Evolution of Perturbations Imposed on 1D Unsteady Shear in a Viscous Half-Plane with Oscillating Boundary

We study unsteady shear flows realized in a half-plane with viscous incompressible fluid, where the law of motion of the boundary oscillating along itself is given. Either the longitudinal velocity of the boundary or the shear stress on it can be specified. The statement of the linearized problem with respect to small initial perturbations imposed on the kinematics in the entire half-plane is presented. For a flat picture of perturbations, the statement consists of a single biparabolic equation with variable coefficients with respect to the complex-valued stream function that generalizes the Orr-Sommerfeld equation to the nonstationary case and of four homogeneous boundary conditions. Using the method of integral relations, we derive exponential estimates for the decay of perturbations. The result is compared with the three-dimensional picture of variations.