Narrow cavities with bounded sections

A solution is sought for the limit $\mbox{$\mathcal A$}\rightarrow\infty$; the method is essentially similar to that used in chapter 4 for the narrow cavity limit ( $\mbox{$\mathcal A$}, \mbox{$\mathcal S$}\rightarrow\infty$), but here $\mathcal S$ is allowed to remain arbitrary.

It is convenient to decompose the velocity into horizontal and vertical components:

$$\mathbf{u} \equiv \mathbf{u}_{\perp} + v\mbox{\boldmath$\hat{\jmath}$}.$$ (7.6)

In the limit $\mbox{$\mathcal A$}\rightarrow\infty$, with Gr, $N$, Pr, Sc and $\mathcal S$ fixed and finite, the equations reduce to:

$$\mbox{\boldmath$\nabla$}_{\perp}\cdot\mathbf{u}_{\perp} = 0$$ (7.7)

$$\mbox{\textit{Gr}}(1+N)\mbox{\textit{Sc}}\;\mathbf{u}_{\perp}\cdot\mbox{\boldmath$\nabla$}_{\perp} m = \nabla_{\perp}^2 m$$ (7.8)

$$0 = -{}\frac{\upartial p}{\upartial X}$$ (7.9)

$$\mbox{\textit{Gr}}(1+N)\;\mathbf{u}_{\perp}\cdot\mbox{\boldmath$\nabla$}_{\perp}v = -{}\frac{\partial p}{\partial Y}+ \frac{T+Nm}{1+N} + \nabla_{\perp}^{2}v$$ (7.10)

$$0 = -{}\frac{\partial p}{\partial Z}$$ (7.11)

$$\mbox{\textit{Gr}}(1+N)\mbox{\textit{Pr}}\;\mathbf{u}_{\perp}\cdot\mbox{\boldmath$\nabla$}_{\perp}T = \nabla_{\perp}^{2}T$$ (7.12)

in $\Omega^{\Box}\equiv\{\mathbf{X}:\vert X\vert<1/2, \vert Z\vert<1/2\}$, subject to (7.1), (7.5) and (7.2) on $\upartial \Omega^{\Box}$, the boundary of $\Omega^{\Box}$.

There should also be boundary conditions at $Y=\pm 1/2$, corresponding to the floor and ceiling of the cavity, but these cannot be met by the solution of equations (7.7)-(7.12). As in the narrow cavity limit (ch. 4), the problem is singular in the sense of perturbation theory; the basic solution cannot be uniformly valid over the domain. For very large $\mathcal A$, however, the region of nonuniformity should be limited to thin layers near $Y=\pm 1/2$, corresponding to a small proportion of the unscaled cavity in terms of $y$. The full set of equations (2.52)-(2.55) would have to be solved to obtain the flow in these regions. This restriction is an advantage for the present work, though, as it means that the same fully developed solution will apply to both ducts and cavities. Some distinction between these will be made in §7.3.3.

Since the horizontal components of the momentum equation (2.54) have degenerated to a statement of the horizontal uniformity of the pressure (equations 7.9 and 7.11), $u$ and $w$ are determined only by the continuity equation (7.7) and the boundary conditions (7.1). An obvious solution is

$$u=w=0.$$ (7.13)

Accepting this solution (uniqueness follows from Theorem 2, p. , if the velocity is assumed to be independent of $Y$), the species (7.8) and energy (7.12) equations reduce to:

$$\nabla_{\perp}^{2}m = 0; \quad\mbox{and}$$ (7.14)

$$\nabla_{\perp}^{2}T = 0.$$ (7.15)

The unique solution satisfying these and the boundary conditions (7.2), for all values of $C$, $I$, $C_m$ and $I_m$ is

$$m = T = X.$$ (7.16)

Substituting this into the vertical momentum equation (7.10) leads to:

$$\nabla_{\perp}^{2}v=\frac{\,\mathrm{d}p}{\,\mathrm{d}Y}-X,$$ (7.17)

subject to (7.1) on $X=\pm 1/2$ and $Z=\pm1/2$. This is conveniently decomposed into two simpler Poisson equations by defining the forced and natural contributions, $v_f$ and $v_n$, satisfying

$$v_f + v_n = v$$ (7.18)

$$\nabla_{\perp}^{2}v_f = \frac{\,\mathrm{d}p}{\,\mathrm{d}Y}$$ (7.19)

$$\nabla_{\perp}^{2}v_n = -X$$ (7.20)

in $\Omega^{\Box}$ and

$$v_f=v_n=0\qquad\mbox{on}\quad\upartial \Omega^{\Box}.$$ (7.21)