Conclusions

Vapour transport across a vertical cavity or duct of bounded horizontal section was considered in the limit as the vertical length scale tends to infinity. Exact solutions were presented for rectangular and elliptic sections at low mass transfer rates.

The mass fraction and temperature in all cases vary linearly with the transverse coordinate. The unique exact solutions for the purely vertical velocity for rectangular and elliptic sections are infinite hyperbolic-trigonometric series and bivariate polynomials of degree 3, respectively. These match all the conditions of the full problem for vertical cavities or ducts of finite height except the boundary conditions at the top and bottom and are valid for all values of Gr, $N$, Pr, Sc and $\mathcal S$, though naturally their stability cannot be guaranteed for large Grashof numbers.

For large sectional spans, the flows approach the familiar odd cubic profile, which also exists in the plane of spanwise symmetry of an elliptic section of arbitrary $\mathcal S$. In the rectangular section, the velocity profile for $Z=0$ is distorted as $\mathcal S$ decreases, with the location of the extrema moving outward from $X=\pm\sqrt{3}/6$ toward the hot and cold walls; for $\mbox{$\mathcal S$}\geq 1.7$, however, the profile and magnitude are practically independent of $\mathcal S$.

The cavities found in the walls built from hollow concrete masonry blocks common in North Queensland are often characterized by a large vertical aspect ratio, $\mathcal A$, and a roughly rectangular section of spanwise aspect ratio $\mbox{$\mathcal S$}\approx 1.3$. Although the Grashof number in these cavities will often be too high for the unicellular flow described here to be stable, it is clear that a two-dimensional analysis would be inadequate. In particular, it would be necessary to consider the influence of the end-walls in predicting the critical Grashof number for the onset of multicellular convection.

Although this work deals with simultaneous heat and mass transfer, it may be pointed out that the corresponding results for the conduction regime in a cavity of bounded section in the analogous single fluid heat transfer problem (trivially obtained by setting the buoyancy ratio, $N$, to zero) are also new.