Viscous heating and instability of the adiabatic buoyant flows in a horizontal channel

The stability of buoyant flows occurring in the mixed convection regime for a viscous fluid in a horizontal plane-parallel channel with adiabatic walls is investigated. The basic flow features a parallel velocity field under stationary state conditions. There exists a duality of flows, for every prescribed value of the mass flow rate across the channel cross-section, caused by the combined actions of viscous dissipation and the buoyancy force. As pointed out in a previous study, only the primary branch of the dual solutions is compatible with the Oberbeck-Boussinesq approximation. Thus, the stability analysis will be focused on the stability of such flows. The onset of the thermal instability with small-amplitude perturbations of the basic flow is investigated by assuming a very large Prandtl number, which is equivalent to a creeping flow regime. The neutral stability curves and the critical parametric conditions for the onset of instability are determined numerically.