Linear stability analysis of a non-slipping mean flow in a 2D-straight lined duct with respect to modes type initial (instantaneous) perturbations

In this paper a necessary and sufficient condition is found for the existence of non-zero modes, which satisfy the Pridmore–Brown equation and the mass-spring-damper impedance boundary condition. The flowing fluid is assumed to be inviscid, non-slipping and compressible. The mean flow velocity profile in the equation is assumed to be function of y only. The condition which is found defines in fact a dispersion relation, which has to be used in the linear stability analysis of the flow also by Briggs–Bers method. As far as we know, the dispersion relation reported in the present paper is new and it is not an obvious consequence of other results already reported in the literature. The numerical illustration shows that the dispersion-relation is effective and for the considered numerical data reveals the existence of mode type initial perturbations whose amplitude increase exponentially in time showing linear instability. In the same time the numerical illustration reveals the existence of mode type initial perturbations whose amplitude decrease exponentially tending to zero for t tending to plus infinity.