x + yi = c[root][sin ([xi] + [eta]i)] (17)
the function
[psi] = Qc sh ½([eta] - [alpha]) sin ½([xi] - [beta]) (18)
will give motion streaming past the fixed cylinder [eta] = [alpha],
and dividing along [xi] = [beta]; and then
x² - y² = c² sin [xi] ch [eta], 2xy = c² cos [xi] sh [eta]. (19)
In particular, with sh [alpha] = 1, the cross-section of [eta] =
[alpha] is
x^4 + 6x²y² + y^4 = 2c^4, or x^4 + y^4 = c^4 (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 [psi]1 is to be made to satisfy the conditions
(i.) [nabla]²[psi]1 = 0,
(ii.) [psi]1 + ½Rx² = ½Ra², or [psi]1 = 0 when x = ±a,
(iii.) [psi]1 + ½Rx² = ½Ra², [psi]1 = ½R(a² - x²), when y = ± b.
Expanded in a Fourier series,
32 __ cos (2n + 1) ½[pi]x/a
a² - x² = ----- a² \ ---------------------, (1)
[pi]³ /__ (2n + 1)³
so that
16 __ cos (2n + 1) ½[pi]x/a · ch(2n + 1) ½[pi]y/a)
[psi]1 = ----- a² \ ---------------------------------------------,
[pi]³ /__ (2n + 1)^3 · ch(2n + 1) ½[pi]b/a
16 __ cos (2n + 1) ½[pi]z/a
w1 = [phi]1 + [psi]1i = iR ----- \ ------------------------------, (2)
[pi]³ /__ (2n + 1)^3 ch(2n + 1) ½[pi]b/a
an elliptic-function Fourier series; with a similar expression for
[psi]2 with x and y, a and b interchanged; and thence [psi] = [psi]1 +
[psi]2.
_Example 4._--Parabolic cylinder, axial advance, and liquid streaming
past.