The motion of vortex rings with bilaterally symmetric initial shape is investigated theoretically and experimentally. The induced velocity at each point on the vortex ring is computed from the Biot-Savart law. The induced velocity is related to the motion of the ring according to two different concepts: (1) Hydrodynamic vortex—the ring moves with the same velocity as the local fluid; (2) Rankine vortex—the local relative velocity produces lift and drag forces on the ring which serve to distort the ring. Observable vortex rings are produced by pulsing dyed fluid through a rectangular orifice and by staining the starting vortex behind ring wings of various shapes. Good qualitative agreement between the analyses and experiments is achieved.
A basic mechanism for the wandering of freely falling spheres is shown to be coupling between rocking of the spheres and their motions perpendicular to the free fall direction. The rocking frequency is determined by small displacements of the spheres' centers of mass from their geometric centers. A phenomenological model of the motion, in which the rocking of the spheres is described by the nonlinear pendulum equation with damping, arid the coupling between the rocking and the lateral force on the spheres involves a Reynolds number dependent phase shift, leads to fairly good agreement with observations. The wandering mechanism discussed here is most effective at Reynolds numbers in the range 10 3 < Re < 10 5 , for sphere-to-fluid mass ratios between 0.8 and 1.2.
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