Nous étudions numériquement l'influence d'un champ magnétique vertical, du nombre de Reynolds et d'une stratification de température sur la stabilité de l'écoulement de Hartmann chauffé par le bas. Pour Pr = 0,001 et Ha 2,5, nos résultats montrent que le champ magnétique stabilise aussi bien les modes transverses oscillatoires progressifs (T ) que les modes longitudinaux stationnaires (L). Quant à la stratification de température, elle est à l'origine d'une déstabilisation de l'écoulement de Hartmann par rapport au cas isotherme et de l'apparition des modes (L) inexistants dans le cas Ra = 0. On note aussi que l'étendue du domaine des valeurs de Re où les modes transverses (T ) sont les plus dangereux se rétrécit lorsque Ha augmente et qu'elle s'élargit quand Ra augmente à Ha donné. Quant à l'étendue du domaine des valeurs de Re où les modes (L) prévalent, elle augmente quand Ha croît.
The stability of a pressure driven flow in an electrically conducting fluid heated from below and subjected to a spanwise constant magnetic field is investigated through a linear stability analysis. The numerical calculations show that such a magnetic field only affects the longitudinal stationary modes ͑L͒, which are stabilized, and has no effect on the transverse traveling modes ͑T͒. A direct consequence is the expansion of the domain where the transverse traveling ͑T͒ modes prevail. This expansion is controlled by the strength of the spanwise magnetic field, expressed through the Hartmann number ͑Ha, ratio of magnetic to viscous dissipation forces͒. Moreover, when Ha exceeds a limiting value depending on the Prandtl number, the ͑T͒ modes become the only dominant modes in the whole Re range. Particular attention was dedicated to the oblique modes ͑O͒ corresponding to fully three-dimensional disturbances in order to verify that they never become the dominant modes when the magnetic field is applied, similarly to what was found without magnetic field. From a practical point of view, these results could be of a great interest since it is known that the optimization of several processes involving the Poiseuille-Rayleigh-Bénard flow is achieved when the traveling ͑T͒ modes prevail.
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