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Coaxial axisymmetric vortex rings: 150 years after Helmholtz. (English) Zbl 1191.76004

Summary: This article addresses the fascinating 150 years history of the classical Helmholtz paper that laid the foundation of the vortex dynamics. Among general theorems on vortex motion, this memoir contains the special section on circular vortex filaments and axisymmetric vortex rings, in particular. The objective of this article is both to clarify some purely mathematical questions connected with the Dyson model of coaxial vortex rings in inviscid incompressible fluid and to provide a historical overview of achievements in experimental, analytical, and numerical studies of vortex rings interactions. The model is illustrated by several examples both of regular and chaotic motion of several vortex rings in an unbounded fluid.

MSC:

76-03 History of fluid mechanics
76B47 Vortex flows for incompressible inviscid fluids
01A55 History of mathematics in the 19th century
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[1] Acheson D.J.: Elementary Fluid Dynamics. Clarendon, Oxford (1990) · Zbl 0719.76001
[2] Ackeret J.: Über die Bildung von Wirbeln in reibungslosen Flüssigkeit. Z. Angew. Math. Mech. 15, 3–4 (1935) · JFM 61.0920.02 · doi:10.1002/zamm.19350150103
[3] Acton E.: A modelling of large eddies in an axisymmetric jet. J. Fluid Mech. 98, 1–31 (1980) · Zbl 0441.76069 · doi:10.1017/S0022112080000018
[4] Albring W.: Elementarvorgänge fluider Wirbelbewegungen. Akademie, Berlin (1981)
[5] Alekseenko S.V., Kuibin P.A., Okulov V.L.: Theory of Concentrated Vortices: An Introduction. Springer, Berlin (2007) · Zbl 1132.76001
[6] Alkemade, A.J.Q.: Vortex atoms and vortons. Ph.D. thesis. Technische Universiteit Delft, Delft (1994)
[7] Appell P.: Traité de mécanique rationnelle. Tome III. Équilibre et mouvement des milieux continus. Gauthier-Villars, Paris (1903) · JFM 34.0727.01
[8] Auerbach D.: Some open questions on the flow of circular vortex rings. Fluid Dyn. Res. 3, 209–213 (1988) · doi:10.1016/0169-5983(88)90067-6
[9] Auerbach F.: Wirbelbewegung. In: Winkelmann, A. (eds) Handbuch der Physik, Band 1, SS, pp. 1047–1074. Barth, Leipzig (1908)
[10] Auerbach F.: Wirbelbewegung. In: Auerbach, F., Hort, W. (eds) Handbuch der Physikalischen und Technischen Mechanik, Band 5, SS, pp. 115–156. Barth, Leipzig (1927) · JFM 53.0725.01
[11] Bagrets A.A., Bagrets D.A.: Nonintegrability of two problems in vortex dynamics. Chaos 7, 368–375 (1997) · Zbl 0933.37025 · doi:10.1063/1.166210
[12] Ball R.S.: Account of experiments upon the retardation experienced by vortex rings of air when moving through air. Trans. R. Irish Acad. 25, 135–155 (1872)
[13] Basset A.B.: A Treatise on Hydrodynamics. Deighton Bell, Cambridge (1888) · JFM 20.0970.01
[14] Batchelor G.K.: An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge (1967) · Zbl 0152.44402
[15] Bauer G.: Die Helmholtzsche Wirbeltheorie für Ingenieure Bearbeitet. Oldenbourg, München-Berlin (1919) · JFM 47.0769.08
[16] Betz A.: Wirbelbildung in idealen Flüssigkeiten und Helmholtzscher Wirbelsatz. Z. Angew. Math. Mech. 10, 413–415 (1930) · JFM 56.0705.04 · doi:10.1002/zamm.19300100412
