a² + [lambda] = r², a² + [lambda]1 = a1², (1)
A = B = C = a³/3r³,
a³ a³ a³ a³
-- + 2 --- -- + 2 ---
r³ a1³ r³ a1³
[phi] = ½Ux -----------, [psi] = ½Uy² ----------; (2)
1 - a^4/a1² 1 - a³/a1³
and the effective inertia of the liquid in the interspace is
A0 + 2A1 a1³ + 2a³
--------- W´ = ½ --------- W´. (3)
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 [mu] at S outside a sphere of radius
a is a source of strength [mu]a/[f] at H, where OS = [f], OH = a²/f,
and a line sink reaching from the image H to the centre O of line
strength - [mu]/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 [mu]
cos PSx, and of the source H and line sink OH is [mu](a/[f]) cos PHx
and -([mu]/a)(PO - PH); so that
/ a PO - PH \
[psi] = [mu] (cos PSx + --- cos PHx - ------- ), (4)
\ [f] a /
and [psi] = -[mu], 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.