A possible analytical approach

Daniels's (1985) treatment of the destruction of the conduction regime in the analogous single fluid heat transfer problem focused on the large Prandtl number limit; nevertheless, he outlined, in passing, an approach that might be of value to the present problem: an expansion of the solution in powers of Ra.

This had in fact already been attempted by Batchelor in 1954, though, unfortunately, that analysis is flawed and should be repeated. Batchelor employed `Grashof's formula' for the approximation of the solution of the biharmonic equation, which the first order stream-function satisfies, and also appears in the theory of elasticity. He adduces Love (1944) for this, saying that it is applicable for $\mathcal A$ `not too different from unity'. Love (1944, p. 495), however, writes that `the formula, though devoid of theoretical foundation, has often been treated with respect...it will be seen that Grashof's formula leads to a serious over-estimate of the strength of a plate which is at all nearly square'. Since it is possible to obtain solutions for this equation to any desired degree of accuracy (see Love 1944, or the references given by Daniels 1985), there would not seem to be any need to persist with Grashof's approximation.

A very similar problem was solved by Cormack, Leal and Imberger (1974): the single fluid heat transfer problem at low vertical aspect ratio and Rayleigh number. The techniques applied there would require hardly any modification to be used for the large $\mathcal A$ problem.

The extension to the nonisothermal vapour transport problem would entail a double expansion in powers of Ra; or, equivalently Gr or $\mbox{\textit{Gr}}(1+N)$; and the mass transfer rate factor, $\varPhi$. Note that the first order approximations will be independent (i.e. simply additive) whereas cross-effects will enter at higher orders (Van Dyke 1964, p. 35). I certainly recommend this approach to the problem as worthy of future investigation.