v2 = 2gh, and ω/Ω = cc;
∴ ω/Ω = 1⁄2 = cc;
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 GhΩ, 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.
| Fig. 40. | Fig. 41. |
§ 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 = pa + Gh1. At m, p = pa, the atmospheric pressure round the jet. Hence, inserting these values,
0 + pa/G + h1 − h1 = v2/2g + pa / G − h;
v2/2g = h;