A = B = C = a3 / 3r3,
| φ = ½ Ux | a3/r3 + 2 a3/a13 | , ψ = ½ Uy2 | a3/r3 − a3/a13 | ; |
| 1 − a4/a12 | 1 − a3/a13 |
(2)
and the effective inertia of the liquid in the interspace is
| A0 + 2A1 | W′ = ½ | 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 |