Stability of the Couette-Poiseuille flow by the Reynolds-Orr energy equation

TL;DR

This paper revisits the stability analysis of Couette-Poiseuille flow using the Reynolds-Orr energy equation, computing minimum Reynolds numbers for various profiles and demonstrating that three-dimensional disturbances are generally more destabilizing than two-dimensional ones.

Contribution

It provides a comprehensive analysis of flow stability via the energy method, extending previous work by including three-dimensional disturbances and comparing results across different flow profiles.

Findings

Minimum Reynolds number is generally lower for three-dimensional disturbances.

Three-dimensional disturbances are more destabilizing than two-dimensional ones for most profiles.

The study offers new stability thresholds for Couette-Poiseuille flows.

Abstract

The normal-mode analysis of the Reynolds-Orr energy equation governing the stability of viscous motion for general three-dimensional disturbances has been revisited. The energy equation has been solved as an unconstrained minimization problem for the Couette-Poiseuille flow. The minimum Reynolds number for every Couette-Poiseuille velocity profile has been computed and compared with those available in the literature. For fully three-dimensional disturbances, it is shown that the minimum Reynolds number is in general smaller than the corresponding two-dimensional counterpart for all the Couette-Poiseuille profiles except plane Couette flow.