ψ = −½ Uy2a4 μ/r4, ψ′ = ½ Uy2 (1 − a4 μ/r4),
(22)
φ′ = Ux + φ, φ = −1⁄3 U (a4 / r3) P2, P2 = 3⁄2 μ2 − ½,
(23)
representing a stream past the surface r4 = a4μ.
35. A circular vortex, such as a smoke ring, will set up motion symmetrical about an axis, and provide an illustration; a half vortex ring can be generated in water by drawing a semicircular blade a short distance forward, the tip of a spoon for instance. The vortex advances with a certain velocity; and if an equal circular vortex is generated coaxially with the first, the mutual influence can be observed. The first vortex dilates and moves slower, while the second contracts and shoots through the first; after which the motion is reversed periodically, as if in a game of leap-frog. Projected perpendicularly against a plane boundary, the motion is determined by an equal opposite vortex ring, the optical image; the vortex ring spreads out and moves more slowly as it approaches the wall; at the same time the molecular rotation, inversely as the cross-section of the vortex, is seen to increase. The analytical treatment of such vortex rings is the same as for the electro-magnetic effect of a current circulating in each ring.
36. Irrotational Motion in General.—Liquid originally at rest in a singly-connected space cannot be set in motion by a field of force due to a single-valued potential function; any motion set up in the liquid must be due to a movement of the boundary, and the motion will be irrotational; for any small spherical element of the liquid may be considered a smooth solid sphere for a moment, and the normal pressure of the surrounding liquid cannot impart to it any rotation.
The kinetic energy of the liquid inside a surface S due to the velocity function φ is given by
| T = ½ρ ∫ ∫ ∫ [ ( | dφ | ) | 2 | + ( | dφ | ) | 2 | + ( | dφ | ) | 2 | ] dx dy dz, |
| dx | dy | dz |
| = ½ρ ∫ ∫ φ | dφ | dS |
| dν |