The ceiling and floor regions

In the solution for the flow near a nonslip surface at high Reynolds numbers, the next stage after finding the basic inviscid flow is to expand the coordinate perpendicular to the surface where the boundary condition was dropped. Here the coordinate is $Y$ and the obvious choice for the expansion factor is $\mathcal A$, so that the full equations have to be solved in these regions. This will be the subject of chapter 5. Once solutions are found for these regions, a composite solution can be constructed for a cavity with $\mathcal A$ arbitrary above a minimum determined by the size of the end regions.

Similarly, to obtain solutions valid near the front and back walls, the coordinate $Z$ must be expanded by the factor $\mathcal S$. This is treated analytically in chapter 7 for the low mass transfer rate limit.