Page:EB1911 - Volume 14.djvu/143

HYDRODYNAMICS] 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

(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 = 1, a slight extension will give the velocity function of the liquid in the interspace as the ellipsoid = 0 is passing with velocity U through the confocal position; must now take the form x( + N), and will satisfy the conditions in the shape

,  B0 + C0 − B1 − C1  (1)

and any confocal ellipsoid defined by, internal or external to = 1, may be supposed to swim with the liquid for an instant, without distortion or rotation, with velocity along Ox

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

(2)

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

(3)

and the Stokes’ current function can be written down

(4)

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

(5)

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

(6)

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

(7)

(8)

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

′ = U [y2 − p √ (x2 + y2) + px]; (9)

= − Up[√ (x2 + y2) − x]; (10)

= − Up log [√ (x2 + y2) + x]. (11)

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

′ = Uc2, a constant, (12)

(13)

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

(14)

y2 − c2 = a2 e−x/p. (15)

46. Between two concentric spheres, with

a2 + = r&#8202;2, a2 + 1 = a12, (1)

A = B = C = a3 / 3r3,

(2)

and the effective inertia of the liquid in the interspace is

(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 at S outside a sphere of radius a is a source of strength a/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 −/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 cos PSx, and of the source H and line sink OH is (a/ƒ) cos PHx and −(/a)(PO − PH); so that

(4)

and = −, 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

(5)

and for its image at H the Stokes’ function is

(6)

so that for the combination

(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 ′ to be inferred from a distribution of doublets, finite in number when the surface is composed of two spheres intersecting at an angle /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

(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

(9)

With a2 = &infin;, these reduce to

(10)

for a sphere or cylinder, and a diametral plane.

Two equal spheres, intersecting at 120°, will require

(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

= xy (1)

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

(2)

and that the continuity of the liquid is secured if

(3)

(4)

and at the surface = 0,

(5)

(6)