Long waves instabilities

TL;DR

This paper provides a detailed analysis of long wave instabilities in shear layers for Navier Stokes equations, focusing on Orr Sommerfeld equations, and proves their existence under specific conditions.

Contribution

It offers a clear presentation of Orr Sommerfeld analysis and establishes the existence of instabilities in various fluid regimes with boundary conditions.

Findings

Existence of long wave instabilities in slowly rotating fluids.

Instabilities in slightly compressible fluids.

Results under Navier boundary conditions.

Abstract

The aim of this paper is to give a detailed presentation of long wave instabilities of shear layers for Navier Stokes equations, and in particular to give a simple and easy to read presentation of the study of Orr Sommerfeld equation and to detail the analysis of its adjoint. Using these analyses we prove the existence of long wave instabilities in the case of slowly rotating fluids, slightly compressible fluids and for Navier boundary conditions, under smallness conditions.