(20)
when the axes are turned through 45°.
33. Example 3.—Analysing in this way the rotation of a rectangle filled with liquid into the two components of shear, the stream function ψ1 is to be made to satisfy the conditions
| (i.) ∇2ψ1 = 0, (ii.) ψ1 + ½Rx2 = ½Ra2, or ψ1 = 0 when x = ±a, (iii.) ψ1 + ½Rx2 = ½Ra2, ψ1 = ½R (a2 − x2), when y = ± b. |
Expanded in a Fourier series,
| a2 − x2 = | 32 | a2 Σ | cos (2n + 1) ½ πx/a | , |
| π3 | (2n + 1)3 |
(1)
so that
| ψ1 = R | 16 | a2 Σ | cos (2n + 1) ½πx/a · ch (2n + 1) ½πy/a) | , |
| π3 | (2n + 1)3 · ch (2n + 1) ½πb/a |
| w1 = φ1 + ψ1i = iR | 16 | a2 Σ | cos (2n + 1) ½πz/a | , |
| π3 | (2n + 1)3 ch (2n + 1) ½πb/a |