v² = 2gh, and [omega]/[Omega] = c_c;
.: [omega]/[Omega] = ½ = c_c;
a result confirmed by experiment with mouthpieces of this kind. A
similar theoretical investigation is not possible for orifices in
plane surfaces, because the velocity along the sides of the vessel in
the neighbourhood of the orifice is not so small that it can be
neglected. The resultant horizontal pressure is therefore greater than
Gh[Omega], and the contraction is less. The experimental values of the
coefficient of discharge for a re-entrant mouthpiece are 0.5149
(Borda), 0.5547 (Bidone), 0.5324 (Weisbach), values which differ
little from the theoretical value, 0.5, given above.
[Illustration: FIG. 40.]
§ 38. _Velocity of Filaments issuing in a Jet._--A jet is composed of
fluid filaments or elementary streams, which start into motion at some
point in the interior of the vessel from which the fluid is
discharged, and gradually acquire the velocity of the jet. Let Mm,
fig. 40 be such a filament, the point M being taken where the velocity
is insensibly small, and m at the most contracted section of the jet,
where the filaments have become parallel and exercise uniform mutual
pressure. Take the free surface AB for datum line, and let p1, v1, h1,
be the pressure, velocity and depth below datum at M; p, v, h, the
corresponding quantities at m. Then § 29, eq. (3a),
v1²/2g + p1/G - h1 = v²/2g + p/G - h (1)
But at M, since the velocity is insensible, the pressure is the
hydrostatic pressure due to the depth; that is v1 = 0, p1 = p_a + Gh1.
At m, p = p_a, the atmospheric pressure round the jet. Hence,
inserting these values,