1 Introduction
1.1 Statement of the problem
1.2 Practical implications for membrane modules
1.3 Purely numerical solution of the wedge problem and motivation for asymptotic analysis
1.4 Overview of the paper
2 Generalized similarity solution
3 Outer expansion in powers of 1 / r
\(\varTheta \)
|
\(v_{r}^{\infty }(r,0) - \sigma \tau _{\theta r}^{\infty }(r,0)\)
|
\(v_{\theta }^{\infty }(r,0) + p^{\infty }(r,0)\)
|
---|---|---|
\(3 \pi / 4\)
|
\(16 \sigma (9 \pi ^{2} - 8)^{-1} r^{-1}\)
|
\(-8 (3 \pi - 2) (9 \pi ^{2} - 8)^{-1} r^{-1}\)
|
\( \pi / 2\)
|
\(8 \sigma (\pi ^{2} - 4)^{-1} r^{-1}\)
|
\(-4 \pi (\pi ^{2} - 4)^{-1} r^{-1}\)
|
\( \pi / 4\)
|
\(16 \sigma (\pi ^{2} - 8)^{-1} r^{-1}\)
|
\(-8 (\pi + 2) (\pi ^{2} - 8)^{-1} r^{-1}\)
|
3.1 Stokes building blocks
\(\varTheta \)
|
i
|
\(\hat{\tau }_{\theta r}^{(i)}(r,0;-1)\)
|
\(\hat{p}^{(i)}(r,0;-1)\)
|
---|---|---|---|
\(3 \pi / 4\)
| 1 |
\(r^{-2}\)
|
\(r^{-2}\)
|
\(3 \pi / 4\)
| 3 |
\(8 (4 + 3 \pi )^{-1} r^{-2} \ln r\)
|
\(-8 (4 + 3 \pi )^{-1} (1 + \ln r) r^{-2}\)
|
\(3 \pi / 4\)
| 5 |
\(r^{-2}\)
|
\(-r^{-2}\)
|
\(\pi / 2\)
| 1 | 0 |
\(r^{-2}\)
|
\(\pi / 2\)
| 3 |
\(2 r^{-2} \ln r\)
|
\((\pi /2) r^{-2}\)
|
\(\pi / 2\)
| 5 |
\(r^{-2}\)
| 0 |
\(\pi / 4\)
| 1 | 0 |
\(2 r^{-2}\)
|
\(\pi / 4\)
| 3 |
\((4 - \pi )^{-1} [8 \ln r - (88 + \pi ) / 7] r^{-2}\)
|
\((4 - \pi )^{-1} [8 \ln r - (32 + \pi ) / 7] r^{-2}\)
|
\(\pi / 4\)
| 5 |
\(r^{-2}\)
|
\(r^{-2}\)
|
3.2 Outer basis functions obtained with an iteration scheme
4 Inner power series
\(\omega \)
|
i
|
\(\check{v}_{r}^{(i)}(r,0;\omega )\)
|
\(\check{v}_{\theta }^{\{i\}}(r,0;\omega )\)
|
---|---|---|---|
1/2 | 3 |
\( (1/2) r^{1/2}\)
|
\(r^{1/2}\)
|
3/2 | 1 |
\(-(1/6) r^{3/2}\)
|
\( (2/3) r^{3/2}\)
|
3/2 | 2 |
\( (1/4) r^{3/2}\)
|
\( (1/6) r^{3/2}\)
|
5/2 | 1 |
\( (1/20) r^{5/2}\)
|
\( (3/10) r^{5/2}\)
|
5/2 | 2 |
\( (3/8) r^{5/2}\)
|
\(-(1/20) r^{5/2}\)
|
7/2 | 1 |
\(-(1/42) r^{7/2}\)
|
\( (4/21) r^{7/2}\)
|
7/2 | 2 |
\( (5/12) r^{7/2}\)
|
\( (1/42) r^{7/2}\)
|
9/2 | 1 |
\( (1/72) r^{9/2}\)
|
\( (5/36) r^{9/2}\)
|
9/2 | 2 |
\( (7/16) r^{9/2}\)
|
\(-(1/72) r^{9/2}\)
|
11/2 | 1 |
\(-(1/110) r^{11/2}\)
|
\( (54/495) r^{11/2}\)
|
11/2 | 2 |
\( (9/20) r^{11/2}\)
|
\( (1/110) r^{11/2}\)
|
13/2 | 1 |
\( (1/156) r^{13/2}\)
|
\( (19,943/222,222) r^{13/2}\)
|
13/2 | 2 |
\( (11/24) r^{13/2}\)
|
\(-(1/156) r^{13/2}\)
|
\(\varTheta \)
|
\(\sigma \)
|
\(R_{1}\)
|
\(R_{2}\)
|
\(C^{\mathrm{inner}}\)
|
\(C^{\mathrm{outer}}\)
|
---|---|---|---|---|---|
\(3 \pi / 4\)
| 20 | 0.01 | 200 | 0.6867 | 2.1526 |
\( \pi / 2\)
| 20 | 0.05 | 500 | 0.5833 | 34.7123 |
\( \pi / 4\)
| 20 | 0.5 | 500 | 0.1331 | 337.4507 |
\( \pi / 2\)
| 40 | 0.05 | 500 | 0.6040 | 76.8044 |
\( \pi / 4\)
| 40 | 0.5 | 500 | 0.1537 | 1084.7693 |