7. To find the time in which a paraboloid of revolution, full of fluid, will empty itself through a small orifice in its vertex, the axis of the paraboloid being vertical.

If h denote the altitude of the vessel and 7 the latus rectum, the time of efflux will be equal to

3 (2g) k

Encycl. Metrop. Mix. Sc., vol 1. p. 205.

8. To find the time in which the fluid, in a vertical prismatic vessel, will discharge itself through a small orifice in its base. If a denote the initial depth of the fluid, and A the area of a horizontal section of the prism, the time required will be equal to

2 Aa

k(2g)

Bossut Traité d'Hydrodynamique, tom. 1. p. 259, 260.

Encycl. Metrop. Mix. Sc., vol. 1. p. 204.

9. To find the time in which the surface of the fluid in a conical vessel will subside to half its original altitude above the vertex through a small orifice in the vertex, the axis of the cone being vertical.

If h denote the height of the cone and r the radius of its base, the required time will be equal to

wr2h3 (2a — 1).

20kgs

10. Fluid issues from innumerable holes in the surface of a vertical cylinder kept constantly full: to determine the form of the bounding surface of the issuing fluid.

The bounding surface will be a truncated cone, its narrow end being the upper end of the cylinder; and the vertex of the entire cone will be in the axis of the cylinder produced upwards, at a distance from the upper end of the cylinder equal to its radius.

11. To find the time in which a vessel, formed by the revolution of a cycloid about its axis, which is vertical, will empty

itself, after being filled with fluid, through a small orifice at its

vertex.

If a denote the radius of the generating circle, the time of efflux will be equal to

Moseley's Hydrostatics, p. 154.

12. To find the time in which the fluid contained in a vessel, formed by the revolution of the curve y1 = a3x about the axis of x, will descend through any proposed space, the axis of the vessel being vertical, supposing a small orifice to be made at the

vertex.

If s denote the space through which the fluid sinks in a time t,

which shews that equal spaces are described in equal times. John Bernoulli: Hydraulica Pars 2, Opera, tom. IV. p. 481. Mariotte: Mouv. des Eaux, Part. III. disc. 3.

Varignon: Mem. Acad. Par. 1694, 1699.

Bossut: Traité d' Hydrodynamique, tom. I. p. 255.

SECTION II.

Small Orifices. General Conditions.

In the preceding section we have considered the problem of the efflux of fluids through small orifices, supposing gravity to be the only acting force and the fluid to issue into the same atmosphere with which its surface is in contact. In this section we shall consider the problem under a more general point of view.

Let p denote the pressure at any point in the fluid, u the velocity of the fluid at this point, X, Y, Z, the components of the accelerating force on the molecule of fluid parallel to three

fixed rectangular axes. Then, by the equation of steady motion, which, the orifice being very small, may be here applied without sensible error,

Ρ

=

= P(Xda + Ydy + Zdz) - \pi + C

C denoting the constant introduced by the integration.

If p', p", denote the units of pressure to which the surface of the fluid and the issuing fluid are respectively subject; then, u being of inconsiderable magnitude at the surface, which is supposed to be nearly of the form of equilibrium during the whole motion,

p = P(Xưa + +
p = pl(Xd + Ydy + Zdã)+C..........(1),

x, y, z, in the integral representing the coordinates of any point in the surface of the fluid; and

p = |Xdz + Ydy + Zdz) - xp + C...... (2),

-

pv2

x, y, z, denoting the coordinates of the orifice, and the velocity of efflux.

The equation (2) gives v in terms of C; suppose accordingly that, (C) denoting a function of C,

and, from the equation (1) together with the equation to the vessel, we may obtain A in terms of C: let A = ƒ (C), a function of C. Hence, for the determination of the time of efflux of any portion of the fluid,

C, denoting the value of C, when A has its initial value, and C, the value of C when A has the value corresponding to the state. of the fluid at the end of the time t.

If the fluid be revolving about a fixed line, we may consider it, as far as the motion of rotation is concerned, as being at rest

in regard to three axes fixed in the fluid, provided that the centrifugal force be taken into consideration together with the other forces to which the fluid is subject, the accelerating forces being supposed to act either parallel to the axis of revolution or towards fixed points within it.

1. The fluid in an open vessel, in the form of a paraboloid of revolution, revolves round its axis, which is vertical, with such an angular velocity that it just reaches the rim: the volume of the fluid is such that, if it were at rest, its surface would bisect the axis of the paraboloid: to compare the time in which the vessel will empty itself into the atmosphere through a small orifice at the vertex with the time which would be required if the fluid were not rotating.

Let x represent the altitude of any molecule of the fluid above the tangent plane at the vertex, and y its distance from the axis of the cylinder. Then, by the equation of steady motion,

u being the component of the velocity of the fluid molecule which does not depend upon the rotation of the fluid.

At the surface of the fluid u may evidently be regarded as a very small quantity, and therefore, approximately,

p = p({w3y3 – gx) + C,

p representing in this equation the atmospheric pressure. Let z denote the value of x at the lowest point of the free surface of the fluid: then

and therefore

p = pgz + C,
w3y2 = 2g (x − z)...

the equation to the free surface of the fluid.

(2),

We proceed now to determine the value of w and the initial value of z. Let a denote the length of the axis of the vessel, and the latus rectum of the generating parabola. Then, initially, at the curve of intersection of the vessel and the surface of the fluid,

w'la = 2g (a – z).

Also, observing that the volume of a paraboloid of revolution is equal to half that of the circumscribing cylinder, we see that the initial volume of the fluid is equal to

but, by the hypothesis, the volume is also equal to

The equation to the free surface of the fluid will accordingly be, at any time during the motion,

y2 = 1 (x − z).

Again, at the beginning of the motion, putting a, la, for x, y3, respectively, we have

la = 1 l (a − z),

At the curve of intersection of the free surface of the fluid and the vessel, at any instant of the motion, there is

Hence, A denoting the volume of the fluid in the vessel at this instant, and x representing 4z,

But, v representing the velocity of the issuing fluid, it is plain