Prediction of pattern selection due to an interaction between longitudinal rolls and transverse modes in a flow through a rectangular channel heated from below

Convection patterns in a flow through a horizontal channel that is heated from below are predicted on the basis of a weakly nonlinear theory. At a certain value of the Reynolds number and the Rayleigh number, the conduction state with steady shear flow becomes linearly unstable to both longitudinal rolls and transverse modes, simultaneously. The longitudinal rolls align along the streamwise direction whereas the transverse modes are periodic in the streamwise direction. Amplitude equations for the interaction between the longitudinal rolls and the transverse modes are derived in a consistent manner. Coefficients in the equations are determined numerically for a wide range of parameters. The longitudinal rolls are found to bifurcate super-critically. On the other hand, the transverse modes bifurcate subcritically or supercritically, depending on the Prandtl number, the aspect ratio of the channel, and the boundary conditions on the sidewalls. Stable convection patterns are classified in a parameter space. A mixed mode pattern, which is a mixture of the components of the longitudinal rolls and the transverse modes, is found to be stable for some sets of parameters.