Dmitri Georgievsky

This paper is concerned with obtaining the parameters of a nonsteady shear rigid viscoplastic flow in a half-plane initially at rest. Beginning with the initial time moment, the constant tangent stress exceeding a yield stress is given on the boundary. The diffusion-vortex solution holds true inside an extending layer with an a priori unknown boundary. The remaining half-plane is immovable in this case. A two-dimensional picture of disturbances is imposed on the obtained flow; the picture may then evolve over time. The upper estimates of velocity disturbances by the integral measure of the space $H_2$ are constructed. It is shown that, in a certain range of parameters, the estimating function may decrease up to some point of minimum and only then increase exponentially. The fact of its initial decrease is interpreted as a stabilization of the main flow on a finite time interval.

This work deals with stability relative to three-dimensional disturbances of a compound rotationalaxial shear flow of Newtonian viscous fluid inside a cylindrical clearance. The corresponding linearized problem on stability is stated with the sticking conditions. On the basis of the integral relation method permitting to obtain sufficient estimates of stability as well as lower estimates for critical Reynolds numbers, the general upper estimate of real part of a spectral parameter (responding to stability) is derived. This estimate is defined more exactly for cases of both threedimensional axially symmetric disturbances and two-dimensional non-axially symmetric ones.

This work deals with tensor-nonlinear constitutive relations connecting the deviators of stress tensor and strain rate tensor in incompressible isotropic media which are called in continuum mechanics as Reiner–Rivlin fluids. The connections of quadratic and cubic invariants of two tensors, where two material functions involve, are presented. The main attention is given to one-dimensional shear flows in various curvilinear coordinate systems. The scheme of obtaining of the material functions for shear on the basis of the steady Poiseuille flow in a plane layer is described. The self-similar solutions corresponding to the generalized diffusion of vortex layer both in plane and axially symmetric cases are derived.