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N. V. Burmasheva, E. Yu. Prosviryakov


A new exact solution of the Oberbeck–Boussinesq equation system, which describes the unidirectional convective flow of a viscous incompressible fluid in an infinite horizontal layer, is obtained. The fluid velocity depends on the vertical (transverse) coordinate. Pressure and temperature are the linear forms relative to the horizontal (longitudinal) coordinate with coefficients depending on the vertical coordinate. The fluid layer is bounded by a rigid infinite plane (lower boundary). In the study of fluid convection, it is assumed that the deformation of the free (upper) boundary of the layer is neglected. The thermocapillary effect inducing a convective flow is taken into account at the upper boundary. The contact of the moving fluid with the lower boundary occurs with slippage. The fluid slippage is described by the Navier boundary slip condition. The paper focusses on the study of the temperature field, which is spatially inhomogeneous. The temperature field is a seventh-degree polynomial with respect to the vertical coordinate. When studying the temperature distribution in the fluid layer, particular cases of the Navier slip condition are discussed. At the zero slip length, the boundary condition is transformed into the no-slip condition. When the slip length tends to infinity, there is a perfect slip boundary condition. It is demonstrated that the temperature field can be stratified into several zones relative to the reference value. In all the considered cases, the number of stratification zones does not exceed two. It is also shown that, in the case of perfect slip, the number of temperature field stratification zones is strictly equal to two, and the position of the temperature field stratification point depends neither on the physical parameters of the fluid nor on the conditions of heating of its boundaries.

Keywords: Oberbeck–Boussinesq equation system, unidirectional flow, layered flow, exact solution, Marangoni convection, counterflows, Navier slip conditionAcknowledgment



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