H. Bénard and D. Avsec, Travaux récents sur les tourbillons cellulaires et les tourbillons en bandes; applications a ` l'astrophysique et a ` la météorologie, J. Phys. Rad, vol.9, p.1938

D. Brunt, Experimental cloud formation, Compendium of Meteorology American Meteorological Society, pp.1255-1262, 1951.

M. E. Braaten and S. V. Patankar, Analysis of laminar mixed convection in shrouded arrays of heated rectangular blocks, Int. J. Heat Mass Transfer, vol.28, 16991985.

M. Hasnaoui, E. Bilgen, P. Vasseur, and L. Robillard, Mixed convective heat transfer in a horizontal channel heated periodically from below, vol.297, 1991.

G. Evans and R. Greif, A study of traveling wave instabilities in a horizontal channel flow with applications to chemical vapor deposition, Int. J. Heat Mass Transfer, vol.32, 1989.

G. Evans and R. Greif, Unsteady three-dimensional mixed convection in a heated horizontal channel with applications to chemical vapor deposition, Int. J. Heat Mass Transfer, vol.34, 20391991.

G. Evans and R. Greif, Thermally unstable convection with applications to chemical vapor deposition channel reactors, Int. J. Heat Mass Transfer, vol.36, 27691993.

H. K. Moffat and K. F. Jensen, Complex flow phenomena in MOCVD reactors-1. Horizontal reactors, J. Cryst. Growth, vol.77, 1986.

K. C. Chiu and F. Rosenberger, Int. J. Heat Mass Transfer, vol.30, 16451987.

K. C. Chiu, J. Ouazzani, and F. Rosenberger, Mixed convection between horizontal plates-2. Fully developed flow, Int. J. Heat Mass Transfer, vol.30, 16551987.

J. M. Luijkx, J. K. Platten, and J. C. Legros, On the existence of thermoconvective rolls, transverse to a superimposed mean Poiseuille flow, Int. J. Heat Mass Transfer, vol.24, 1981.

J. M. Luijkx, Influence de la présence de parois latérales sur l'apparition de la convection libre, forcée et mixte, 1983.

J. K. Platten and J. C. Legros, Convection in Liquids Springer-Verlag, vol.6, 1984.

M. T. Ouazzani, Transferts thermiques et mécanique des e ´coulements de convection mixte, 1991.

S. S. Chen and A. S. Lavine, Laminar, buoyancy induced flow structures in a bottom heated, aspect ratio 2 duct with throughflow, Int. J. Heat Mass Transfer, vol.39, 1996.

M. T. Ouazzani, J. K. Platten, and A. Mojtabi, Etude expérimentale de la convection mixte entre deux plans horizontaux a ` températures différentes-2, Int. J. Heat Mass Transfer, vol.33, 14171990.

M. T. Ouazzani, J. K. Platten, and A. Mojtabi, Intermittent patterns in mixed convection, Appl. Sci. Res, vol.51, 1993.

E. Schröder and K. Bühler, Three-dimensional convection in rectangular domains with horizontal throughflow, Int. J. Heat Mass Transfer, vol.38, 12491995.

H. R. Brand, R. J. Deissler, and G. Ahlers, Simple model for the Bénard instability with horizontal flow near threshold, Phys. Rev. A, vol.43, 1991.

H. W. Müller, M. Tveitereid, and S. Trainoff, Rayleigh-Bénard problem with imposed weak through-flow: Two coupled Ginzburg-Landau equations, Phys. Rev. E, vol.48, 1993.

H. W. Müller, M. Lücke, and M. Kamps, Convective patterns in horizontal flow, Europhys. Lett, vol.10, p.1989

R. J. Deissler, Spacially growing waves, intermittency and convective chaos in an open flow system, Physica D, vol.25, 1987.

M. T. Ouazzani, J. K. Platten, H. W. Müller, and M. Lücke, Etude de la convection mixte entre deux plans horizontaux a ` des températures différentes-3, Int. J. Heat Mass Transfer, vol.38, 1995.

H. W. Müller, Thermische Konvection in horizontaler Scherströmung, 1990.

M. T. Ouazzani, J. P. Caltagirone, G. Meyer, and A. Mojtabi, Etude numérique et expérimentale de la convection mixte entre deux plans horizontaux a ` températures différentes, Int. J. Heat Mass Transfer, vol.32, 1989.

, 23 and used for comparisons in the present paper. Geometrical characteristics of the channel: length in the x direction L115 mm, width in the y direction l15.05 mm, height in the z direction h4.15 mm, longitudinal aspect ratio AL/h27.7, lateral aspect ratio Bl/h3.63. Physical characteristics of water at T23 °C: thermal diffusivity 0.14510 6 m 2 /s, thermal expansion coefficient 237, vol.16

M. Fortin and R. Glowinski, Applications a ´ la Résolution Numérique de Problèmes ausc Limites, Collection Méthodes Mathématique de l'Informatique-9, Méthodes de Lagrangien Augmenté, 1982.

R. Glowinski, Numerical Methods for Non-linear Variational Problems, 1984.

K. J. Arrow, L. Hurwicz, and H. Uzawa, Studies in Nonlinear Programming, 1958.

X. Nicolas, P. Traore, A. Mojtabi, and J. P. Caltagirone, Augmented Lagrangian method and open boundary conditions in the 2D simulation of the Poiseuille-Bénard channel flow, Int. J. Numerical Methods

I. Orlanski, A simple boundary condition for unbounded hyperbolic flows, J. Comput. Phys, vol.21, pp.251-1976

P. Bergé, Rayleigh-Bénard instability: Experimental findings obtained by light scattering and other optical methods,'' in Fluctuation, Instabilities and Phase Transition, Riste NATO Advanced Study Institutes, vol.11, pp.323-352, 1975.

C. Normand, Y. Pomeau, and M. G. Velarde, Convective instability: A physicists approach, Rev. Mod. Phys, vol.49, p.1977

A. Schlüter, D. Lortz, and F. Busse, On the stability of steady finite amplitude convection, J. Fluid Mech, vol.23, p.1965