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.
REFERENCES
1.
K. R. Atkins, Liquid Helium (Cambridge U.P., Cambridge, 1959), Chap. 4. This is the Landau theory applied to vortex rings.
2.
S. V.
Iordanskii
, Zh. Eksp. Teor. Fiz.
48
, 708
(1965
)[
S. V.
Iordanskii
, Sov. Phys.—JETP
21
, 467
(1965
)].3.
4.
H. Lamb, Hydrodynamics (Dover, New York, 1945). Secs. 119 and 152.
5.
C. C. Lin, Liquid Helium, edited by G. Careri (Academic, New York, 1963), p. 93.
6.
J. S. Langer and J. D. Reppy, Progress in Low Temperature Physics, edited by C. J. Gorter (Wiley, New York, 1970), Vol. 6, p. 1.
7.
V. L.
Ginsburg
and L. P.
Pitaevskii
, Zh. Eksp. Teor. Fiz.
34
, 1240
(1958
)[
V. L.
Ginsburg
and L. P.
Pitaevskii
, Sov. Phys.—JETP.
7
, 858
(1958
)].8.
9.
See
W. I.
Glaberson
and R. J.
Donnelly
, Phys. Rev.
141
, 208
(1966
), for a study of the process depicted in Fig. 2.10.
This ambiguity was pointed out by W. F. Vinen, Quantum Fluids, edited by D. F. Brewer (North‐Holland, Amsterdam, 1966), p. 74,
and explored in some detail by
E. R.
Huggins
, Phys. Rev. Lett.
17
, 1284
(1966
). I thank the referee for bringing it to my attention.
This content is only available via PDF.
© 1976 American Institute of Physics.
1976
American Institute of Physics
You do not currently have access to this content.