Page:EB1911 - Volume 14.djvu/139

HYDRODYNAMICS] rotating about the centre with components of angular velocity , , ; then

u = − y + z, v = − z + x, w = − x + y. (2)

Now suppose the liquid to be melted, and additional components of angular velocity 1, 2, 3 communicated to the ellipsoidal case; the additional velocity communicated to the liquid will be due to a velocity-function

(3)

as may be verified by considering one term at a time.

If u′, v′, w′ denote the components of the velocity of the liquid relative to the axes,

(4)

(5)

(6)

P = 1 +, Q = 2 + , R = 3 +. (7)

Thus

(8)

so that a liquid particle remains always on a similar ellipsoid.

The hydrodynamical equations with moving axes, taking into account the mutual gravitation of the liquid, become

(9)

where

P2 = 4 (a2 + ) (b2 + ) (c2 + ). (10)

With the values above of u, v, w, u′, v′, w′, the equations become of the form

(11)

(12)

(13)

and integrating

p−1 + 2 (Ax2 + By2 + Cz2)

+ (x2 + y2 + z2 + 2fyz + 2gzx + 2hxy) = const., (14)

so that the surfaces of equal pressure are similar quadric surfaces, which, symmetry and dynamical considerations show, must be coaxial surfaces; and f, g, h vanish, as follows also by algebraical reduction; and

(15)

with similar equations for and.

If we can make

(4A + ) x2 = (4B + ) b2 = (4C + ) c2, (16)

the surfaces of equal pressure are similar to the external case, which can then be removed without affecting the motion, provided, , remain constant.

This is so when the axis of revolution is a principal axis, say Oz; when

1 = 0, 2 = 0, = 0,  = 0.  (17)

If 3 = 0 or 3 = in addition, we obtain the solution of Jacobi’s ellipsoid of liquid of three unequal axes, rotating bodily about the least axis; and putting a = b, Maclaurin’s solution is obtained of the rotating spheroid.

In the general motion again of the liquid filling a case, when a = b, 3 may be replaced by zero, and the equations, hydrodynamical and dynamical, reduce to

(18)

(19)

of which three integrals are

(20)

(21)

(22)

and then

(23)

where Z is a quadratic in 2, so that is an elliptic function of t, except when c = a, or 3a.

Put 1 = cos, 2 = − sin ,

(24)

,  dt (a2 − c2)  (25)

·  d ,  √Z a2 − c2  √Z (26)

which, as Z is a quadratic function of 2, are non-elliptic integrals; so also for, where =  cos ,  = − sin.

In a state of steady motion

(27)

= = nt, suppose, (28)

1 + 2 =, (29)

(30)

(31)

(32)

(33)

and a state of steady motion is impossible when 3a > c > a.

An experiment was devised by Lord Kelvin for demonstrating this, in which the difference of steadiness was shown of a copper shell filled with liquid and spun gyroscopically, according as the shell was slightly oblate or prolate. According to the theory above the stability is regained when the length is more than three diameters, so that a modern projectile with a cavity more than three diameters long should fly steadily when filled with water; while the old-fashioned type, not so elongated, would be highly unsteady; and for the same reason the gas bags of a dirigible balloon should be over rather than under three diameters long.

40. A Liquid Jet.—By the use of the complex variable and its conjugate functions, an attempt can be made to give a mathematical interpretation of problems such as the efflux of water in a jet or of smoke from a chimney, the discharge through a weir, the flow of water through the piers of a bridge, or past the side of a ship, the wind blowing on a sail or aeroplane, or against a wall, or impinging jets of gas or water; cases where a surface of discontinuity is observable, more or less distinct, which separates the running stream from the dead water or air.

Uniplanar motion alone is so far amenable to analysis; the velocity function and stream function  are given as conjugate functions of the coordinates x, y by

w = ƒ(z) where z = x + yi, w = + i, (1)

and then

(2)

so that, with u = q cos, v = q sin , the function

(3)

gives as a vector representing the reciprocal of the velocity q in direction and magnitude, in terms of some standard velocity Q.

To determine the motion of a jet which issues from a vessel with plane walls, the vector must be Constructed so as to have a constant