At large Prandtl number, the mixed-convection instability in a vertical annulus receives most of its energy from the thermal field. This is the thermal-buoyant instability. The evolution of this instability is determined by three parameters, the wave amplitude, the amplification rate, and Ra-Rac, the Rayleigh number difference measured from the neutral curve. A consistent weakly nonlinear instability theory must obey the orderings among these three parameters. We find that theories of Stuart and Stewartson & Stuart provide asymptotically identical results for the problem studied in this paper. On the other hand, most amplitude-expansion methods for this problem are inconsistent since their formulations do not consider the ordering between the wave amplitude and the amplification rate. Theoretical results confirm that the instability is supercritical, in agreement with experimental observations, and show that the velocity waves exchange energy with the thermal field at the same wavelength and in the direction of the body force. Predicted Nusselt numbers and friction coefficients agree well with experiment.
The Royal Society is a self-governing Fellowship of many of the world's most distinguished scientists drawn from all areas of science, engineering and medicine, and is the oldest scientific academy in continuous existence. The Society’s fundamental purpose, reflected in its founding Charters of the 1660s, is to recognise, promote, and support excellence in science and to encourage the development and use of science for the benefit of humanity. The Society has played a part in some of the most fundamental, significant, and life-changing discoveries in scientific history and Royal Society scientists continue to make outstanding contributions to science in many research areas.
This item is part of a JSTOR Collection.
For terms and use, please refer to our
Proceedings: Mathematical and Physical Sciences
© 1992 Royal Society