Small spanwise aspect ratios

For small spanwise aspect ratios,

$$v_n\sim v_n^{=} \qquad (\mbox{$\mathcal S$}\rightarrow 0)$$ (7.36)

but not uniformly. The region of nonuniformity adjacent to the hot wall, $X=1/2$, may be treated by the method of matched asymptotic expansions (Van Dyke 1964, ch. 5); i.e. introducing a coordinate there stretched by the factor suggested by Stokes's (1899) estimate of the length scale for the influence of the short walls:

$$\xi = \frac{\frac{1}{2}-X}{\mbox{$\mathcal S$}} = \frac{\frac{b}{2}-x}{\mbox{$\mathcal S$}b},$$ (7.37)

(as illustrated in figure 7.3a)

Figure 7.3: The stretched coordinates (a) $(\xi,Z)$ and (b) $(X,\zeta)$ for the viscous layers adjacent to the hot wall for small $\mathcal S$ and the front wall for large $\mathcal S$, respectively.

and assuming an asymptotic expansion there of the form:

$$v_n(\xi,Z;\mbox{$\mathcal S$}) \sim \sum_i \varDelta _i(\mbo... ...x{$\mathcal S$}\rightarrow 0), \quad (\xi,Z)\quad\mbox{fixed}.$$ (7.38)

The $\varDelta _i$ form an asymptotic sequence to be determined along with the $\mathcal{V}_i$ in the course of the solution. The series (7.38) is called the inner solution.

Any attempt to extend the basic solution, $v_n^{=}$, in terms of $X$ and $Z$, to form an outer solution with a series

$$v_n(X,Z;\mbox{$\mathcal S$}) \sim v_n^{=} + \sum_i \delta_i(\mbox{$\mathcal S$}) v_i(X,Z)$$ (7.39)

leads to the homogeneous problems

$$\frac{\upartial ^2 v_i}{\upartial Z^2} = 0$$ (7.40)

for which the unique solution satisfying the no-slip conditions at $Z=\pm1/2$ is

$$v_i = 0$$ (7.41)

regardless of the choice of $\delta_i$. Thus

$$v_n = v_n^{=}+o(\mbox{$\mathcal S$}^K)\qquad(\mbox{$\mathcal... ...),\quad(X,Z)\;\mbox{fixed}, \quad\forall K\in\{0,1,2,\ldots\}.$$ (7.42)

For nondegenerate $\mathcal{V}_1$, choose

$$\varDelta _1 = \mbox{$\mathcal S$}^2$$ (7.43)

whence $\mathcal{V}_1$ satisfies

$$\frac{\upartial ^2\mathcal{V}_1}{\upartial \xi^2} + \frac{\upartial ^2\mathcal{V}_1}{\upartial Z^2} = -\frac{1}{2}$$ (7.44)

subject to

$$\mathcal{V}_1 = 0\qquad\quad\mbox{at}\quad\xi=0$$ (7.45)

$$\mathcal{V}_1 = 0\qquad\quad\mbox{at}\quad Z=\pm1/2$$ (7.46)

and a condition arising from the asymptotic matching process:

$$\mathcal{V}_1\sim\frac{1-4Z^2}{16} + o(\xi)\qquad\quad(\xi\rightarrow\infty).$$ (7.47)

Noticing that the limiting form (7.47) satisfies the field equation (7.44) and the boundary conditions at the end-walls (7.46), take it as a particular integral and solve by Fourier's method to obtain:

$$\mathcal{V}_1(\xi,Z) = \frac{1-4Z^2}{16}$$ (7.48)

$$- \frac{2}{\upi ^3}\sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)^3} \exp[-(2k+1)\upi \xi] \cos[(2k+1)\upi Z].$$

For $\varDelta _2=S^3$, $\mathcal{V}_2$ satisfies

$$\frac{\upartial ^2\mathcal{V}_2}{\upartial \xi^2} + \frac{\upartial ^2\mathcal{V}_2}{\upartial Z^2} = \xi$$ (7.49)

subject to

$$\mathcal{V}_2 = 0\qquad\quad\mbox{at}\quad\xi=0$$ (7.50)

$$\mathcal{V}_2 = 0\qquad\quad\mbox{at}\quad Z=\pm1/2$$ (7.51)

and a condition arising from the asymptotic matching process:

$$\mathcal{V}_2\sim-\frac{\xi(1-4Z^2)}{8} + o(\xi)\qquad\quad(\xi\rightarrow\infty).$$ (7.52)

Here the limiting form (7.52) satisfies the field equation (7.49) and all the boundary conditions; (7.50), (7.51) and (7.52); it must itself, therefore, be the solution.

Further terms in the series (7.39) lead only to trivial solutions, so that the inner asymptotic expansion to any order is:

$$v_n(\xi,Z;\mbox{$\mathcal S$}) \sim \mbox{$\mathcal S$}^2\left\{\frac{1-4Z^2}{16} \right.$$ (7.53)

$$+ \left. \frac{2}{\upi ^3}\sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)^3} \exp[-(2k+1)\upi \xi] \cos[(2k+1)\upi Z]\right\}$$

$$- \mbox{$\mathcal S$}^3\frac{\xi(1-4Z^2)}{8} + o(\mbox{$\mathcal S$}^K),$$

$$\qquad(\mbox{$\mathcal S$}\rightarrow 0),\quad(\xi,Z)\;\mbox{fixed}, \quad\forall K\in\{0,1,2,\ldots\}.$$

Rewriting the inner solution (7.53) in terms of $(X,Z)$ coordinates gives:

$$v_n(X,Z;\mbox{$\mathcal S$}) \sim v_n^{=} \mbox{\hspace*{60mm}}$$ (7.54)

$$- \mbox{$\mathcal S$}^2\frac{2}{\upi ^3}\sum_{k=0}^{\infty}... ...rac{1}{2}-X)}{\mbox{$\mathcal S$}}\right] \cos[(2k+1)\upi Z]$$

which satisfies the full field equation (7.20) for all values of $\mbox{$\mathcal S$}\in(0,\infty)$, and all the boundary conditions except $v_n=0$ at the cold wall ($X=-1/2$).

To convert the inner solution (7.54) into a full solution the effect of the cold wall msut be incorporated. A simple way to do this is to subtract the difference between itself and the known full solution (7.28).

The same solution can also be used near the cold wall by taking advantage of the odd symmetry of $v_n$ with respect to $X$.