The dispersion relation of Tollmien-Schlichting waves

TL;DR

This paper presents a clear analysis of the dispersion relation for Tollmien-Schlichting waves, shedding light on their stability properties in shear flows at low viscosity.

Contribution

It provides a straightforward presentation and mathematical analysis of the dispersion relation for Tollmien-Schlichting waves, enhancing understanding of flow instability mechanisms.

Findings

Dispersion relation characterizes Tollmien-Schlichting wave stability.

Instability occurs for small viscosity and specific wave numbers.

Mathematical properties of the dispersion relation are elucidated.

Abstract

It is well-known that shear flows in a strip or in the half plane are unstable for the Navier-Stokes equations if the viscosity $\nu$ is small enough, provided the horizontal wave number $\alpha$ lies in a small interval, between the so called lower and upper marginal stability curves. The corresponding instabilities are called Tollmien-Schlichting waves. In this letter, we give a simple presentation of the dispersion relation of these waves and study its mathematical properties.