(1)
representing vortex motion circulating round the annulus of liquid.
Considered by itself, with the cylinders held fixed, the vortex sets up a circumferential velocity m/r on a radius r, so that the angular momentum of a circular filament of annular cross section dA is ρm dA, and of the whole vortex is ρmπ (b2 − a2).
Any circular filament can be started from rest by the application of a circumferential impulse πρm dr at each end of a diameter; so that a mechanism attached to the cylinders, which can set up a uniform distributed impulse πρm across the two parts of a diameter in the liquid, will generate the vortex motion, and react on the cylinder with an impulse couple −ρmπa2 and ρmπb2, having resultant ρmπ (b2 − a2), and this couple is infinite when b = ∞, as the angular momentum of the vortex is infinite. Round the cylinder r = a held fixed in the U current the liquid streams past with velocity
q′ = 2U sin θ + m/a;
(2)
and the loss of head due to this increase of velocity from U to q′ is
| q′2 − U2 | = | (2U sin θ + m/a)2 − U2 | , |
| 2g | 2g |
(3)
so that cavitation will take place, unless the head at a great distance exceeds this loss.