| U = −ψ − 2 (a2 + λ) | dψ | , |
| dλ |
(11)
and so the boundary condition is satisfied; moreover, any ellipsoidal surface λ may be supposed moving as if rigid with the velocity in (11), without disturbing the liquid motion for the moment.
The continuity is secured if the liquid between two ellipsoids λ and λ1, moving with the velocity U and U1 of equation (11), is squeezed out or sucked in across the plane x = 0 at a rate equal to the integral flow of the velocity ψ across the annular area α1 − α of the two ellipsoids made by x = 0; or if
| αU − α1U1 = ∫λ1λ ψ | dα | dλ, |
| dλ |
(12)
α = π√ (b2 + λ.c2 + λ).
(13)
Expressed as a differential relation, with the value of U from (11),
| d | [ αψ + 2 (a2 + λ) α | dψ | ] − ψ | dα | = 0, |
| dλ | dλ | dλ |