The linear stability of Hunt-Rayleigh-Bénard flow

The stability of a pressure driven flow in a duct heated from below and subjected to a vertical magnetic field (Hunt-Rayleigh-Bénard flow) is studied. We use the Chebyshev collocation approach to solve the eigenvalue problem for the small-amplitude perturbations. It is demonstrated that the magnetic field can stabilize the flow, while the temperature field can disturb the flow. There exists a threshold for the Hartmann number below which the growth rate changes with the Prandtl number non-monotonously (first increases and then decreases) with a critical Prandtl number for the maximum growth rate. By comparing the [Formula: see text] neutral curves at different Rayleigh numbers, we find that the critical Reynolds number decreases with the increase in the Rayleigh number, which has an obvious influence on the long-wave instability and a little influence on the short-wave instability. The dominant mode of the long-wave instability changes from the boundary layer instability to the inflectional instability with the increase in the growth rate, which forms a new flow map. We also compare the [Formula: see text] curves and find that the critical Rayleigh number decreases with the increase in the Reynolds number. The obtained results gain an insight into the flow stability affected by the temperature field and the magnetic field.