Eugeny Prosviryakov
Publications:
Burmasheva N. V., Prosviryakov E. Y.
Exact Solutions to the Navier – Stokes Equations for Describing the Convective Flows of Multilayer Fluids
2022, Vol. 18, no. 3, pp. 397410
Abstract
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.

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.

Aristov S. N., Prosviryakov E. Y.
Stokes waves in vortical fluid
2014, Vol. 10, No. 3, pp. 309318
Abstract
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.

Aristov S. N., Prosviryakov E. Y.
Inhomogeneous Couette flow
2014, Vol. 10, No. 2, pp. 177182
Abstract
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.

Aristov S. N., Prosviryakov E. Y.
On laminar flows of planar free convection
2013, Vol. 9, No. 4, pp. 651657
Abstract
New exact steadystate 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 twodimensional: there is no linear transformation allowing the flows to be transformed into onedimensional 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.

Berestova S. A., Prosviryakov E. Y.
Abstract
An exact solution of the Oberbeck – Boussinesq equations for the description of the steadystate
Bénard – Rayleigh convection in an infinitely extensive horizontal layer is presented. This
exact solution describes the largescale motion of a vertical vortex flow outside the field of the
Coriolis force. The largescale fluid flow is considered in the approximation of a thin layer with
nondeformable (flat) boundaries. This assumption allows us to describe the largescale fluid
motion as shear motion. Two velocity vector components, called horizontal components, are taken
into account. Consequently, the third component of the velocity vector (the vertical velocity) is
zero. The shear flow of the vertical vortex flow is described by linear forms from the horizontal
coordinates for velocity, temperature and pressure fields. The topology of the steady flow of
a viscous incompressible fluid is defined by coefficients of linear forms which have a dependence
on the vertical (transverse) coordinate. The functions unknown in advance are exactly defined
from the system of ordinary differential equations of order fifteen. The coefficients of the forms
are polynomials. The spectral properties of the polynomials in the domain of definition of the
solution are investigated. The analysis of distribution of the zeroes of hydrodynamical fields has
allowed a definition of the stratification of the physical fields. The paper presents a detailed study
of the existence of steady reverse flows in the convective fluid flow of Bénard – Rayleigh – Couette
type.
