Extreme spanwise aspect ratios

While (7.25) and (7.28) are exact solutions of the equations of motion, both are difficult to evaluate numerically for very large or small values of $\mathcal S$. There is a physical reason for this which may be explained by analogy with the theory of elasticity.

Consider (7.25) for large $\mathcal S$. The problem solved by $v_n-v_n^{\Vert}$ is formally identical to that for the sum of the bending moments about two perpendicular axes in a uniform elastic plate subjected to a distributed bending moment about the edge proportional to $-v_n^{\Vert}$ (Timoshenko & Woinowsky-Krieger 1959, p. 93). The applied moment vanishes along the edges $X=\pm 1/2$ and is odd-symmetric on the edges $Z=\pm1/2$. Thus, by Saint-Venant's principle (Love 1944, p. 131), the effects of the applied moment are of negligible magnitude at distances which are large compared with the width of the edges $Z=\pm1/2$: 1/$\mathcal S$ in terms of $Z$. For $\mbox{$\mathcal S$}\gg2$, then, each end-wall should be sensibly independent of the other's presence. The exact solution (7.25), however, combines the effects of the two.

For small $\mathcal S$, Stokes (1899) has pointed out that, for the general Hele-Shaw problem, the velocity field will only differ from the limiting form (7.26) over a distance from obstacles or walls comparable with the wall separation.

In this section, solutions are derived which account for the effect of the nearest short wall first. The basic solutions to be perturbed are (7.24) and (7.27) for large and small $\mathcal S$, respectively. Since neither of the basic solutions satisfy the condition (7.21) at all parts of the boundary, both the large and small $\mathcal S$ asymptotic expansions are singular perturbation problems.