The Analisys of Instabilities of Generalised Kolmogorov Flows

The treatment of (in)stability of the classical Kolmogorov flow of viscous incompressible fluid on the plane torus T²={(x,y)∈ IR² : x∈ [0, 2π/α], y∈ [0, 2/π]} immediately leads to the analysis of solutions of the Navier-Stokes system in the infinite domain K={(x,y)∈ IR² : -\infty < x < \infty , 0 < y < 2π} as the minimal critical Reynolds number of the loss of stability of the Kolmogorov flow corresponds to α=0. In this paper we demonstrate that under some restrictions on the velocity field the critical eigenvalue is real and hence it is natural to analyze the spatial dynamics of the problem.