Completely enclosed flows

Since $v_n$ is odd in $X$, its integral over the section is zero and the only contribution to a net vertical flow rate comes from $v_f$. If the physical system of finite $\mathcal A$, modelled by equations (7.7)-(7.12) when $\mathcal A$ is large, has solid surfaces at $y_*=\pm b\mbox{$\mathcal A$}/2$, the net vertical flow rate must vanish to satisfy conservation of mass (2.52) so that:

$$\frac{\,\mathrm{d}p}{\,\mathrm{d}Y} = 0;$$ (7.29)

$$v_f(X,Z) = 0;$$ (7.30)

$$v(X,Z) = v_n(X,Z).$$ (7.31)

A similar line of reasoning leads to the conclusion that $\,\mathrm{d}p/\,\mathrm{d}Y$ is uniform in the case of a duct, where the planes $Y=\pm 1/2$ now represent orifices. Indeed, $\,\mathrm{d}p/\,\mathrm{d}Y$ is proportional to the net vertical flow rate (Dryden et al. 1956, p. 197).