Abstract

We consider the stability of symmetric flows in a two-dimensional channel (including the Poiseuille flow). In 2015 Grenier, Guo, and Nguyen have established instability of these flows in a particular region of the parameter space, affirming formal asymptotics results from the 1940's. We prove that these flows are stable outside this region in parameter space. More precisely we show that the Orr-Sommerfeld operator $$ {\mathcal B} =\Big(-\frac{d^2}{dx^2}+i\beta(U+i\lambda)\Big)\Big(\frac{d^2}{dx^2}-\alpha^2\Big) -i\beta U^{\prime\prime}\,, $$ which is defined on $$ D({\mathcal B})=\{u\in H^4(0,1)\,,\, u^\prime(0)=u^{(3)}(0)=0 \mbox{ and }\, u(1)=u^\prime(1)=0\}. $$ is bounded on the half-plane $\Re \lambda \geq 0$ for $\alpha \gg \beta^{-1/10}$ or $\alpha \ll \beta^{-1/6}$.