Partial Fourier and Laplace transform in the linear stability analysis of a non slipping mean flow in a 2D straight lined duct with respect to perturbations of the initial value by disturbances spatially distributed in a bounded neighborhood of a point in the duct
AbstractIn 1D case (when the liner effect is neglected) it is shown that the partial Fourier, Laplace transform technique, can be applied in the stability analysis with respect to perturbations of the initial value by disturbances spatially distributed in a bounded neighborhood of a point in the duct without any supplementary hypothesis. The obtained results are similar to those obtained for time harmonic permanent or instantaneous point source produced perturbations. In 2D case (for wall-perturbation interaction of mass-spring-damper type) when the same Fourier, Laplace transform technique, leads to boundary conditions which are different from those generally accepted and used by engineers, disturbances of the initial value, which support is completely included in the interior of the duct are considered. For such perturbations the boundary conditions coincide with those commonly accepted. In this case for symmetric mean flow velocity profile the partial Fourier transform with respect to the spatial variable x and the Laplace transform with respect to the time variable t of the evolving perturbation is analyzed and dispersion relations are derived. Lyapunov stability criteria are given in terms of the zeros of the dispersion relations.