The effect of an external circulation of vortex motion on the motion of a cylinder has been investigated in § 29; a similar procedure will show the influence of circulation through a hole in a solid, taking as the simplest illustration a ring-shaped figure, with uniplanar motion, and denoting by ξ the resultant axial linear momentum of the circulation.
As the ring is moved from O to O′ in time t, with velocity Q, and angular velocity R, the components of liquid momentum change from
αM′U + ξ and βM′V along Ox and Oy
to
αM′U′+ ξ and βM′V′ along O′x′ and O′y′,
(1)
the axis of the ring changing from Ox to O′x′; and
| U = Q cos θ, V = Q sin θ, U′ = Q cos (θ − Rt), V′ = Q sin (θ − Rt), |
(2)
so that the increase of the components of momentum, X1, Y1, and N1, linear and angular, are