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    Eugeny Prosviryakov

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


    Burmasheva N. V., Prosviryakov E. Y.
    In this paper, we report on several classes of exact solutions for describing the convective flows of multilayer fluids. We show that the class of exact Lin – Sidorov – Aristov solutions is an exact solution to the Oberbeck – Boussinesq system for a fluid discretely stratified in density and viscosity. This class of exact solutions is characterized by the linear dependence of the velocity field on part of coordinates. In this case, the pressure field and the temperature field are quadratic forms. The application of the velocity field with nonlinear dependence on two coordinates has stimulated further development of the Lin – Sidorov – Aristov class. The values of the degrees of the forms of hydrodynamical fields satisfying the Oberbeck – Boussinesq equation are determined. Special attention is given to convective shear flows since the reduced Oberbeck – Boussinesq system will be overdetermined. Conditions for solvability within the framework of these classes are formulated.
    Keywords: exact solution, multilayer fluids, Oberbeck – Boussinesq equations, shear flows, self-similar flows with spatial acceleration
    Citation: Burmasheva N. V., Prosviryakov E. Y.,  Exact Solutions to the Navier – Stokes Equations for Describing the Convective Flows of Multilayer Fluids, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 3, pp.  397-410
    Privalova V., Prosviryakov E. Y., Simonov M. A.
    A new exact solution to the Navier – Stokes equations is obtained. This solution describes the inhomogeneous isothermal Poiseuille flow of a viscous incompressible fluid in a horizontal infinite layer. In this exact solution of the Navier – Stokes equations, the velocity and pressure fields are the linear forms of two horizontal (longitudinal) coordinates with coefficients depending on the third (transverse) coordinate. The proposed exact solution is two-dimensional in terms of velocity and coordinates. It is shown that, by rotation transformation, it can be reduced to a solution describing a three-dimensional flow in terms of coordinates and a two-dimensional flow in terms of velocities. The general solution for homogeneous velocity components is polynomials of the second and fifth degrees. Spatial acceleration is a linear function. To solve the boundaryvalue problem, the no-slip condition is specified on the lower solid boundary of the horizontal fluid layer, tangential stresses and constant horizontal (longitudinal) pressure gradients specified on the upper free boundary. It is demonstrated that, for a particular exact solution, up to three points can exist in the fluid layer at which the longitudinal velocity components change direction. It indicates the existence of counterflow zones. The conditions for the existence of the zero points of the velocity components both inside the fluid layer and on its surface under nonzero tangential stresses are written. The results are illustrated by the corresponding figures of the velocity component profiles and streamlines for different numbers of stagnation points. The possibility of the existence of zero points of the specific kinetic energy function is shown. The vorticity vector and tangential stresses arising during the flow of a viscous incompressible fluid layer under given boundary conditions are analyzed. It is shown that the horizontal components of the vorticity vector in the fluid layer can change their sign up to three times. Besides, tangential stresses may change from tensile to compressive, and vice versa. Thus, the above exact solution of the Navier – Stokes equations forms a new mechanism of momentum transfer in a fluid and illustrates the occurrence of vorticity in the horizontal and vertical directions in a nonrotating fluid. The three-component twist vector is induced by an inhomogeneous velocity field at the boundaries of the fluid layer.
    Keywords: Poiseuille flow, gradient flow, exact solution, counterflow, stagnation point, vorticity
    Citation: Privalova V., Prosviryakov E. Y., Simonov M. A.,  Nonlinear Gradient Flow of a Vertical Vortex Fluid in a Thin Layer, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 3, pp.  271-283
    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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