On the Rayleigh-Bénard-Poiseuille problem with internal heat generation

The Rayleigh-Bénard-Poiseuille flow system with a uniform internal heat source is analyzed. A horizontal plane channel is bounded by two plane isothermal walls with unequal temperatures. The boundary heating from below and theinternal heating are parametrized by the Rayleigh number and by the internal Rayleigh number, respectively. Other governing parameters are the Prandtl number of the fluid and the Reynolds number associated with the basic Poiseuille flow. The linear stability to small-amplitude disturbances arbitrarily inclined to the basic flow direction is studied. A range of sufficiently small Reynolds numbers is investigated, where the thermoconvective instability has no interplay with the hydrodynamic (Orr-Sommerfeld) instability of the Poiseuille flow. In this range, the wavelike disturbances with a wave vector perpendicular to the Poiseuille flow direction, i.e. the longitudinal rolls, are the least stable modes. These modes are non-travelling, and they are not affected either by the Reynolds number or by the Prandtl number. On the other hand, the critical values of the wave number and of the Rayleigh number change with the internal Rayleigh number. The critical Rayleigh number can be even zero or negative, meaning heating from above, when the internal Rayleigh number is equal or greater than 37325.17.