On one class of analytic solutions of the stationary axisymmetric convection B\'enard--Marangoni viscous incompressible fluid

S. N. Aristov ; E. Yu. Prosviryakov

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, no. 3 (2013), pp. 110-118

Résumé

The purpose of this work is to find solutions for the system of equations Oberbeck–Boussinesq flat convection Bénard–Marangoni a viscous incompressible fluid. In this viscous incompressible fluid the radial component of the temperature gradient may become zero. It is shown that the initial system may be reduced to the system of equations of ordinary differential equations of the eleventh order. We obtain the exact solution at the point of the extremum of the temperature (at zero including Grasgof's). Integration of equations is carried out in dimensionless variables, which are non-classical way: put the scale factor for each variable, and not by linear characteristic size of the layer. The solution is the initial approximation to the solution of convection Bénard–Marangoni in numbers Grasgof's, the big zero.

Détail

Keywords: axisymmetric thermocapillary convection (convection Bénard–Marangoni), localized parabolic heaters, isolines, eigenvalues, localization of polynomials roots, localization of eigenvalues of the matrix.
Mots-clés : exact solution, Hessian matrix