In a multiply connected space, like a ring, with a multiply valued velocity function φ, the liquid can circulate in the circuits independently of any motion of the surface; thus, for example,
φ = mθ = m tan−1 y/x
(5)
will give motion to the liquid, circulating in any ring-shaped figure of revolution round Oz.
To find the kinetic energy of such motion in a multiply connected space, the channels must be supposed barred, and the space made acyclic by a membrane, moving with the velocity of the liquid; and then if k denotes the cyclic constant of φ in any circuit, or the value by which φ has increased in completing the circuit, the values of φ on the two sides of the membrane are taken as differing by k, so that the integral over the membrane
| ∫ ∫ φ | dφ | dS = k ∫ ∫ | dφ | dS, |
| dν | dν |
(6)
and this term is to be added to the terms in (1) to obtain the additional part in the kinetic energy; the continuity shows that the integral is independent of the shape of the barrier membrane, and its position. Thus, in (5), the cyclic constant k = 2πm.
In plane motion the kinetic energy per unit length parallel to Oz
| T = ½ρ ∫ ∫ [ ( | dφ | ) | 2 | + ( | dφ | ) | 2 | ] dx dy = ½ρ ∫ ∫ [ ( | dψ | ) | 2 | + ( | dψ | ) | 2 | ] dx dy |
| dx | dy | dx | dy |