Along the jet surface A´J´, q = Q,
/b - a´ /a - u
ch n[Omega] = cos n[theta] = / ------ / ------, (36)
\/ a - a´ \/ b - u´
/a - b /a´ - u
sh n[Omega] = i sin n[theta] = i / ------ / ------, (37)
\/ a - a´ \/ b - u´
a´ > u = a´e^([pi]/sc) > -[oo], (38)
giving the intrinsic equation.
41. The first problem of this kind, worked out by H. v. Helmholtz, of
the efflux of a jet between two edges A and A1 in an infinite wall, is
obtained by the symmetrical duplication of the above, with n = 1, b =
0, a´ = -[oo], as in fig. 5,
/u - a / -a
ch [Omega] = / -----, sh [Omega] = / ---; (1)
\/ u \/ u
and along the jet APJ, [oo] > u = ae^([pi]s/c) > a,
sh [Omega] = i sin [theta] - i[root](a/u) = ie^(-½[pi]s/c), (2)
_ _
/ [oo] / c c
PM = | sin [theta] ds = | e^(-½[pi]s/c) ds = ----- e^(-½[pi]s/c) = ----- sin [theta], (3)
_/ s _/ ½[pi] ½[pi]
so that PT = c/½[pi], and the curve AP is the tractrix; and the
coefficient of contraction, or
breadth of the jet [pi]
---------------------- = --------. (4)
breadth of the orifice [pi] + 2
A change of [Omega] and [theta] into n[Omega] and n[theta] will give
the solution for two walls converging symmetrically to the orifice AA1
at an angle [pi]/n. With n = ½, the reentrant walls are given of
Borda's mouthpiece, and the coefficient of contraction becomes ½.
Generally, by making a´ = - [oo], the line x´A´ may be taken as a
straight stream line of infinite length, forming an axis of symmetry;
and then by duplication the result can be obtained, with assigned n,
a, and b, of the efflux from a symmetrical converging mouthpiece, or
of the flow of water through the arches of a bridge, with wedge-shaped
piers to divide the stream.
[Illustration: FIG. 5.]
[Illustration: FIG. 6.]
42. Other arrangements of the constants n, a, b, a´ will give the
results of special problems considered by J. M. Michell, _Phil.
Trans._ 1890.
Thus with a´ = 0, a stream is split symmetrically by a wedge of angle
[pi]/n as in Bobyleff's problem; and, by making a = [oo], the wedge
extends to infinity; then
/ b / n
ch n[Omega] = / -----, sh n[Omega] = / -----. (1)
\/ b - u \/ b - u
Over the jet surface [psi] = m, q = Q,
u = - e^([pi][phi]/m) = - be^([pi]²/c),
/ 1 / e^([pi]²/c)
ch [Omega] = cos n[theta] = / ---------------, sh [Omega] = i sin n[theta] = i / ---------------, (2)
\/ e^([pi]²/c) + 1 \/ e^([pi]²/c) + 1
½[pi] ds 2n
e^(½[pi]²/c) = tan n[theta], ----- -------- = -------------. (3)
c d[theta] sin 2n[theta]
For a jet impinging normally on an infinite plane, as in fig. 6, n = 1,
e^(½[pi]²/c) = tan [theta], ch (½[pi]s/c) sin 2[theta] = 1, (4)
sh ½[pi]x/c = cot [theta], sh ½[pi]y/c = tan [theta],
sh ½[pi]x/c sh ½[pi]y/c = 1, e^(½[pi](x + y)/c) = e^(½[pi]x/c) + e^(½[pi]y/c) + 1. (5)
With n = ½, the jet is reversed in direction, and the profile is the
catenary of equal strength.
In Bobyleff's problem of the wedge of finite breadth,
/ b /u - a /b - a / u
ch n[Omega] = / --- / -----, sh n[Omega] = / ----- / -----, (6)
\/ a \/ u - b \/ a \/ u - b
/ b /a - b
cos n[alpha] = / ---, sin n[alpha] = / -----, (7)
\/ a \/ a
and along the free surface APJ, q = Q, [psi] = 0, u = e^(-[pi][phi]/m) = ae^([pi]s/c),
/ e^([pi]²/c) - 1
cos n[theta] = cos n[alpha] / ---------------------------,
\/ e^([pi]²/c) - cos² n[alpha]
cos² n[alpha] sin² n[theta]
e^([pi]²/c) = -----------------------------, (8)
sin² n[theta] - sin² n[alpha]
the intrinsic equation, the other free surface A´P´J´ being given by
cos² n[alpha] sin² n[theta]
e^([pi]²/c) = -----------------------------. (9)
sin² n[alpha] - sin² n[theta]
Putting n = 1 gives the case of a stream of finite breadth disturbed
by a transverse plane, a particular case of Fig. 7.
When a = b, [alpha] = 0, and the stream is very broad compared with
the wedge or lamina; so, putting w = w´(a - b)/a in the penultimate
case, and
u = ae^(-w) [asympt] a - (a - b)w´, (10)
/w´ + 1 / 1
ch n[Omega] = / ------, sh n[Omega] = / --------, (11)
\/ w´ \/ [root]w´
in which we may write
w´ = [phi] + [psi]i. (12)
Along the stream line xABPJ, [psi] = 0; and along the jet surface APJ,
-1 > [phi] > -[oo]; and putting [phi] = -[pi]s/c - 1, the intrinsic
equation is
[pi]s/c = cot² n[theta], (13)
which for n = 1 is the evolute of a catenary.
[Illustration: FIG. 7.]
43. When the barrier AA´ is held oblique to the current, the stream
line xB is curved to the branch point B on AA´ (fig. 7), and so must
be excluded from the boundary of u; the conformal representation is
made now with
d[Omega] [root](b - a·b - a´)
-------- = - ---------------------------- (1)
du (u - b) [root](u - a·u - a´)
dw m 1 m´ 1 m + m´ u - b
-- = - ---- ----- - ---- -----, = - ------ · -------------,
du [pi] u - j [pi] u - j [pi] u - j·u - j´
mj´ + m´j
b = ---------, (2)
m + m´
taking u = [oo] at the source where [phi] = [oo], u = b at the branch
point B, u = j, j´ at the end of the two diverging streams where [phi]
= -[oo]; while [psi] = 0 along the stream line which divides at B and
passes through A, A´; and [psi] = m, -m´ along the outside boundaries,
so that m/Q, m´/Q is the final breadth of the jets, and (m + m´)/Q is
the initial breadth, c1 of the impinging stream. Then
/b - a´ /u - a /b - a /u - a´
ch ½[Omega] = / ------ / -----, sh ½[Omega] = / ------ / ------, (3)
\/ a - a´ \/ u - b \/ a - a´ \/ u - b
2b - a - a´ N
ch [Omega] = ----------- - -----,
a - a´ u - b
/ [root](2·a - u·u - a´)
sh [Omega] = / N----------------------,
\/ u - b
a - b·b - a´
N = 2------------. (4)
a - a´