Stability of natural convection in an inclined square duct with perfectly conducting side walls

We have performed three-dimensional linear stability analyses for natural convection in an inclined square duct. The duct is heated from the bottom, while the lateral walls are assumed to be perfectly thermal conducting. Three-dimensional transverse rolls whose axes are normal to the axis of the duct occur from the motionless state when the Rayleigh number exceeds a critical value and the duct is placed horizontally (θ = 0°). However, it is found that when the duct is placed inclined (θ = 0.01°), a two-dimensional longitudinal roll which is unchanged in the axis of the duct occurs and is stable if the Rayleigh number is small.