The two-dimensional equations

In terms of the Cartesian components of velocity and the normalized coordinates (2.74), the conservation equations for the gas-vapour mixture (2.52)-(2.55) are

$${\frac{\partial u}{\partial X} +\frac{1}{\mbox{$\mathcal A$}}\fra... ...+\frac{1}{\mbox{$\mathcal S$}}\frac{\partial w}{\partial Z} = 0,}\hspace{-12mm}$$

$${\mbox{\textit{Gr}}(1+N)\mbox{\textit{Sc}}\left(u\frac{\partial m... ...ac{w}{\mbox{$\mathcal S$}}\frac{\partial m}{\partial Z}\right) = }\hspace{20mm}$$

$$\frac{\partial^2 m}{\partial X^2} +\frac{1}{\mbox{$\mathcal A$}^2... ...partial Y^2} +\frac{1}{\mbox{$\mathcal S$}^2}\frac{\partial^2 m}{\partial Z^2},$$ (4.2)

$${\mbox{\textit{Gr}}(1+N) \left(u\frac{\partial u}{\partial X} +\f... ...ac{w}{\mbox{$\mathcal S$}}\frac{\partial u}{\partial Z}\right) = }\hspace{20mm}$$

$$-{}\mbox{$\mathcal A$}\frac{\partial p}{\partial X}+\frac{\partia... ...partial Y^2} +\frac{1}{\mbox{$\mathcal S$}^2}\frac{\partial^2 u}{\partial Z^2},$$ (4.3)

$${\mbox{\textit{Gr}}(1+N) \left(u\frac{\partial v}{\partial X} +\f... ...ac{w}{\mbox{$\mathcal S$}}\frac{\partial v}{\partial Z}\right) = }\hspace{20mm}$$

$$-{}\frac{\partial p}{\partial Y}+\frac{T+Nm}{1+N}+\frac{\partial^... ...partial Y^2} +\frac{1}{\mbox{$\mathcal S$}^2}\frac{\partial^2 v}{\partial Z^2},$$ (4.4)

$${\mbox{\textit{Gr}}(1+N) \left(u\frac{\partial w}{\partial X} +\f... ...ac{w}{\mbox{$\mathcal S$}}\frac{\partial w}{\partial Z}\right) = }\hspace{20mm}$$

$$-{}\frac{\mbox{$\mathcal A$}}{\mbox{$\mathcal S$}}\frac{\partial ... ...partial Y^2} +\frac{1}{\mbox{$\mathcal S$}^2}\frac{\partial^2 w}{\partial Z^2},$$ (4.5)

and

$${\mbox{\textit{Gr}}(1+N)\left[\mbox{\textit{Pr}}_r+\mbox{\textit{... ...\frac{w}{\mbox{$\mathcal S$}}\frac{\partial T}{\partial Z}\right)}\hspace{57mm}$$

$${ - {}\frac{\mbox{\textit{Pr}}_I}{\mbox{\textit{Sc}}}\left(1-\mat... ...c{\partial m}{\partial Z}\frac{\partial T}{\partial Z}\right) = } \hspace{45mm}$$

$$\frac{\partial^2 T}{\partial X^2} +\frac{1}{\mbox{$\mathcal A$}^2... ...partial Y^2} +\frac{1}{\mbox{$\mathcal S$}^2}\frac{\partial^2 T}{\partial Z^2}.$$ (4.6)

Restricting attention for the moment (until chapter 7) to two dimensions by considering only cavities very extensive in the spanwise direction, in the limit $\mbox{$\mathcal S$}\rightarrow\infty$; the above equations (4.1)-(4.6) become

$$\frac{\partial u}{\partial X} +\frac{1}{\mbox{$\mathcal A$}}\frac{\partial v}{\partial Y} = 0,$$ (4.7)

$$\mbox{\textit{Gr}}(1+N)\mbox{\textit{Sc}}\left(u\frac{\partial m}{\partial X} +\frac{v}{\mbox{$\mathcal A$}}\frac{\partial m}{\partial Y}\right) = \frac{\partial^2 m}{\partial X^2} +\frac{1}{\mbox{$\mathcal A$}^2}\frac{\partial^2 m}{\partial Y^2},$$ (4.8)

$$\mbox{\textit{Gr}}(1+N)\left(u\frac{\partial u}{\partial X} +\frac{v}{\mbox{$\mathcal A$}}\frac{\partial u}{\partial Y}\right) = -{}\mbox{$\mathcal A$}\frac{\partial p}{\partial X}$$

$$+\frac{\partial^2 u}{\partial X^2} +\frac{1}{\mbox{$\mathcal A$}^2}\frac{\partial^2 u}{\partial Y^2},$$ (4.9)

$$\mbox{\textit{Gr}}(1+N)\left(u\frac{\partial v}{\partial X} +\frac{v}{\mbox{$\mathcal A$}}\frac{\partial v}{\partial Y}\right) = -{}\frac{\partial p}{\partial Y} +\frac{T+Nm}{1+N}$$

$$+\frac{\partial^2 v}{\partial X^2} +\frac{1}{\mbox{$\mathcal A$}^2}\frac{\partial^2 v}{\partial Y^2},$$ (4.10)

$$\mbox{\textit{Gr}}(1+N)\left(u\frac{\partial w}{\partial X} +\frac{v}{\mbox{$\mathcal A$}}\frac{\partial w}{\partial Y}\right) = \frac{\partial^2 w}{\partial X^2} +\frac{1}{\mbox{$\mathcal A$}^2}\frac{\partial^2 w}{\partial Y^2},$$ (4.11)

and

$${\mbox{\textit{Gr}}(1+N)\left[\mbox{\textit{Pr}}_r+\mbox{\textit{... ...frac{v}{\mbox{$\mathcal A$}}\frac{\partial T}{\partial Y}\right)=}\hspace{50mm}$$

$${\frac{\mbox{\textit{Pr}}_I}{\mbox{\textit{Sc}}}\left(1-\mathrm{e... ...frac{\partial m}{\partial Y}\frac{\partial T}{\partial Y} \right)}\hspace{40mm}$$

$$+ \frac{\partial^2 T}{\partial X^2} +\frac{1}{\mbox{$\mathcal A$}^2}\frac{\partial^2 T}{\partial Y^2}.$$ (4.12)

It is seen that apart from (4.11), none of the equations depend on $w$, so that it may be ignored and (4.11) dropped; note that $w=0$ is a solution of the large $\mathcal S$ limit of the spanwise component of the momentum equation (4.11). It may then be assumed that $m,u,v,p$ and $T$ are independent of $Z$, so long as this is consistent with the boundary conditions on all walls with the possible exception of the front and back walls, since these are now ( $\mbox{$\mathcal S$}\rightarrow\infty$) infinitely removed.