then, taking has the depth of the orifice below the surface, and II as the atmospheric pressure, But, if A be the area of the surface, and K of the orifice, and if the motions of all the issuing particles be supposed perpendicular to the plane of the orifice, AU= Ku, since the quantity of fluid poured in at the surface in any time is equal to the quantity which passes through the orifice in the same time; K If the orifice be very small, the ratio may be neglected, and, approximately, u=√ A 128. In any case of steady motion, if gravity be the only force in action, we have Suppose the orifice not in the base of the vessel, and so small that the velocities of all the particles passing through it are sensibly the same; we then have, as in the previous case, u2 = U2+2gh, AU= Ku, and approximately, u=√(2gh). If the vessel be not kept constantly full, the motion will not be steady, but when the orifice is very small, it may be taken as being approximately steady, and the expression √(2gh) may be employed as the velocity of the issuing fluid. Looking upon the issuing fluid as a series of particles in motion under the action of gravity, every particle moves in a parabolic path, and the issuing fluid takes the form of a parabolic arc. Moreover sinee the velocity at the orifice is, approximately, that due to the height h, the directrix of the parabola is approximately coincident with the surface of the fluid. 129. PROP. To find the time in which a given quantity of fluid will flow through a small orifice. At the time t, let x be the height of the surface above the orifice, and X its area. X being a known function of x, this equation gives t in terms of x, and therefore x in terms of t. It will be seen hereafter that, in certain cases, particularly when the containing vessel is formed of a thin substance, a considerable modification of the value of k, employed in the preceding process, is requisite, in order to obtain results in approximate accordance with observations. 130. Ex. 1. A hollow cone, having its axis vertical, is filled with water; required to find the time in which it will be emptied through a small aperture at its vertex. In this case, X = x2 tan2 a, taking 2a as the vertical angle; and, if h be the height of the cone, the time (t) in which it will If the cone had been kept constantly full, the velocity at the orifice would have been always √(2gh), and the same quantity of liquid would have flowed out in a time 7, such that тK √(2gh) = }πh3 tan2 a ; hence we obtain t 7:: 6 : 5. Ex. 2. A vessel, in the form of a surface of revolution, has a small aperture at its lowest point; determine its form so that the surface of water, contained in it, may descend uniformly. We must have dx constant, and therefore constant; but, if y=f(x) be the generating curve, X=πу2, and therefore constant: hence the generating curve is one of the class y* = a3x, is the velocity of descent being determined by the value of a. This example contains the theory of the Clepsydra or waterclock. The Hypothesis of Parallel Sections. 131. Suppose the interior of a vessel to be a surface of revolution, the axis of which is vertical; and suppose moreover that the inclination to the vertical of the generating curve is always small, and does not change rapidly. If such a vessel contain fluid, which is allowed to flow out through a horizontal aperture in its base, it is evident that the fluid particles will move in directions nearly vertical, and the velocities of all particles in the same horizontal plane will be very nearly the same. The discussion of the real motions in such a case would be excessively complicated, but an approxi mate solution may be obtained by means of the hypothesis, that the successive horizontal laminæ of fluid descend vertically, and replace each other in succession, that is, that the motions of all the fluid particles in a horizontal plane are the same, and all vertical. This is the hypothesis of parallel sections, and it is clearly equivalent to the neglecting of all horizontal motions, and of the changes which take place in the component particles of the descending laminæ of fluid. If the orifice be much less than the horizontal base of the vessel, the motions of the particles near the base cannot be all vertical, and the same, in the same horizontal plane; the hypothesis therefore will not even approximately hold good. In order however to obtain a solution of the question, the hypothesis will be made throughout, and a large allowance must therefore be made for the probable error arising from this cause. Under this head we shall discuss the following problems. 132. I. A vase in the form of a surface of revolution, and having a finite horizontal aperture in its base, is kept constantly full; required to determine the rate at whick fluid must be poured in. Let A be the area of the top of the vase, K of the aperture, and h the depth of the vase. At a depth z below the surface, where Z is the area of the horizontal section, let v be the velocity at the time t, and, at the same time, let U be the velocity at the surface and u at the aperture. Then, the fluid being supposed incompressible, the same quantity must pass through any horizontal section in the same element of time St; .. Udt. AudtK = vôt. Z, or AU-Ku = Zv. B. H. 18 |