The vertical pressure gradient

A more general solution that still satisfies equations (4.13)-(4.17) and boundary conditions (4.18)-(4.21) is one where $c$ varies with $Y$, but this must be rejected on physical grounds as it leads to a net vertical mass flux that varies with height. This violates the conservation of mass, as by equation (4.39) the cold and hot wall Sherwood numbers are equal, so that there is no net addition of mass to the cavity through the hot and cold walls at any horizontal section for which this fully developed solution applies.

The value of the constant $c$ remains indeterminate. It is clearly related to the net vertical mass flux, which is proportional to the integral of $v$ across the cavity. This integral depends on $c$.

In the small mass transfer rate factor limit, the integral from $X=0$ to 1 of $v$ (4.36) is $-c/12$. For the cavity, the net vertical mass flux in this limiting case must be zero (assuming3 there is no mass flux at the floor or ceiling, or front or back walls) so that $c=0$. For an open channel, $c$ is determined by the pressure boundary conditions at the inlet and outlet.

No such simple treatment is possible for a cavity with $\varPhi \neq 0$. An inspection of figures 4.2 or 4.3, for which $c=0$ in all the plotted profiles, reveals that the condition of the pressure gradient balancing the mean density perturbation (i.e. $c=0$, from equation 4.28); does not imply a net zero vertical mass flux. Further, if $\mbox{\textit{Gr}}(1+N) \neq 0$, a recirculating flow would be expected, which would certainly cause the net mass added to the cavity through the hot and cold walls to be different in the top and bottom end regions. This difference cannot be determined without the solutions valid for these regions. These are obtained numerically in chapter 5.