Convective instability of a spherical fluid inclusion

Abstract

When a vertical temperature gradient is applied to a large solid containing a spherical fluid inclusion, the temperature in the fluid is a function only of height. The stability of this fluid against convection is investigated and it is found that the principle of exchange of stabilities applies. The linear differential system governing stability is then solved; the results show that the thermal conductivity of the surrounding solid is always stabilizing and that the most unstable mode is the first asymmetric mode, for which the critical Rayleigh number is given. The energy method can be applied, with due modifications to account for heat conduction in the surrounding solid. The same mathematical governing differential system would then be obtained, giving the same number for the upper bound of the Rayleigh numbers below which the fluid is stable. This number is then truly critical: The fluid is stable or unstable according to whether the Rayleigh number is below or above it, whatever the magnitude of the disturbance. The results are discussed in the context of the movement of the spherical inclusion in a soluble solid. The greater instability of the asymmetric mode indicates that when instability occurs, the fluid inclusion will have a sidewise component, which is greater for a greater supercritical Rayleigh number. The effect of double diffusion is also discussed.