[Omega] = log [zeta] = log(Q/q) + [theta]i (4)
so that the polygon representing [Omega] conformally has a boundary
given by straight lines parallel to the coordinate axes; and then to
determine [Omega] and w as functions of a variable u (not to be
confused with the velocity component of q), such that in the conformal
representation the boundary of the [Omega] and w polygon is made to
coincide with the real axis of u.
[Illustration: FIG. 4.]
It will be sufficient to give a few illustrations.
Consider the motion where the liquid is coming from an infinite
distance between two parallel walls at a distance xx´ (fig. 4), and
issues in a jet between two edges A and A´; the wall xA being bent at
a corner B, with the external angle [beta] = ½[pi]/n.
The theory of conformal representation shows that the motion is given
by
_ _
| [root](b - a´·u - a) + [root](b - a·u - a´) |^1/n
[zeta] = | ------------------------------------------- | , u = ae^(-[pi]w/m); (5)
|_ [root](a - a´·u - b) _|
where u = a, a´ at the edge A, A¹; u = b at a corner B; u = 0 across
xx´ where [phi] = [oo]; and u = [oo], [phi] = [oo] across the end JJ´
of the jet, bounded by the curved lines APJ, A´P´J´, over which the
skin velocity is Q. The stream lines xBAJ, xA´J´ are given by [psi] =
0, m; so that if c denotes the ultimate breadth JJ´ of the jet, where
the velocity may be supposed uniform and equal to the skin velocity Q,
m = Qc, c = m/Q.
If there are more B corners than one, either on xA or x´A´, the
expression for [zeta] is the product of corresponding factors, such as
in (5).
Restricting the attention to a single corner B,
/ Q \^n [root](b - a´·u - a) + [root](b - a·u - a´)
[zeta]^n = ( --- ) (cos n[theta] + i sin n[theta] = -------------------------------------------, (6)
\ q / [root](a - a´.u - b)
/ Q \^n / Q \^n
ch n[omega] = ch log( --- ) cos n[theta] + i sh log ( --- ) sin n[theta]
\ q / \ q /
/b - a´ /u - a´
= ½([zeta]^n + [zeta]^(-n)) = / ------ / ------ (7)
\/ a - a´ \/ u - b´
/ Q \ / Q \^n
sh n[Omega] = sh log ( --- ) cos n[theta] + i ch log ( --- ) sin n[theta]
\ q / \ q /
/b - a´ /u - a´
= ½([zeta]^n - [zeta]^(-n)) = / ------ / ------ (8)
\/ a - a´ \/ u - b´
[oo] > a > b > 0 > a´ > -[oo] (9)
and then
d[Omega] 1 [root](b - a´·b - a´) dw m
-------- = - -- ---------------------------, -- = - ------ (10)
du 2n (u - b)[root](a - a·u - a´) du [pi]u´
the formulas by which the conformal representation is obtained.
For the [Omega] polygon has a right angle at u = a, a´, and a zero
angle at u = b, where [theta] changes from 0 to ½[pi]/n and [Omega]
increases by ½i[pi]/n; so that
d[Omega] A [root](b - a·b - a´)
-------- = ---------------------------, where A = --------------------. (11)
du (u - b)[root](u - a·u - a´) 2n
And the w polygon has a zero angle at u = 0, [oo], where [psi] changes
from 0 to m and back again, so that w changes by im, and
dw B m
-- = ---, where B = - ----. (12)
du u [pi]
Along the stream line xBAPJ,
[psi] = 0, u = ae^(-[pi][phi]/m); (13)
and over the jet surface JPA, where the skin velocity is Q,
d[phi]
------ = -q = -Q, u = ae^([pi]sQ/m) = ae^([pi]s/c), (14)
ds
denoting the arc AP by s, starting at u = a;
/b - a´ /u - a´
ch n[Omega] = cos n[theta] = / ----- / ------ (15)
\/ a - a´ \/ u - b´
/a - b /u - a´
sh n[Omega] = i sin n[theta] = i / ------ / ------ (16)
\/ a - a´ \/ u - b´
[oo] > u = ae^([pi]s/c) > a, (17)
and this gives the intrinsic equation of the jet, and then the radius
of curvature
ds 1 d[phi] i dw i dw / d[Omega]
[rho] = - -------- = --- -------- = --- ------- = --- -- / -------
d[theta] Q d[theta] Q d[Omega] Q du / du
c u - b [root](u - a·u - a´)
= ----·2n----- --------------------, (18)
[pi] u [root](a - b·b - a´)
not requiring the integration of (11) and (12)
If [theta] = [alpha] across the end JJ´ of the jet, where u = [oo], q
= Q,
/b - a´ /a - b
ch n[Omega] = cos n[alpha] = /-------, sh n[Omega] = i sin n[alpha]= / ------, (19)
\/ a - a´ \/ a - a´