International Conference
"Modern Problems of Applied Mathematics and Mechanics: Theory, Experiment and Applications", devoted to the 90th anniversary of professor Nikolai N. Yanenko

Пухначев В.В.   Гончарова О.Н.  

Construction of the exact solutions of the three-dimensional problems of convection

Reporter: Пухначев В.В.

The exact solutions of the three dimensional problems of convection of two immiscible viscous fluids in an infinite channel with a rectangular cross-section are studied (Pukhnachov V.V., 2000). The stationary solution can be called a three dimensional generalization of the well-known exact solution (Birikh R.V., 1966) because of a constant relative to the longitudinal coordinate temperature gradient given along the interface. Convective fluid flows are described by the Oberbeck-Boussinesq equations. Under assumption of non-deformability of interface by thermocapillary forces, the kinematic and dynamic conditions and also the conditions of continuity of temperature and heat flux are fulfilled exactly. The group-theoretical nature of the constructed solutions and the questions of solvability of the initial boundary value problems are investigated.

Analytical construction of the exact solutions of the stationary problems is added by the numerical investigation. Reduction to the two dimensional problem statements is performed. By construction of the numerical algorithm the new input functions are introduced: they are the analogues of the stream function and vorticity instead of the transversal components of the velocity. A possibility of a control of the convection mechanisms is investigated under conditions of normal gravity, microgravity and weightlessness. The catalogues of the three dimensional flows in a channel are presented in the case with heat-insulated solid boundaries and in the case of heating of one of the vertical walls.

The research has been supported by the Joint Integrated Project No.116 of Siberian, Ural and Far East Branches of the Russian Academy of Sciences, by the Russian Foundation for Basic Research (Grants No. 10-01-00007, No. 09-08-01127) and by the Russian Federal Program “Scientific and Pedagogical Staff” (Contract No. 14.740.11.0355).