Linear stability of a non-slipping gas, flowing in a rectangular lined duct, with respect to perturbation produced by a prescribed frequency time harmonic source

Abstract

The homogeneous nonlinear Euler equations are linearized around the mean flow. The functions describing the perturbation produced by a given frequency time harmonic source are added as right hand members to the homogeneous linearized Euler equations. For the obtained non homogeneous equations solution corresponding to the zero initial value is researched. It is required that the solution be defined for all positive values of the time variable t, to have Laplace transform with respect to t and Fourier transform with respect to the spatial variable x and satisfy the boundary condition corresponding to the liner-perturbation interaction at the duct wall. Conditions assuring existence and uniqueness of bounded solution and dispersion relations are derived. The Lyapunov stability of the gas flow with respect to the considered perturbations is discussed.