Linear Instability of the Plane Couette and Plane Poiseuille Flows

TL;DR

This paper demonstrates the linear instability of Plane Couette and Plane Poiseuille flows at specific Reynolds numbers by challenging traditional assumptions on disturbance periodicity, aligning theoretical predictions with experimental data.

Contribution

It introduces a new approach that abandons the assumption of longitudinal periodicity, leading to more accurate predictions of flow instability thresholds.

Findings

Plane Couette flow becomes unstable at Re > 140.

Plane Poiseuille flow's threshold Re is approximately 1040.

New theoretical thresholds align closely with experimental results.

Abstract

We show possibility of the Plane Couette (PC) flow instability for Reynolds number Re>Reth=140. This new result of the linear hydrodynamic stability theory is obtained on the base of refusal from the traditionally used assumption on longitudinal periodicity of the disturbances along the direction of the fluid flow. We found that earlier existing understanding on the linear stability of this flow for any arbitrary large Reynolds number is directly related with an assumption on the separation of the variables of the spatial variability for the disturbance field and their periodicity in linear theory of stability. By the refusal from the pointed assumptions also for the Plane Poiseuille (PP) flow, we get a new threshold Reynolds value Reth=1040 that with 4% accuracy agrees with the experiment contrary to more than 500% discrepancy for the earlier known estimate Reth=5772 obtained in the frame of the linear theory but when using the "normal" disturbance form (S. A. Orszag, 1971).