The weakly nonlinear theory for modelling flows away from the bifurcation point developed by the authors in their previous work (Suslov & Paolucci 1997) is generalized for flows of variable-density fluids in open systems. It is shown that special treatment of the continuity equation is necessary to perform the analysis of such flows and to account for the potential total fluid mass variation in the domain. The stability analysis of non-Boussinesq mixed convection flow of air in a vertical channel is then performed for a wide range of temperature differences between the walls, and Grashof and Reynolds numbers. A cubic Landau equation, which governs the evolution of a disturbance amplitude, is derived and used to identify regions of subcritical and supercritical bifurcations to periodic flows. Equilibrium disturbance amplitudes are computed for regions of supercritical bifurcations.