(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)