LAUSR

We study how the pattern of perturbations superimposed on a plane-parallel time-periodic flow of a Newtonian viscous fluid evolves in a layer in which one of the boundaries performs longitudinal harmonic vibrations along itself, with the zero-friction slip of material allowed on the other boundary. We pose a generalized Orr–Sommerfeld problem as a linearized problem of hydrodynamic stability of unsteady-state viscous incompressible flows. Using the integral relation method, based on variational inequalities for quadratic functionals and developed as applied to unsteady-state flows, we derive integral estimates sufficient for the exponential decay of the initial perturbations. For each wave number, these estimates are inequalities relating three constant dimensionless quantities, viz., period-average depth-maximum shear velocity in the layer, boundary vibration amplitude, and the Reynolds number. We compare the established stability estimates for the planar and three-dimensional perturbation patterns.