An Inhomogeneous Steady-State Convection of a Vertical Vortex Fluid
Received 15 May 2022; accepted 16 December 2022; published 24 February 2023
Author(s): Berestova S. A., Prosviryakov E. Y.
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 large-scale motion of a vertical vortex flow outside the field of the
Coriolis force. The large-scale fluid flow is considered in the approximation of a thin layer with
nondeformable (flat) boundaries. This assumption allows us to describe the large-scale 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.
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