The onset of convective instability in the thermal entrance region of plane Poiseuille flow heated uniformly from below

The onset condition of regular longitudinal vortex rolls in the thermal entrance region of plane Poiseuille flow heated from below is analyzed. Under propagation theory the stability equations are produced self-similarly, based on scale analysis. The onset position of secondary flow, which represents the starting point of mixed convection, is predicted as a function of the Prandtl number, Reynolds number and Rayleigh number. As expected, the critical position moves upstream as the Rayleigh number increases and an increase in Reynolds number makes the system more stable. The present predictions compare favorably with existing experimental data of water and air.