a - b·b - a´ a - a´
cos 2n[alpha] - cos 2n[theta] = 2------------ = ½ sin² 2n[alpha]------
a - a´·u - b u - b
[root](a - b.b - a´)[root](u - a·u - b´)
sin 2n[theta] = 2---------------------------------------- (20)
a - a´·u - b
[root](a - a·b - a´)
= sin 2n[alpha]--------------------;
u - b
2n c / b \ [root](a - b·b - a´)
----- ----- = ( 1 + ----- ) -------------------- (21)
[phi] [rho] \ u - b/ [root](u - a·u - a´)
a - a´ + (a + a´) cos 2n[alpha] - [a + a´ + (a - a´) cos 2n[alpha] cos 2n[theta]
= --------------------------------------------------------------------------------
(a - a´) sin² 2n[alpha]
cos 2n[alpha] - cos 2n[theta]
× -----------------------------
sin 2n[theta]
Along the wall AB, cos n[theta] = 0, sin n[theta] = 1,
a > u > b, (22)
/ Q \^n /b - a´ /a - u
ch n[Omega] = i sh log ( --- ) = i / ------ / ------ (23)
\ q / \/ a - a´ \/ u - b´
/ Q \^n /a - b /u - a´
sh n[Omega] = i ch log ( --- ) = i / ------ / ------ (24)
\ q / \/ a - a´ \/ u - b´
ds ds d[phi] m c Q
-- = ------ ------ = ------ = --- -- (25)
du d[phi] dt [pi]qu [pi] qu
_
AB / a Q du
[pi]-- = | --- --
c _/ b q u
_ _ _
/ | [root](a - b)[root](u - a´) + [root](b - a´)[root](a - u) |^1/n du
= | | --------------------------------------------------------- | --. (26)
_/ |_ [root](a - a´)[root](u - b´) _| u
Along the wall Bx, cos n[theta] = 1, sin n[theta] = 0,
b > u > 0 (27)
/ Q \^n /b - a´ /a - u
ch n[Omega] = ch log ( --- ) = / ------ / ------ (28)
\ q / \/ a - a´ \/ b - u´
/ Q \^n /a - b /u - a´
sh n[Omega] = sh log ( --- ) = / ------ / ------. (29)
\ q / \/ a - a´ \/ b - u
At x where [phi] = [oo], u = 0, and q = q0,
/ Q \^n /b - a´ / a /a - b / -a´
( --- ) = / ------ / --- + / ------ / ---. (30)
\ q0 / \/ a - a´ \/ b \/ a - a´ \/ q
In crossing to the line of flow x´A´P´J´, [psi] changes from 0 to m,
so that with q = Q across JJ´, while across xx´ the velocity is q0, so
that
m = q0.xx´ = Q.JJ´ (31)
_ _
JJ´ q0 | /b - a´ / a /a - b / -a´ |^1/n
-- = --- = | / ------ / --- - / ------ / --- | , (32)
xx´ Q |_ \/ a - a´ \/ b \/ a - a´ \/ q _|
giving the contraction of the jet compared with the initial breadth of
the stream.
Along the line of flow x´A´P´J´, [psi] = m, u = a´e^(-[pi][phi]/m),
and from x´ to A´, cos n[theta] = 1, sin n[theta] = 0,
/ Q \^n /b - a´ /a - u
ch n[Omega] = ch log ( --- ) = / ------ / ------, (33)
\ q / \/ a - a´ \/ b - u´
/ Q \^n /a - b /u - a´
sh n[Omega] = sh log ( --- ) = / ------ / ------. (34)
\ q / \/ a - a´ \/ b - u´