Stability of mixed-convection flow in a tall vertical channel under non-Boussinesq conditions

We have examined the linear stability of the fully developed mixed-convection flow in a differentially heated tall vertical channel under non-Boussinesq conditions. The three-dimensional analysis of the stability problem was reduced to an equivalent two-dimensional one by the use of Squire's transformation. The resulting eigenvalue problem was solved using an integral Chebyshev pseudo-spectral method. Although Squire's theorem cannot be proved analytically, two-dimensional disturbances are found to be the most unstable in all cases. The influence of the non-Boussinesq effects on the stability was studied. We have investigated the dependence of the critical Grashof and Reynolds numbers on the temperature difference. The results show that four different modes of instability are possible, two of which are new and due entirely to non-Boussinesq effects.