The fully developed solution

A solution satisfying all of the above equations and conditions (see §5.1.1) except the velocity boundary conditions on the horizontal surfaces (5.1)-(5.2) and which, if stable, can be expected to apply sufficiently far from these surfaces, is the narrow cavity limiting solution derived in chapter 4, equations (4.23)-(4.24), (4.26)-(4.27).

In two dimensions ($x$ and $y$), the conduction-diffusion regime is defined as a state of the system in which the limiting solution applies for some values of $y$.

The constant $c$ depends on the net vertical mass flow rate through the fully developed region, arising from the transversely asymmetric convective mass transfer in the end-zones. In the analogous single fluid heat transfer problem, the net vertical mass flow rate is known (§4.4.2) so that there are no undetermined constants in the fully developed solution of Jones and Furry (1946).

The well-known (Gill 1966) centrosymmetry of the analogous single fluid heat transfer problem allows a considerable simplification: an end-zone can be treated as a semi-infinite rectangular strip, with the Jones-Furry solution as a boundary condition at infinity (Batchelor 1954; Daniels 1985; Daniels & Wang 1994). With finite mass transfer rates ($\varPhi \neq 0$), the solutions are no longer centrosymmetric so that the advantage of this approach is lost and the domain is here taken as the full cavity. This also avoids the difficulty of truncating an unbounded domain. The a priori indeterminacy of $c$ and the loss of centrosymmetry are discussed further once the numerical solutions are presented in the next section.