subject to the condition, from (4) § 34,
y−2 ∇2ψ′ = −ƒ′(ψ′), y−2 ∇2ψ = −ƒ′ (ψ + ½Uy2).
(12)
Thus, for example, with
ψ′ = ¾U y2 (r2a−2 − 1), r2 = x2 + y2,
(13)
for the space inside the sphere r = a, compared with the value of ψ′ in § 34 (13) for the space outside, there is no discontinuity of the velocity in crossing the surface.
Inside the sphere
| 2ζ = | d | ( | 1 | dψ′ | ) + | d | ( | 1 | dψ′ | ) = | 15 | U | y | , | ||
| dx | y | dx | dy | y | dy | 2 | a2 |
(14)