The equations of motion for convection in dilute3He-superfluid-4He mixtures are the same as those for convection in a conventional pure fluid with the addition of several correction terms. Fetter has considered, for a horizontally infinite layer with realistic boundary conditions, the effect of these corrections on the critical Rayleigh number, Rc. The results are a perturbation expansion for Rc to lowest order in three perturbation terms, ε1, ε2, ε3. In order to make a comparison with recent precise experiments which have yielded Rc as a function of the layer height d, we have carried out several calculations. First we show that the analysis can be recast as an expansion in inverse powers of d2. We then carry out a complete expansion to O(d-6). Up to O(d-4), the expansion involves only the ratio (λ0/d) where λ0 is a length scale which is intrinsic to superfluid mixtures. We consider the effect of the superfluid perturbations on both the critical Rayleigh numbers and wavevectors. These are shifted very little as long as λ0/d is small; the crossover from large to small occurs for λ0/d∼0.1. We also solve a simplified version of the stability problem which contains the dominant superfluid effect. The simplified problem is Hermitian, and is therefore amenable to an exact solution. A comparison with experimental data for Rc and the simplified model shows excellent agreement with the calculations.
Metcalfe, G., & Behringer, R. P. (1993). Superfluid effects at the onset of convection in 3He-superfluid-4He mixtures. Journal of Low Temperature Physics, 90(1-2), 95-117. https://doi.org/10.1007/BF00682012