Page:EB1911 - Volume 14.djvu/139

rotating about the centre with components of angular velocity ξ, η, ζ; then

u = − yζ + zη, v = − zξ + xζ, w = − xη + yξ.

(2)

Now suppose the liquid to be melted, and additional components of angular velocity Ω₁, Ω₂, Ω₃ communicated to the ellipsoidal case; the additional velocity communicated to the liquid will be due to a velocity-function

φ = − Ω1	b2 − c2	yz − Ω2	c2 − a2	zx − Ω3	a2 − b2	xy,
	b2 + c2		c2 + a2		a2 + b2

(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,

u′ = u + yR − zQ =	2a2	Ω3y −	2a2	Ω2z,
	a2 + b2		c2 + a2

(4)

v′ = v + zP − xR =	2b2	Ω1z −	2b2	Ω3x,
	b2 + c2		a2 + b2

(5)

w′ = w + xQ − yP =	2c2	Ω2x −	2c2	Ω1y,
	c2 + a2		b2 + c2

(6)

P = Ω₁ + ξ, Q = Ω₂ + η, R = Ω₃ + ζ.

(7)

Thus

u′	x	+ v′	y	+ w′	z	= 0,
	a2		b2		c2

(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

1		dp	+ 4πρAx +	du	− vR + wQ + u′	du	+ v′	du	+ w′	du	= 0, ... , ... ,
ρ		dx		dt		dx		dy		dz

(9)

where

A, B, C = $\int _{0}^{\infty }$	abcdλ
	(a2 + λ, b2 + λ, c2 + λ) P

P² = 4 (a² + λ) (b² + λ) (c² + λ).

(10)

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

1		dp	+ 4πρ Ax + αx + hy + gz = 0,
ρ		dx

(11)

1		dp	+ 4πρBy + hx + βy + fz = 0,
ρ		dy

(12)

1		dp	+ 4πρCz + gx + fy + γz = 0,
ρ		dz

(13)

and integrating

pρ⁻¹ + 2πρ (Ax² + By² + Cz²)

+ 1/2 (αx² + βy² + γz² + 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

α =	4c2(c2 − a2)	Ω22 − (	c2 − a2	Ω2 − η )	2
	(c2 + a2)2		c2 + a2

−	4b2(a2 − b2)	Ω32 − (	a2 − b2	Ω3 − ζ )	2	,
	(a2 + b2)2		a2 + b2

(15)

with similar equations for β and γ.

If we can make

(4πρA + α) x² = (4πρB + β) b² = (4πρC + γ) c²,

(16)

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

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

Ω₁ = 0, Ω₂ = 0, ξ = 0, η = 0.

(17)

If Ω₃ = 0 or θ₃ = ζ 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, Ω₃ may be replaced by zero, and the equations, hydrodynamical and dynamical, reduce to

dξ	= −	2c2	Ω2 ζ,	dη	=	2a2	Ω1 ζ,	dζ	=	2c2	(Ω2 ξ − Ω2 η)
dt		a2 + c2		dt		a2 + c2		dt		a2 + c2

(18)

dΩ1	= Ω2 ζ +	a2 + c2	ηζ,	dΩ2	= −Ω1 ζ −	a2 + c2	ξζ;
dt		a2 − c2		dt		a2 − c2

(19)

of which three integrals are

ξ2 + η2 = L −	a2	ζ2,
	c2

(20)

Ω12 + Ω22 = M +	(a2 + c2)2	ζ2,
	2c2 (a2 − c2)

(21)

Ω1 ξ + Ω2 ηN = +	a2 + c2	ζ2;
	4c2

(22)

and then

(	dζ	)	2	=	4c4	(Ω2ξ − Ω12η)2⁠
	dt				(a2 + c2)

=	4c4	[ (ξ2 + η2) (Ω12 + Ω22) − (Ω1ξ + Ω2η)2 ]⁠
	(a2 + c2)2

=	4c4	[ LM − N2 + {	(a2 + c2)2	− M	a2	− N	a2 + c2	} ζ2
	(a2 + c2)2		2c2 (a2 + c2)		c2		2c2

−	(a2 + c2) (9a2 − c2)	ζ4 ] = Z,
	16c4 (a2 − c2)

(23)

where Z is a quadratic in ζ², so that ζ is an elliptic function of t, except when c = a, or 3a.

Put Ω₁ = Ω cos φ, Ω₂ = −Ω sin φ,

Ω2	dφ	=	dΩ1	Ω2 − Ω1	dΩ2	= Ω2ζ −	(a2 + c2)	(Ω1ξ + Ω2η) ζ,
	dt		dt		dt		(a2 − c2)

(24)

dφ

= ζ −

(a² + c²)

·

N +	a2 + c2
	4c2

,

dt

(a² − c²)

M +	(a2 + c2)2	ζ2
	2c2 (a2 − c2)

(25)

φ = ∫

ζ dζ

−

a² + c²

∫

N +	a2 + c2
	4c2

·

ζ dζ

,

√Z

a² − c²

M +	(a2 + c2)2	ζ2
	2c2 (a2 − c2)

√Z

(26)

which, as Z is a quadratic function of ζ², are non-elliptic integrals; so also for ψ, where ξ = ω cos ψ, η = −ω sin ψ.

In a state of steady motion

dζ	= 0,	Ω1	=	Ω2	,
dt		ξ		η

(27)

φ = ψ = nt, suppose,

(28)

Ω₁ξ + Ω₂η = Ωω,

(29)

dφ	= ζ −	a2 + c2		ω	ζ,
dt		a2 − c2		Ω

(30)

dψ	= −	2a2		Ω	ζ,
dt		a2 + c2		ω

(31)

1 −	a2 + c2		ω	= −	2a2		Ω	,
	a2 − c2		Ω		a2 + c2		ω

(32)

(	ω	− 1/2	a2 + c2	)	2	=	(a2 − c2) (9a2 − c2)	,
	Ω		a2 − c2				4 (a2 + c2)

(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 φ and stream function ψ are given as conjugate functions of the coordinates x, y by

w = ƒ(z) where z = x + yi, w = φ + ψi,

(1)

and then

dw	=	dφ	+ i	dψ	= −u + vi;
dz		dx		dx

(2)

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

ζ = −Q	dz	=	Q	=	Q	(u + vi) =	Q	(cos θ + i sin θ),
	dw		(u − vi)		q2		q

(3)

gives ζ 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 ζ must be Constructed so as to have a constant