The aim of this study was to clarify the structure and dynamic behaviour of the linear system (Navier-Stokes equation). A numerical method for calculating the eigenvalues is proposed together with a measure of accuracy. The distribution of eigenvalues and the mode of perturbations for the Poiseuille pipe flow are discussed. The wave perturbations for various azimuthal and axial wave numbers were investigated with a fixed Reynolds number. It is shown that the distribution of eigenvalues in a complex phase velocity plane assumes a tree like shape. The mode of perturbations is divided into three classes: slow, fast and mean modes by the axial phase velocity, or wall, centre and neutral modes by the radial distribution of the magnitude of the eigenfunction. For each mode, the location of the corresponding eigenvalue in the complex phase velocity plane and the dependence of the eigenvalue on the original linear dynamic system was clarified.
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