Linear stability of mixed convection flows in horizontal rectangular channels of finite transversal extension heated from below

We present a study of the linear hydrodynamic stability of purely conductive Poiseuille flow in three-dimensional horizontal rectangular channels uniformly heated from below. Its originality is to modellize the flow using realistic conditions on all the boundaries of the channel, in particular, no-slip conditions on the vertical lateral boundaries. The aim of this paper is to provide quantitative results for the values of the critical dimensionless parameters. When the Rayleigh number Ra is higher than the critical value Ra∗, the instabilities are shown to be three-dimensional horseshoe-shaped transversal rolls for Reynolds numbers Re smaller than the critical value Re∗, and longitudinal rolls for Re>Re∗. The results are given for a wide range of the Prandtl number (10−6≤Pr≤105) and of the transversal aspect ratio B of the duct (0.1≤B≤5.4 and B→∞).