φ = { (n + 1) Arn − nBr−n−1 } Pn,
(10)
where Pn denotes the zonal harmonic of the nth order; also, in the exceptional case of
| ψ = A0 cos θ, φ = A0/r; ψ = B0r, φ = −B0 log tan ½θ = −½B0 sh−1 x/y. |
(11)
Thus cos θ is the Stokes’ function of a point source at O, and PA − PB of a line source AB.
The stream function ψ of the liquid motion set up by the passage of a solid of revolution, moving with axial velocity U, is such that
| 1 | dψ | = −U | dy | , ψ + ½Uy2 = constant, | |
| y | ds | ds |
(12)
over the surface of the solid; and ψ must be replaced by ψ′ = ψ + ½Uy2 in the general equations of steady motion above to obtain the steady relative motion of the liquid past the solid.