Generalizations of the Orr-Sommerfeld problem for the case in which the unperturbed shear motion is nonsteady

Abstract

We consider problems of the linearized theory of hydrodynamic stability for the case in which the unperturbed plane-parallel-flow of a viscous incompressible fluid in a layer is substantially unsteady. We analyze the Orr-Sommerfeld equation, which is generalized for this case, with different combinations of the four boundary conditions specified on the straight parts of the boundaries of the layer. Using the apparatus of integral relations, including, in particular, the analysis of the minimization problem for quadratic functionals, we derive upper bounds for the growth or decay of kinematic perturbations with respect to the integral measure. A special attention is paid to the longitudinal oscillation mode of the layer, to the power-law acceleration or deceleration, and also to the process similar to the diffusion of the vortex layer. An investigation of the reducibility of the three-dimensional picture of perturbations imposed on a plane-parallel unsteady shift to a two-dimensional picture in the plane of this shift is carried out. Generalizations of the Squire theorem are established.