General model

In the low mass transfer rate limit, the field equations are (6.16)-(6.19), and the velocity vanishes on all walls (cf. equation 6.20):

$$\mathbf{u=0}.$$ (7.1)

Since Gill's (1966) centrosymmetry properties are regained in the low mass transfer rate limit, it is convenient to translate the origin of the coordinates to the centroid of the cuboid, and, as in chapter 4, use normalized coordinates (see equation 2.74, fig. 7.1b).

Figure 7.1: Geometry for cavities and ducts of rectangular section, in (a) primitive and (b) central normalized coordinates.

A general thermal boundary condition is:

$$C(T - X) + I\mbox{\boldmath$\hat{n}$}\cdot\left(\mbox{\boldmath$\nabla$}_{\perp}T-\mbox{\boldmath$\hat{\imath}$}\right)=0$$ (7.2)

where $\hat{n}$ is the unit outward normal, $\hat{\imath}$ is the unit vector in the $X$-direction and

$$\mbox{\boldmath$\nabla$}_{\perp} \equiv \frac{\partial }{\pa... ...hcal S$}}\frac{\partial }{\partial Z}\mbox{\boldmath$\hat{k}$}$$ (7.3)

is a horizontal gradient operator. $C$ and $I$ are dimensionless scalars defined on the walls, which may vary with position so long as $C$ and $I$ do not both vanish at any point. If $C=0$ everywhere, the temperature is rendered determinate by the further condition

$$\int\!\!\!\int\!\!\!\int T \,\mathrm{d}X\,\mbox{$\mathcal A$}\,\mathrm{d}Y\,\mbox{$\mathcal S$}\,\mathrm{d}Z = 0$$ (7.4)

where the integral extends over the entire domain.

On the end-walls, $I=0$ leads to a linear temperature variation, while $C=0$ gives an adiabatic condition. On the hot and cold walls, $C=0$ implies a uniform heat flux while $I=0$ gives an isothermal condition. If the solid presumed to be surrounding the fluid has a higher thermal conductivity, $I=0$ everywhere is a simple and consistent idealization (Batchelor 1954; Leong et al. 1998) , although the condition $I=0$ on the heated and cooled walls and $C=0$ on the connecting walls appears frequently in the literature (Mallinson & de Vahl Davis 1973, 1977; Schladow, Patterson & Street 1989; Fusegi, Hyun & Kuwahara 1991).

A boundary condition for $m$ analogous to (7.2) is:

$$C_m(m-X)+I_m\mbox{\boldmath$\hat{n}$}\cdot(\mbox{\boldmath$\nabla$}_{\perp} m-\mbox{\boldmath$\hat{\imath}$})=0.$$ (7.5)

It will be noticed that the reference levels, $m_{*r}$ and $T_{*r}$ are no longer evaluated at the cold wall, but are rather taken as the means.