Energy stability of plane Couette and Poiseuille flows and Couette paradox

TL;DR

This paper investigates the nonlinear stability of plane Couette and Poiseuille flows, revealing that streamwise perturbations are stable at all Reynolds numbers, which challenges previous findings and offers insights into the Couette-Sommerfeld paradox.

Contribution

It demonstrates that streamwise perturbations are L2-energy stable for any Reynolds number, contradicting earlier studies and suggesting a new perspective on the paradox.

Findings

Streamwise perturbations are stable at all Reynolds numbers.

Critical nonlinear Reynolds numbers are associated with spanwise perturbations.

Provides a potential resolution to the Couette-Sommerfeld paradox.

Abstract

We study the nonlinear stability of plane Couette and Poiseuille flows with the Lyapunov second method by using the classical L2-energy. We prove that the streamwise perturbations are L2-energy stable for any Reynolds number. This contradicts the results of Joseph [10], Joseph and Carmi [12] and Busse [4], and allows us to prove that the critical nonlinear Reynolds numbers are obtained along two-dimensional perturbations, the spanwise perturbations, as Orr [16] had supposed. This conclusion combined with recent results by Falsaperla et al. [8] on the stability with respect to tilted rolls, provides a possible solution to the Couette-Sommerfeld paradox.