Valentina Privalova
ul. Komsomolskaya 34, Yekaterinburg, 620049, Russia
Institute of Ingineering Science UB RAS
Publications:
Privalova V., Prosviryakov E. Y., Simonov M. A.
Nonlinear Gradient Flow of a Vertical Vortex Fluid in a Thin Layer
2019, Vol. 15, no. 3, pp. 271283
Abstract
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 twodimensional in terms
of velocity and coordinates. It is shown that, by rotation transformation, it can be reduced to
a solution describing a threedimensional flow in terms of coordinates and a twodimensional 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 noslip 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 threecomponent twist vector is induced by an inhomogeneous velocity
field at the boundaries of the fluid layer.

Privalova V., Prosviryakov E. Y.
Steady convective Coutte flow for quadratic heating of the lower boundary fluid layer
2018, Vol. 14, no. 1, pp. 6979
Abstract
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.

Aristov S. N., Privalova V., Prosviryakov E. Y.
Stationary nonisothermal Couette flow. Quadratic heating of the upper boundary of the fluid layer
2016, Vol. 12, No. 2, pp. 167178
Abstract
A new exact solution of the twodimensional 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 largescale 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.
