Page:EB1911 - Volume 14.djvu/143

so that the effective inertia of a sphere is increased by half the weight of liquid displaced; and in frictionless air or liquid the sphere, of weight W, will describe a parabola with vertical acceleration

W − W′	g.
W + 1/2W′

(30)

Thus a spherical air bubble, in which W/W′ is insensible, will begin to rise in water with acceleration 2g.

45. When the liquid is bounded externally by the fixed ellipsoid λ = λ₁, a slight extension will give the velocity function φ of the liquid in the interspace as the ellipsoid λ = 0 is passing with velocity U through the confocal position; φ must now take the form x(ψ + N), and will satisfy the conditions in the shape

φ = Ux

A + B₁ + C₁

= Ux

abc	+ $\int _{\lambda }^{\lambda _{1}}$	abcdλ
a1b1c1		(a2 + λ) P

,

B₀ + C₀ − B₁ − C₁

1 −	abc	− $\int _{0}^{\lambda _{1}}$	abcdλ
	a1b1c1		(a2 + λ) P

(1)

and any confocal ellipsoid defined by λ, internal or external to λ = λ₁, may be supposed to swim with the liquid for an instant, without distortion or rotation, with velocity along Ox

U	Bλ + Cλ − B1 − C1	.
	B0 + C0 − B1 − C1

Since − Ux is the velocity function for the liquid W′ filling the ellipsoid λ = 0, and moving bodily with it, the effective inertia of the liquid in the interspace is

A0 + B1 + C1	W′.
B0 + C0 − B1 − C1

(2)

If the ellipsoid is of revolution, with b = c,

φ = 1/2Ux	A + 2B1	,
	B0 − B1

(3)

and the Stokes’ current function ψ can be written down

ψ = − 1/2 Uy2	B − B1	;
	B0 − B1

(4)

reducing, when the liquid extends to infinity and B₁ = 0, to

φ = 1/2 Ux	A	,   ψ = − 1/2 Uy2	B	;
	B0		B0

(5)

so that in the relative motion past the body, as when fixed in the current U parallel to xO,

φ′ = 1/2Ux ( 1 +	A	),   ψ′ = 1/2Uy2 ( 1 −	B	).
	B0		B0

(6)

Changing the origin from the centre to the focus of a prolate spheroid, then putting b² = pa, λ = λ′a, and proceeding to the limit where a = ∞, we find for a paraboloid of revolution

B = 1/2	p	,	B	=	p	,
	p+ λ′		B0		p+ λ′

(7)

y2	= p + λ′ − 2x,
p+ λ′

(8)

with λ′ = 0 over the surface of the paraboloid; and then

ψ′ = 1/2 U [y² − p √ (x² + y²) + px];

(9)

ψ = −1/2 Up[√ (x² + y²) − x];

(10)

φ = −1/2 Up log [√ (x² + y²) + x].

(11)

The relative path of a liquid particle is along a stream line

ψ′ = 1/2 Uc², a constant,

(12)

x =	p2y2 − (y2 − c2)2	,   √ (x2 + y2) =	p2y2 − (y2 − c2)2
	2p (y2 − c2)		2p (y2 − c2)

(13)

a C₄; while the absolute path of a particle in space will be given by

dy	= −	r − x	=	y2 − c2	,
dx		y		2py

(14)

y² − c² = a² e^−x/p.

(15)

46. Between two concentric spheres, with

a² + λ = r ², a² + λ₁ = a₁²,

(1)

A = B = C = a³ / 3r³,

φ = 1/2 Ux	a3/r3 + 2 a3 /a13	,   ψ = 1/2 Uy2	a3/r3 − a3 /a13	;
	1 − a3/a13		1 − a3/a13

(2)

and the effective inertia of the liquid in the interspace is

A0 + 2A1	W′ = 1/2	a13 + 2a3	W′.
2A0 − 2A1		a13 − a3

(3)

When the spheres are not concentric, an expression for the effective inertia can be found by the method of images (W. M. Hicks, Phil. Trans., 1880).

The image of a source of strength μ at S outside a sphere of radius a is a source of strength μa/ƒ at H, where OS = ƒ, OH = a²/ƒ, and a line sink reaching from the image H to the centre O of line strength −μ/a; this combination will be found to produce no flow across the surface of the sphere.

