V. V. Privalova, E. Yu. Prosviryakov

EXACT SOLUTIONS FOR A COUETTE–HIEMENZ CREEPING CONVECTIVE FLOW WITH LINEAR TEMPERATURE DISTRIBUTION ON THE UPPER BOUNDARY

A new exact solution is found for the plane convection of a viscous incompressible fluid describable by the Oberbeck–Boussinesq equation system in an infinite thin layer. The solution of the boundary-value problem is obtained for the fluid flow arising in the case of inhomogeneous velocity distribution and a linear heat source at the upper boundary of an infinite layer of a viscous incompressible fluid. A creeping convective flow is studied by the generalization of the isothermal class of the Hiemenz exact solutions. The temperature and pressure fields are linear with respect to the horizontal coordinate in this class of solutions. The analysis of polynomial solutions describing natural fluid convection in the Stokes approximation is presented. The paper shows the existence of points where the velocity field vanishes inside the fluid layer. These points are termed stagnation points and indicate the presence of counterflows in the fluid. Similar investigations are carried out for the temperature and pressure fields. The isotherms and isobars are shown to be always locally parabolic or hyperbolic, i.e. to have an infinitely distant point, due to the structure of the discussed exact solution.

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