The energy cross term Ec between an initial fluid flow v0 and the flow from a vortex ring is normally written as v0⋅P, where P is the momentum of the vortex ring. There is some uncertainty about how to calculate P. Ec is computed exactly for flow around a closed circuit (a ring) in an incompressible fluid where circulation is quantized. v0 is allowed to be a function of position. The vortex rings are assumed not to be created impulsively. The result is that Ec is equal to the circulation times the mass of fluid that v0 causes to flow through the vortex ring per second. This reduces to the usual result P=ρκπr2 when v0 is constant in the neighborhood of the vortex ring.
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Research Article| February 01 1976
Frank A. Blood; Energy cross term for a quantum vortex ring. Phys. Fluids 1 February 1976; 19 (2): 179–181. https://doi.org/10.1063/1.861444
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