Stability of internally generated thermal convection in a tall vertical annulus

Abstract

The stabilityof a convective fluid motion generated by internal heat sources in a tall vertical annulus is investigated by means of a mathematical model in the cases of both axisymmetric and asymmetric disturbances. The critical Grasshof numbers are computed for several values of the Prandtl number and different sizes of the gap between the cylinders. It is found that, for low Prandtl numbers and large gaps, asymmetric disturbances lead to instability while, in the case of small gaps, instability is associated with axisymmetric disturbances. In both cases, the critical Grasshof number increases as the gap decreases. For high values of the Prandtl number, instability occurs in the form of thermal running waves. The critical Grasshof numbers decrease as the Prandtl number grows. The neutral stability curve has one or two closed loops for sufficiently high Prandtl numbers. It is found that for high Prandtl numbers instability is associated with axisymmetric perturbations at least in the interval 0.05 < R < 1, where R is the ratio of the inner to the outer radii.