This paper aims 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 the Orr-Sommerfeld equation and to detail the analysis of its adjoint.Using these analyses,we prove the existence of long-wave instabilities in the cases of slowly rotating fluids,slightly compressible fluids,or fluids with Navier boundary conditions,under smallness conditions.
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