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    Eugeny Prosviryakov

    ul. Karla Marksa 10, Kazan, 420111, Russia
    Kazan National Research Technical University named after A.N.Tupolev


    Privalova V., Prosviryakov E. Y.
    This paper presents an exact solution to the Oberbeck – Boussinesq system which describes the flow of a viscous incompressible fluid in a plane channel heated by a linear point source. The exact solutions obtained generalize the isothermal Couette flow and the convective motions of Birikh – Ostroumov. A characteristic feature of the proposed class of exact solutions is that they integrate the horizontal gradient of the hydrodynamic fields. An analysis of the solutions obtained is presented and thus a criterion is obtained which explains the existence of countercurrents moving in a nonisothermal viscous incompressible fluid.
    Keywords: Couette flow, Birikh – Ostroumova flow, planar Rayleigh – Benard convection, quadratic heating, exact solution, counterflow
    Citation: Privalova V., Prosviryakov E. Y.,  Steady convective Coutte flow for quadratic heating of the lower boundary fluid layer, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 1, pp.  69-79
    Aristov S. N., Privalova V., Prosviryakov E. Y.
    A new exact solution of the two-dimensional Oberbeck–Boussinesq equations has been found. The analytical expressions of the hydrodynamic fields, which have been obtained, describe convective Couette flow. Fluid flow occurs in the case of nonuniform distribution of velocities and the quadratic heat source at the upper boundary of an infinite layer of viscous incompressible fluid. Two characteristic scales have been introduced for finding the exact solutions of the Oberbeck–Boussinesq equations. Using the anisotropic layer allows one to explore large-scale flows of liquids for large values of the Grashof number. A connection is shown between solutions describing the quadratic heating of boundaries with boundary problems concerned with motions of fluids in which the temperature is distributed linearly. Analysis of polynomial solutions describing the natural convection of the fluid is presented. The existence of points at which the hydrodynamic fields vanish inside the fluid layer. Thus, the above class of exact solutions allows us to describe the counterflows in the fluid and the separations of pressure and temperature fields.
    Keywords: Couette flow, linear heating, quadratic heating, convection, exact solution, polynomial solution
    Citation: Aristov S. N., Privalova V., Prosviryakov E. Y.,  Stationary nonisothermal Couette flow. Quadratic heating of the upper boundary of the fluid layer, Rus. J. Nonlin. Dyn., 2016, Vol. 12, No. 2, pp.  167-178
    Aristov S. N., Prosviryakov E. Y.
    Stokes waves in vortical fluid
    2014, Vol. 10, No. 3, pp.  309-318
    The solution of the second task of Stokes for the swirled knitting of incompressible liquid is provided. The found solutions represent the elliptic polarized cross waves. The solution of the second Stokes problem for the swirl flow of a viscous incompressible fluid is presented.
    Keywords: second Stokes problem, layered flows, vortical fluid, exact solution, wave amplification, elliptical polarization
    Citation: Aristov S. N., Prosviryakov E. Y.,  Stokes waves in vortical fluid, Rus. J. Nonlin. Dyn., 2014, Vol. 10, No. 3, pp.  309-318
    Aristov S. N., Prosviryakov E. Y.
    Inhomogeneous Couette flow
    2014, Vol. 10, No. 2, pp.  177-182
    We have obtained a solution of the problem within the exact solutions of the Navier–Stokes equations which describes the flow of a viscous incompressible fluid caused by spatially inhomogeneous wind stresses.
    Keywords: Couette flow, redefined boundary-value problem, exact solution, liquid vorticity, stream function, equatorial countercurrent
    Citation: Aristov S. N., Prosviryakov E. Y.,  Inhomogeneous Couette flow, Rus. J. Nonlin. Dyn., 2014, Vol. 10, No. 2, pp.  177-182
    Aristov S. N., Prosviryakov E. Y.
    On laminar flows of planar free convection
    2013, Vol. 9, No. 4, pp.  651-657
    New exact steady-state solutions of the Oberbeck–Boussinesq system which describe laminar flows of the Benard–Marangoni convection are constructed. We consider two types of boundary conditions: those specifying a temperature gradient on one of the boundaries and those specifying it on both boundaries simultaneously. It is shown that when the temperature gradient is specified the problem is essentially two-dimensional: there is no linear transformation allowing the flows to be transformed into one-dimensional ones. The resulting solutions are physically interpreted and dimensions of the layers are found for which there is no friction on the solid surface and a change occurs in the direction of velocity on the free surface.
    Keywords: laminar flow, analytical solution, polynomial solution, decrease in dimension, Benard–Marangoni convection
    Citation: Aristov S. N., Prosviryakov E. Y.,  On laminar flows of planar free convection, Rus. J. Nonlin. Dyn., 2013, Vol. 9, No. 4, pp.  651-657

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