Thermal Instability in Plane Poiseuille Flow

Abstract

The conditions marking the onset of longitudinal columnar vortices due to buoyant forces are studied for fully developed laminar flow between two infinite horizontal parallel plates with nonlinear basic temperature profile. The wall temperatures at the bottom and top plates, T1 and T2, respectively, are assumed to vary linearly in the main flow direction. The nonlinear basic temperature distribution and connective motion due to longitudinal disturbance component give rise to the influence on stability criteria: This influence may be expressed by a characteristic parameter representing the effect of longitudinal temperature gradient. Numerical values for critical Rayleigh numbers based on temperature difference, T1 − T2, are found for various Prandtl numbers and the parameter μ characterizing the effect of longitudinal wall temperature gradient. An increase in value for μ reduces the critical Ra further to a value less than 1708 when T1 > T2, and this tendency becomes pronounced as Pr increases. Results for the cases T1 ≦ T2 also show that the vortex rolls can be caused by the effect of longitudinal temperature gradient. Tentative discussion in terms of Richardson number is made to define the region where columnar vortices have priority of appearance over two-dimensional Tollmien-Schlichting waves. The computed secondary flow streamlines and perturbation temperatures show that the mode of convection motion is also affected by the parameter μ.