Stability of thermal convection rolls in a rectangular channel in the presence of a through-flow

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Secondary instability of longitudinal convection rolls of air in a rectangular channel heated from below is studied, in the presence of a through-flow. The longitudinal convection roll is a steady two-dimensional convection roll aligned along the channel; it is realized as a result of the primary instability of the conduction state above a critical Rayleigh number, when the Reynolds number is above a certain value. Stability boundaries of the longitudinal rolls with respect to three-dimensional infinitesimal disturbance are determined numerically in the Reynolds number-Rayleigh number plane. Several kinds of instabilities are found, which bring about an unsteady three-dimensional convection. One of them, an instability due to the primary transverse mode, is found for a low Reynolds number; it drastically changes its boundary in accordance with the thermal condition on the lateral boundaries.

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