CHAPTER II.

FLOW OF WATER UNDER A VARIABLE HEAD.

83. Flow of Water when the Level is variable upon one or both Faces of the Orifice of Discharge.—When a reservoir, instead of being maintained constantly full, as we have supposed it to be hitherto, receives no supply, or receives less than it discharges through an orifice in the bottom, the surface of the fluid gradually descends, and the tank or reservoir is at length emptied. The laws of the discharge are in this case different from those which have been stated in the first chapter, and the questions to be resolved are of a different character.

The form of the vessels may be also, either prismatic —that is, of identical sections at every height of the surface-or having sides sloping at some known inclination.

84. Ratio between the Velocities at the Orifice and in the Vessel.-Let us suppose that the fluid contained in a prismatic vessel be divided into extremely small horizontal sections, and that they descend parallel to each other, the particles of the fluid in each of the sections must then have the same velocity. This is the hypothesis of the parallelism of the horizontal sections, admitted, and perhaps too much extended, by many hydraulicians.

Let v be the velocity of the particles in the vessel; V that which they have at the orifice; A the horizontal section of the reservoir or vessel containing the water; S, or rather mS, that of the orifice; m being the coefficient of contraction, the volume of water which flows out in the indefinitely small time will be expressed by mSVT.

During this same time the surface of the water descends by a quantity vr, and the corresponding value of the volume of water is Avr = mSVτ, or v : V:: mS : A, giving an example of that hydraulic axiom-namely, that the velocities are in the inverse ratio of the various transverse sections.

85. Head due to the Velocity of the Water at its Point of Discharge. The velocity V of the issuing fluid does not now maintain the same constant rate. It is uniform only for a given instant; for, besides being due to the actual head at the given instant, the velocity Vis a consequence of the velocity v acquired during the descent of the parallel sections above mentioned: the two velocities acting in the same direction, from above downwards, the result Va

is equal to their sum. Thus, if H' =

2g

be the height

due to the velocity of the water at its point of discharge, H being always the actual head in the vessel, we shall have

When mS is small compared with A, as is generally the case, m2 S2, with regard to A2, may be neglected; so that H' = H, that is, the velocity of issue at any given instant is that due to the actual head at that same moment. In this chapter it is always assumed to be so,' although the hypothesis of the parallelism of the horizontal sections, however admissible in their descent, does not hold good when they have arrived near the orifice, the circumstances of the movement of the molecules of the fluid become then very complicated, and are indeed entirely unknown.

86. Nature of the Motion.-Let M (Fig. 30) represent

M

a

H

a vessel of water filled up to AB; let us divide the height from B to the orifice D into a great number of equal parts, Ba, d' ab, bc, &c. Suppose, then, that a body, P, were impelled from ' below upwards with a velocity such that it rises to the point H, PH being equal to DB, and let us divide PH into the same number of equal parts.

Fig. 30.

D

In proportion as the body rises, its velocity will diminish, in such a manner that when it shall have arrived successively at the points a', b', c', the velocities will be respectively Ha', ✔Hb', ✓Hc'

o, as is shown in Recurring to the

works on the Elements of Mechanics. fluid contained in the vessel M, in proportion as it flows out, the surface AB is lowered; and when it shall have successively reached the points a, b, c, the respective velocities of the issuing water will be (§ 85) as ✓Da, √Dỏ, ✓Dc... o, or, according to the construction, as their equals ✓Ha', ✓/Hb',√/Hc' ... o; so that, in proportion as the vessel is emptied, the velocity of the discharge will decrease down to zero, following the same law as the velocity of the body impelled from below upwards, each being an example of an uniformly retarded motion; consequently, the discharge also will be governed by the same law.

It will be the same, also, in the descent of the surface of the water in the vessel, which will be uniformly retarded, its velocity being in a constant ratio to that at the orifice, namely, as the section of the orifice to the area of the surface of the water.

87. Volume discharged.-According to the laws of an

uniformly retarded motion, when a body, starting with a certain velocity, loses it gradually until it is reduced to zero, it only describes one-half the space it would have traversed in the same time if it had moved uniformly with the velocity with which it commenced the motion. Now the volume of water which flows out from any vessel until it is all discharged may be regarded as a prism, whose base is the orifice, and height the space which the first issuing particles would describe, with a uniformly retarded motion identical with that by which the discharge takes place; but if the same particles had always preserved their initial velocity (which is that due to the primary charge), the space described in the same time, or the height of the prism, and, consequently, the volume of water discharged, would have been doubled. Hence this theorem :-The volume of water which passes through an orifice at the bottom of a prismatic vessel, receiving no supply, and therefore becoming empty, is only one-half of that which would be given during the time of complete discharge, if the flow had taken place under a constant charge equal to the primary.

88. Time which is required to empty a vessel.-Let H be this charge; A the horizontal section of the vessel; T the time which it may require to be completely discharged. The volume of water discharged during this time-that is to say, all the water the vessel contains (above the orifice)-is Ax H. The volume, according to the theorem above, which would have been discharged in that time under the constant charge H, would have been 2 (A x H). This same volume, or the discharge during the time T, is also equal to mST √ 2gH.

We may use the Italic capital H instead of the Roman H, conventionally applied hitherto in this formula (844), since the orifice is now supposed to be in the bot

tom of the vessel, and therefore H and H are identical. Equating these two values, we have

2AH =mST √2gH,

and solving for T, we have

and dividing above and below by H, we have, finally—

If we represent by T" the time which the volume AH would take to flow out under the constant head H, we should have had (§ 14)—

consequently, T= 2T'; that is to say, the time which a prismatic vessel takes to be completely discharged is double that in which the same volume would flow out, if the head had remained constantly the same as it was at the commencement of the discharge.

89. Time which the Surface of the Water takes to descend a given Depth.-Let t be the time sought in which the level descends the given depth a: now the time in which the whole volume would be discharged is (§ 88)

the head at the commencement being H; and putting—

H-a-h

for the head at the end of t, we have the time in which