Laminar mixed convection flow in horizontal ducts of circular and rectangular shapes has been investigated numerically for both developed and developing regions. For developing flow, the fully elliptic three-dimensional Navier-Stokes equations together with the continuity and energy equations are solved by a modified SIMPLER algorithm in which the global continuity at each cross-section in the flow is enforced by applying corrections to the axial velocity and the bulk pressure. Comparing with the standard SIMPLER algorithm, the procedure is found more efficient and robust for various cases tested. For the fully developed flow in a circular tube, the dual solution, which was not identified in the earlier studies, has been observed for the range of Prandtl number ($0.2\le{Pr}\le{10}$). The lower end of the dual-solution region, Gr$_\mbox{crit}$ has been located. The upper limit of Gr, on the other hand, appears to be more susceptible to disturbance and is dependent upon the grid being used. The dual solution could be maintained up to $\frac{2}{\pi}Gr\le{10^8}$ for Pr=5, which is the highest Gr examined in the study. The developing flow calculations have been performed for Pr = 0.7 $\&$ 5 at Re = 250 with Gr ranging up to $10^7$. For Gr between $10^6$ and $10^7$, the vortical secondary flow develops almost immediately and increases the heat transfer considerably. The active crossflow motion can cause the flow in the upper region to separate early in the developing region. The natural developed flow is of two-vortex type. However, as the temperature gradient builds up near the bottom surface, a new pair of vortices could emerge. The flow, in the present parameter range, is shown to develop to either type of solution. The four-vortex solution is more likely to occur as Pr and (or) Gr increase(s). The details of the developing process are presented. The developing flow in the rectangular duct has been examined at Re = 250 and Pr = 0.7 for various aspect ratios ($\gamma$) and Gr combinations: for fixed aspect ratio of 2, the Grashof number is varied from $2\times{10^4}$ to $5\times{10^4}$ and, for Gr fixed at $2\times{10^4}$, $\gamma$ is varied between 2 and 8. Irrespective of these conditions, it is found that the thermal instability evolves at the lower corners of the duct and is magnified by the unstable thermal condition across the entire bottom plate. This process results in longitudinal vortices driven by a plume ascending along the sidewall. At $\gamma = 2$, the fully developed secondary flow changes from a two-vortex pattern to a four-vortex one as Gr increases. The developing process shows that, early in the developing region, the cell width of the secondary flow is nearly equal to its height, hence, the number of vortices increases with increasing aspect ratio (e.g., total eight cells exist at $\gamma = 8$). Farther downstream, however, in the region of decaying secondary flow, either one or two pairs of the vortices disappear depending on the flow structure. As a result, the fully developed secondary flow emerges as either a two- or a four-vortex flow. The developing process for a few cases of aspect ratio has been illustrated fully and that illuminates the possibility of the fully developed dual solutions, which were obtained earlier by other researchers. The average Nusselt number distribution, which vividly reflects the secondary flow development, is discussed in detail. The qualitative comparison of the present results with experimental results is also presented.
