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    Valentina Privalova

    ul. Komsomolskaya 34, Yekaterinburg, 620049, Russia
    Institute of Ingineering Science UB RAS


    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

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