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 + 12W′ |
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 |
|
, | |||||
B0 + C0 − B1 − C1 |
|
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 |
If the ellipsoid is of revolution, with b = c,
φ = 12Ux | A + 2B1 | , |
B0 − B1 |
and the Stokes’ current function ψ can be written down
ψ = − 12 Uy2 | B − B1 | ; |
B0 − B1 |
reducing, when the liquid extends to infinity and B1 = 0, to
φ = 12 Ux | A | , ψ = − 12 Uy2 | B | ; |
B0 | B0 |
so that in the relative motion past the body, as when fixed in the
current U parallel to xO,
φ′ = 12Ux ( 1 + | A | ), ψ′ = 12Uy2 ( 1 − | B | ). |
B0 | B0 |
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 = 12 | p | , | B | = | p | , |
p+ λ′ | B0 | p+ λ′ |
y2 | = p + λ′ − 2x, |
p+ λ′ |
with λ′ = 0 over the surface of the paraboloid; and then
The relative path of a liquid particle is along a stream line
x = | p2y2 − (y2 − c2)2 | , √ (x2 + y2) = | p2y2 − (y2 − c2)2 |
2p (y2 − c2) | 2p (y2 − c2) |
a C4; while the absolute path of a particle in space will be given by
dy | = − | r − x | = | y2 − c2 | , |
dx | y | 2py |
46. Between two concentric spheres, with
φ = 12 Ux | a3r3 + 2 a3 a13 | , ψ = 12 Uy2 | a3r3 − a3 a13 | ; |
1 − a3/a13 | 1 − a3/a13 |
and the effective inertia of the liquid in the interspace is
A0 + 2A1 | W′ = 12 | a13 + 2a3 | W′. |
2A0 − 2A1 | a13 − a3 |
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 |
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 |
and for its image at H the Stokes’ function is
m | d | cos PHx = −m | a3 | y2 | ; | |
dƒ | ƒ3 | PH3 |
so that for the combination
ψ = my2 ( | a3 | 1 | − | 1 | ) = m | y2 | ( | a3 | − | ƒ3 | ), | |
ƒ3 | PH3 | PS3 | ƒ3 | PH3 | PS3 |
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′ = 12 Uy2 ( 1 − | a13 | − | a23 | + | a3 | ), |
r13 | r23 | r 3 |
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 |
With a2 = ∞, these reduce to
ψ′ = 12Uy2 ( 1 − | a5 | ) | x | , or Uy ( 1 − | a4 | ) | x | , |
r 5 | a | r 4 | a |
for a sphere or cylinder, and a diametral plane.
Two equal spheres, intersecting at 120°, will require
ψ′ = 12Uy2 [ | x | − | a3 | + | a4 (a − 2x) | + | a3 | − | a4 (a + 2x) | ], |
a | 2r13 | 2r15 | 2r23 | 2r25 |
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
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 + 1b2 + λ) χ + 2 dxdλ | , |
1 / (b2 + λ) − 1 / (a2 + λ) |
and that the continuity of the liquid is secured if
(a2 + λ)3/2 (b2 + λ)3/2 (c2 + λ) 12 | dχ | = constant, |
dλ |
χ = | N dλ | = | N | · | Bλ − Aλ | ; |
(a2 + λ) (b2 + λ) P | abc | a2 − b2 |
and at the surface λ = 0,
R = − | (1a2 + 1b2) Nabc B0 − A0a2 − b2 − Nabc 1a2b2 | , |
1/b2 − 1/a2 |
Nabc | = R | 1/b2 − 1/a2 | , |
1a2b2 − (1a2 + 1b2) B0 − A0a2 − b2 |
= R | (a2 − b2)2 / (a2 + b2) | . |
(a2 − b2) / (a2 + b2) − (B0 − A0) |