Distribution Patterns of Eigenvalues of Laminar Pipe Flows (1st Report, Determination of Approximate Subspace in the Galerkin Method)

As the first step in clarifying the structure and dynamic behavior of a linear system which describes the perturbation of a parallel flow in a pipe, the distribution of the eigenvalues in the system is studied. For this purpose, first the formulation is made in a Hilbert space, which facilitates the geometrical interpretation of the problem. Then, by applying the Galerkin method, a finite dimensional approximate linear system is obtained. In the latter process, a difficulty arises as to how the order of the approximation is to be determined. To give an answer to this, a concrete measure is proposed using the concept of the operator invariance of the subspace. To examine the validity of the measure, the distribution of the eigenvalues of Poiseuille flow is calculated. It is found that the proposed measure is a dequate for obtaining the accurate distribution of the eigenvalues.