Axial Heat Conduction Effects on Thermal Instability of Horizontal Plane Poiseuille Flows Heated From Below

A linear stability analysis is used to study the effect of axial heat conduction on the onset of instability for longitudinal and transverse vortex disturbances for plane Poiseuille flow in the thermal entrance region of a horizontal parallel-plate channel heated from below with a constant temperature difference between two plates. The basic flow solution for temperature (Graetz problem) incorporates axial heat conduction effects and the fluid temperature is taken to be uniform at the far upstream location x = −∞ to allow for upstream heat penetration through the thermal entrance x = 0. Numerical results for the critical Rayleigh numbers are obtained for an entrance temperature parameter θ₀ = 1 (T₀ = T₂) and Peclet numbers 1, 5, 10, 50. It is found that the transverse vortex disturbances are preferred over the longitudinal vortex disturbances for Pe ≲ 1 and Pr ≳ 1 (low Re) in the developing regions upstream and downstream of the thermal entrance. For other conditions, the longitudinal rolls have priority of occurrence. The Prandtl number effect on the onset of the longitudinal vortices is clarified.