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    Dmitri Georgievsky

    Dmitri Georgievsky
    GSP-2, Leninskie Gory, Moscow, 119992, Russian Federation
    Lomonosov Moscow State University


    Georgievsky D. V.
    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.
    Keywords: viscoplastic solid, rigid domain, yield stress, diffusion, vortex layer, nonsteady shear, disturbance, quadratic functional
    Citation: Georgievsky D. V.,  On the diffusion of a rigid viscoplastic vortex layer, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 1, pp.  63-67
    Georgievsky D. V.
    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.
    Keywords: Newtonian fluid, cylindrical clearance, shear flow, rotation, the integral relation method, quadratic functional, variational inequality, stability, critical Reynolds number
    Citation: Georgievsky D. V.,  Evolution of three-dimensional picture of disturbances imposed on a rotational-axial flow in a cylindrical clearance, Rus. J. Nonlin. Dyn., 2014, Vol. 10, No. 3, pp.  345-354
    Georgievsky D. V.
    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.
    Keywords: tensor nonlinearity, invariant, material function, constitutive relation, Reiner–Rivlin fluid, shear, diffusion of vortex, vortex layer
    Citation: Georgievsky D. V.,  Tensor-nonlinear shear flows: Material functions and the diffusion-vortex solutions, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 3, pp.  451-463
    Akulenko L. D., Georgievsky D. V., Nesterov S. V.
    Citation: Akulenko L. D., Georgievsky D. V., Nesterov S. V.,  Preface to the translations of two classic works of Jeffery and Hamel, Rus. J. Nonlin. Dyn., 2009, Vol. 5, No. 1, pp.  99-100

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