The onset of longitudinal vortex rolls in the thermal entrance region of plane Poiseuille flow heated from below with a constant heat flux is investigated theoretically. In the present study, the Boussinesq equations are solved numerically by using the finite volume method to analyze the onset of secondary flow. The local growth rates of the mean temperature and its fluctuations are examined, and the three characteristic distances, which are those to exhibit the onset of intrinsic instability, the detection of secondary flow, and the minimum Nusselt number under mixed convection, are suggested. They are discussed in comparison with available experimental data for water and air. (c) 2006 Elsevier Ltd. All rights reserved.