Stability of Mixed Convection in a Differentially Heated Vertical Channel

In this study, the linear stability of mixed convection in a differentially heated vertical channel is investigated for various Prandtl numbers. The results indicate that this fully developed heated flow can become unstable under appropriate conditions. It is found that both the Prandtl number and Reynolds number hold very important effects on the critical Grashof number, wave number, wave speed, and instability mechanism for higher Prandtl numbers. For low Prandtl numbers, the effects from the Prandtl number and Reynolds number are relatively small. The most significant finding is that the local minimum wave numbers can be as high as eight for Pr = 1000, which is substantially higher than those found before for other heated flows. The existence of multiple local minimum wave numbers is responsible for the sudden jumps of the critical wave number and wave speed and the sudden shift of instability type for higher Prandtl numbers. The energy budget analysis shows that the thermal-shear and shear instabilities dominate at both low and high Reynolds numbers for Pr = 0.7 and 7. It is the thermal-buoyant instability for Re < 1365 and shear instability for Re ≥ 1365 for Pr = 100. The thermal-buoyant and mixed instabilities are the possible instability types for Pr = 1000. In general, for mixed convection channel flows, the instability characteristics of differentially heated flows are found to be substantially different from those of uniformly heated flows.