Quasi-self-similar solutions to some parabolic problems in the theory of viscoplastic flows

The initial-boundary value problems of acceleration from a state of rest of a two-constant viscoplastic medium (Bingham body) in a half-plane is investigated when the tangential stress is given at the boundary as a piecewise continuous monotonically non-decreasing function of time. As an additional condition at an unknown interface between a flow zone that increases with time in thickness and a stationary semi-infinite rigid zone, the requirement is chosen that the solution of this problem with a tendency to zero of the yield strength of the material at each point and at each moment of time tends to the solution of the corresponding viscous flow problem known as the generalized vortex layer diffusion problem. The exact analytical solutions are found for tangential stress and velocity profiles in nonstationary one-dimensional flow. The cases of self-similarity and so-called quasi-self-similarity are distinguished. The nature of the tendency at $t\to \infty $ of the thickness of the layer, in which the shear is realized, to infinity is of particular interest.