Critical Rayleigh numbers for cryogenic experiments

Abstract

We give critical Rayleigh numbers, Rc, and the corresponding critical wavevectors, ac, for the onset of Rayleigh-Benard convection for thermal conditions on the horizontal boundaries that model physical experiments, particularly those carried out at low temperatures with liquid helium. We assume that a fluid layer, satisfying the Boussinesq approximation, is bounded above and below by rigid plates with finite, nonzero vertical thicknesses and finite thermal conductivities. The effect of sidewalls on Rc is not likely to be important for many experiments and so is not considered here; specifically, we assume a horizontally infinite layer. At the top of the top plate and the bottom of the bottom plate, we consider boundary conditions for which a linear combination of the convective temperature field and its vertical derivative vanishes. For these boundary conditions, the growth rates of the linear stability problem are necessarily real. We find that Rc only deviates significantly from 1708 and ac only deviates significantly from 3.11, when the thermal conductivity of the fluid is comparable to or larger than that of the boundaries, or when the plates are exceptionally thin. In particular a fixed heat flux applied to highly conducting plates (a configuration frequently used in cryogenic experiments) does not cause Rc to vary much from the standard value, 1708.