a differential equation from which a may in some cases be found

and, if A' be small compared with A, the second and third terms may be neglected, and an approximate equation obtained for u2,

i.e.

u'2 = 2g (y' -7).

Suppose the tube, as in the figure, to end in an orifice from which the fluid issues. If we take the origin at the orifice, we shall have a'=0, y'=0, A' constant, and y negative, and, as before, the velocity at the lowest point of the tube will be approximately the velocity due to the depth below the surface.

135. The motion of an incompressible fluid in an uniform tube of small section.

In this case v is the same at all points of the tube, and

dv

therefore =0, and the equation of motion is

ds

Let gravity be the only force in action, and measure z ver

and

P = C-gz - at
P

dv

S.

Taking a and a' as the extreme values of s, y and y' of z, we obtain

or, if I be the length of the filament of fluid,

Ex. Liquid rests in a fine tube, the axis of which is a circle in a vertical plane; the fluid being slightly disturbed, required the time of a small oscillation.

Let the filament of fluid subtend an angle 2a at the centre, and at the time t let be the angular distance from the vertical of the middle point of the filament.

Then y y'a {cos (a — 0) — cos (a + 0)} = 2a sin a sin 0,

and, if the original displacement be small, the time of a small oscillation is

136. In each of the three preceding problems we have seen that, when the orifice is small, the velocity of efflux is approximately the velocity due to the depth of the orifice below the surface. This is in accordance with the result of Art. (127), in which it is assumed that the motion is steady in all cases in

B. H.

19

which the orifice is small, and we are therefore now enabled, by observing the quantities neglected in making the approximations, to estimate the amount of error involved in taking the hypothesis of steady motion for such cases.

The case of a small orifice, not in a horizontal plane, may be illustrated by the third problem, Art. (134).

For it may be easily conceived that the fluid below the orifice will be almost entirely at rest, and that the issuing

stream, or the central portion of it, will, before its efflux, have been flowing through the fluid in the vessel in a somewhat tubular form, so that its motion may be considered as the motion of a fluid in a tube, the section of which continually diminishes near the orifice; and therefore the result of the problem referred to may be applied to the confirmation of the result before obtained on the hypothesis of steady motion.

The contracted vein.

137. When fluid issues through a small orifice in the thin base of a vessel, it is observed that the issuing stream is not cylindrical, but, near the orifice, is contracted so that its sectional area is less than the area of the orifice. The stream then expands and afterwards, as it descends, again diminishes gradually in size.

The sudden diminution of the issuing stream forms what is called the 'contracted vein,' and is due to the oblique or nearly horizontal motions of the particles near the edges of the orifice just before their efflux.

The after contraction, which is gradual, is due to the law of continuity, which requires that the mean velocity of the particles

in any horizontal section of the issuing stream should vary inversely as the area of the section, and therefore that, as the velocity increases in the descent, the area of the section should diminish.

138. The discrepancy which exists between the results of theory and experiment is to a great extent accounted for by the contraction of the vein or filament of issuing fluid, and it is found moreover, as would be anticipated, that the amount of difference depends upon the nature of the orifice.

For instance, if the orifice be simply an opening in the side of the vessel, and if the side be very thin, the quantity of fluid which flows out in a given time is about ths of the quantity given by the theory. Again, when the fluid issues through a cylindrical aperture of sensible length, formed by attaching to the orifice, externally, a small hollow cylinder, the ratio is found to be about ths; but, if the cylinder be attached internally, the rate of efflux is about one half the theoretical rate*

The rate of efflux depends upon the area of the orifice and the velocity of the issuing stream; it is shewn by experiment that the latter is, in general, not very different from the theoretical velocity, and the observed error in the rate of efflux is therefore to a great extent accounted for by the formation of the 'contracted vein.'

An account of experiments, made by Bossut and others, on the efflux of fluids through orifices of various kinds, is given in the Encyclopædia Metropolitana, Hydrodynamics, p. 207.

* Poisson, Mécanique, Art. 676.

139. In all the preceding investigations the containing vessel has been supposed to have the form of a surface of revolution, but they are evidently applicable to the case of a vessel of any form, the horizontal section of which does not change very rapidly, and the symbols employed (K, Z, &c.), being perfectly general, no correction is requisite for the application of the formulæ to such cases.

Motion of Elastic Fluids.

140. If elastic fluid move in a tube the section of which does not change rapidly in size, we may make use of the hypothesis of parallel sections as before.

Assuming the motions of all the particles in any one section to be sensibly in the same direction, parallel to the axis of the tube, and neglecting gravity, the action of which will not sensibly affect the pressure, the equation of motion is

where v is the velocity in a section at a distance x from a fixed section.

The equation of continuity, depending on the hypothesis which neglects all motions but those perpendicular to the section, is determined as follows.

Let X be the area of the section, the velocity of the particles passing through which is v, and p the density about this section at the time t.

Then pvXst is the mass of fluid which flows across the section in the time St;

is the quantity which flows across the section defined by the

distance xSx, and

d

dx

(pvX) Sx St is the increase of the

quantity of fluid in the volume Xdx during the time dt, which is also given by the expression