Large spanwise aspect ratios

The procedure for large $\mathcal S$ is similar. The solution of (7.20) as $\mbox{$\mathcal S$}\rightarrow\infty$ is (7.24) but not uniformly in space. The neglected boundary conditions are at the end-walls. Near the front end-wall, a coordinate, $\zeta$, stretched by the factor suggested by Saint-Venant's principle, is introduced:

$$\zeta = \mbox{$\mathcal S$}\left(\frac{1}{2}-Z\right) =\frac{1}{b}\left( \frac{\mbox{$\mathcal S$}b}{2}-z\right),$$ (7.55)

as illustrated in figure 7.3(b).

The outer expansion is:

$$v_n(X,Z;\mbox{$\mathcal S$}) \sim v_n^{\Vert} + o(\mbox{$\mathcal S$}^K)$$ (7.56)

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

The inner expansion valid near the front end-wall, $(Z=1/2)$, is:

$$v_n(X,\zeta;\mbox{$\mathcal S$}) \sim v_n^{\Vert}$$ (7.57)

$$+ \frac{1}{4\upi ^3}\sum_{k=1}^{\infty}\frac{(-1)^k}{k^3} \sin[(2k+1)\upi X]\mathrm{e}^{-2k\upi \zeta} + o(\mbox{$\mathcal S$}^K),$$

$$\qquad (\mbox{$\mathcal S$}\rightarrow\infty), \quad(X,\zeta)\;\mbox{fixed}, \quad\forall K\in\{0,1,2,\ldots\},$$

or, in terms of $(X,Z)$,

$$v_n(X,Z;\mbox{$\mathcal S$}) \sim v_n^{\Vert} +$$ (7.58)

$$\frac{1}{4\upi ^3} \sum_{k=1}^{\infty}\frac{(-1)^k}{k^3} \sin(2k\upi X)\exp[-k\upi \mbox{$\mathcal S$}(1-2Z)].$$

This is a solution of the partial differential equation (7.20) and the boundary conditions at $X=\pm 1/2$ and $Z=1/2$. It is converted to a full solution by subtracting the difference between it and the full solution (7.25):

$$v_n = v_n^{\Vert}$$ (7.59)

$$+ \frac{1}{4\upi ^3}\sum_{k=1}^{\infty} \frac{(-1)^k}{k^3}\sin(2k\upi X)\exp[-k\upi \mbox{$\mathcal S$}(1-2Z)]$$

$$+ \frac{1}{4\upi ^3}\sum_{k=1}^{\infty} \frac{(-1)^k}{k^3}\sin(2k\u... ...}- \mathrm{e}^{-2k\upi \mathcal{S}(1-Z)}]} {1-\mathrm{e}^{-2k\upi \mathcal{S}}}$$

This is now a full solution that is practical for large $\mathcal S$ in the vicinity of the front end-wall. A solution useful near the back end-wall is easily constructed by exploiting the even symmetry of $v_n$ with respect to $Z$. The first term of (7.59) is the Jones-Furry solution, the second gives the effect of the front end-wall and the third the effect of the back end-wall. In the earlier representation for $v_n$ using the Jones-Furry solution as a particular integral (7.25), the effects of the two end-walls are entangled, whereas in (7.59), they are separated.

This representation (7.59) of $v_n$, arrived at by physical reasoning, is identical to that suggested in §7.3.4 on purely numerical grounds.