Taking Ox along OS, the Stokes’ function at P for the source S is μ cos PSx, and of the source H and line sink OH is μ(a/ƒ) cos PHx and −(μ/a)(PO − PH); so that

ψ = μ ( cos PSx +	a	cos PHx −	PO − PH	),
	ƒ		a

(4)

and ψ = −μ, a constant, over the surface of the sphere, so that there is no flow across.

When the source S is inside the sphere and H outside, the line sink must extend from H to infinity in the image system; to realize physically the condition of zero flow across the sphere, an equal sink must be introduced at some other internal point S′.

When S and S′ lie on the same radius, taken along Ox, the Stokes’ function can be written down; and when S and S′ coalesce a doublet is produced, with a doublet image at H.

For a doublet at S, of moment m, the Stokes’ function is

m	d	cos PSx = −m	y2	;
	dƒ		PS3

(5)

and for its image at H the Stokes’ function is

m	d	cos PHx = −m	a3		y2	;
	dƒ		ƒ3		PH3

(6)

so that for the combination

ψ = my2 (

a3

1

−

1

) = m

y2

(

a3

−

ƒ3

),

ƒ3

PH3

PS3

ƒ3

PH3

PS3

(7)

and this vanishes over the surface of the sphere.

There is no Stokes’ function when the axis of the doublet at S does not pass through O; the image system will consist of an inclined doublet at H, making an equal angle with OS as the doublet S, and of a parallel negative line doublet, extending from H to O, of moment varying as the distance from O.

A distribution of sources and doublets over a moving surface will enable an expression to be obtained for the velocity function of a body moving in the presence of a fixed sphere, or inside it.

The method of electrical images will enable the stream function ψ′ to be inferred from a distribution of doublets, finite in number when the surface is composed of two spheres intersecting at an angle π/m, where m is an integer (R. A. Herman, Quart. Jour. of Math. xxii.).

Thus for m = 2, the spheres are orthogonal, and it can be verified that

ψ′ = 1/2 Uy2 ( 1 −	a13	−	a23	+	a3	),
	r13		r23		r 3

(8)

where a₁, a₂, a = a₁a₂/√ (a₁² + a₂²) is the radius of the spheres and their circle of intersection, and r₁, r₂, r the distances of a point from their centres.

The corresponding expression for two orthogonal cylinders will be

ψ′ = Uy ( 1 −	a12	−	a22	+	a2	).
	r12		r22		r 2

(9)

With a₂ = ∞, these reduce to

ψ′ = 1/2Uy2 ( 1 −	a5	)	x	, or Uy ( 1 −	a4	)	x	,
	r 5		a		r 4		a

(10)

for a sphere or cylinder, and a diametral plane.

Two equal spheres, intersecting at 120°, will require

ψ′ = 1/2Uy2 [	x	−	a3	+	a4 (a − 2x)	+	a3	−	a4 (a + 2x)	],
	a		2r13		2r15		2r23		2r25

(11)

with a similar expression for cylinders; so that the plane x = 0 may be introduced as a boundary, cutting the surface at 60°. The motion of these cylinders across the line of centres is the equivalent of a line doublet along each axis.

47. The extension of Green’s solution to a rotation of the ellipsoid was made by A. Clebsch, by taking a velocity function

φ = xyχ

(1)

for a rotation R about Oz; and a similar procedure shows that an ellipsoidal surface λ may be in rotation about Oz without disturbing the motion if

R = −	(1/ a2 + λ + 1/b2 + λ) χ + 2 dx/dλ	,
	1 / (b2 + λ) − 1 / (a2 + λ)

(2)

and that the continuity of the liquid is secured if

(a2 + λ)3/2 (b2 + λ)3/2 (c2 + λ) 1/2	dχ	= constant,
	dλ

(3)

χ = $\int _{\lambda }^{\infty }$	N dλ	=	N	·	Bλ − Aλ	;
	(a2 + λ) (b2 + λ) P		abc		a2 − b2

(4)

and at the surface λ = 0,

R = −	(1/a2 + 1/b2) N/abc B0 − A0/a2 − b2 − N/abc 1/a2b2	,
	1/b2 − 1/a2

(5)

N/abc	= R	1/b2 − 1/a2	,
		1/a2b2 − (1/a2 + 1/b2) B0 − A0/a2 − b2

(6)

= R	(a2 − b2)2 / (a2 + b2)	.
	(a2 − b2) / (a2 + b2) − (B0 − A0)