Small Perturbations of the Diffusion-Vortex Flows of a Newtonian Liquid in a Half-Plane

Abstract

In this paper, we study plane diffusion-vortex flows in the half-plane of a viscous incompressible fluid controlled by the boundary motion. At the boundary, either the longitudinal velocity or shear stress can be specified as functions of time. Classical self-similar solutions take place if these functions coincide with the Heaviside function. A linearized problem for relatively small initial perturbations superimposed on the kinematics in the entire half-plane is formulated. It consists of one biparabolic equation with variable coefficients with respect to the complex-valued current function and four homogeneous boundary conditions. Exponential estimates, which are estimates of the attenuation for some values of the parameters while they indicate the nature of the growth of perturbations for others, are derived using the method of integral relations. Some characteristic cases of specifying the boundary velocity or tangential stress on it are analyzed.