Superpositions of the diffusion-vortex solutions and flow control inside infinite domains

Abstract

Superpositions of self-similar solutions of diffusion-vortex type are constructed and analyzed. These solutions simulate a one-dimensional acceleration from the quiescent state of a Newtonian medium in a half-plane (a planar problem) or in the space without a straight line (an axially symmetric problem) under the action of a combination of force factors, namely, of the pressure drop along the entire infinite thickness of the domain and of the tangent stress at the boundary. For the existence of a classical parabolic self-similar variable, these loadings must change, each according to its own temporal law, which is singular as t → 0. It is proved that, when prescribing, for example, the pressure drop, one can control the tangent stress along the boundary in such a way that the points inside the domain at which the stress either vanishes or has a local extremum move from the boundary deep down with a prescribed rate coefficient.