Thermal instability of plane poiseuille flow in the thermal entrance region

The onset of thermal convection in plane Poiseuille flow is investigated theoretically. New stability equations are derived by using the propagation theory considering the variations of disturbance amplitudes in the main flow direction. In the thermal entrance region an analytical procedure to predict the critical conditions for extremely small Prandtl-number fluids is described, based on the local similarity. For x_c≤0.01 the critical Rayleigh numbers are well represented in the whole domain of the Prandtl number by Ra_c = 200(1 + 0.123Pr^-1)Ra _C =200(1+0.123Pr^-1)x _C ^-1 under the conventional boundary layer theory. It is of much interest that the time-independent, three dimensional disturbances become more stable with a decrease in the Prandil number.