[17] Betz A.: Wie entsteht ein Wirbel in einer wenig zähen Flüssigkeit?. Naturwissenschaften 37, 193–195 (1950) · doi:10.1007/BF00622314
[18] Blackmore D., Brøns M., Goullet A.: A coaxial vortex ring model for vortex breakdown. Physica D 237, 2817–2844 (2008) · Zbl 1155.76017 · doi:10.1016/j.physd.2008.05.012
[19] Blackmore D., Champanerkar J., Wang C.W.: A generalized Poincaré-Birkhoff theorem with applications to coaxial vortex ring motion. Discret. Contin. Dyn. Syst. B 5, 15–33 (2005) · Zbl 1075.37018
[20] Blackmore D., Knio O.: KAM theory analysis of the dynamics of three coaxial vortex rings. Physica D 140, 321–348 (2000) · Zbl 0987.37060 · doi:10.1016/S0167-2789(99)00223-7
[21] Blackmore D., Knio O.: Transition from quasiperiodicity to chaos for three coaxial vortex rings. Z. Angew. Math. Mech. 80(Suppl 1), S173–S176 (2000) · Zbl 0963.76017 · doi:10.1002/zamm.20000801344
[22] Boyarintzev V.I., Levchenko E.S., Savin A.S.: Motion of two vortex rings. Fluid Dyn. 20, 818–819 (1985) · Zbl 0603.76019 · doi:10.1007/BF01050100
[23] Brillouin, M.: Recherches récentes sur diverses questions d’hydrodynamique. Exposé des travaux de von Helmholtz, Kirchhoff, Sir W. Thomson, Lord Rayleigh. Gauthier-Villars, Paris (1891)
[24] Brutyan M.A., Krapivskii P.L.: The motion of a system of vortex rings in an incompressible fluid. J. Appl. Math. Mech. 48, 365–368 (1984) · Zbl 0575.76032 · doi:10.1016/0021-8928(84)90148-5
[25] Cahan D. (eds): Hermann von Helmholtz and Foundations of Nineteenth-Century Science. University of California Press, Berkeley (1993) · Zbl 0868.01019
[26] Cahan D.: Helmholtz and the shaping of the American physics elite in the Gilded Age. Hist. Stud. Phys. Biol. Sci. 35, 1–34 (2004) · doi:10.1525/hsps.2004.35.1.1
[27] Cahan D.: The ”Imperial Chancellor of the Sciences”: Helmholtz between science and politics. Soc. Res. 73, 1093–1128 (2006)
[28] Chorin A.J., Marsden J.E.: Mathematical Introduction to Fluid Mechanics. 3rd edn. Springer, New York (1992) · Zbl 0417.76002
[29] Chu C.-C., Wang C.-T., Chang C.-C., Chang R.-Y., Chang W.-T.: Head-on collision of two coaxial vortex rings: experiment and computation. J. Fluid Mech. 296, 39–71 (1995) · doi:10.1017/S0022112095002060
[30] Cromby A.C.: Helmholtz. Sci. Amer. 198, 94–102 (1958) · doi:10.1038/scientificamerican0358-94
[31] Dabiri J.O., Gharib M.: Fluid entrainment by isolated vortex rings. J. Fluid Mech. 511, 311–331 (2004) · Zbl 1061.76502 · doi:10.1017/S0022112004009784
[32] Darrigol O.: From organ pipes to atmospheric motion: Helmholtz on fluid mechanics. Hist. Stud. Phys. Sci. 29, 1–51 (1998) · doi:10.1016/S0039-3681(97)00022-8
[33] Darrigol O.: Worlds of Flow: A History of Hydrodynamics from the Bernoullis to Prandtl. Oxford University Press, Oxford (2005) · Zbl 1094.76002
[34] Dickinson M.: How to walk on water. Nature 424, 621–622 (2003) · doi:10.1038/424621a
[35] Didden N.: On the formation of vortex rings: rolling-up and production of circulation. Z. Angew. Math. Phys. 30, 101–116 (1979) · doi:10.1007/BF01597484
[36] Dolbear A.E.: Modes of Motion or Mechanical Conceptions of Physical Phenomena. Lee and Shepard, Boston (1897) · JFM 28.0716.05
[37] Donnelly R.J.: Quantized Vortices in Helium II. Cambridge University Press, Cambridge (1991)
[38] Drucker E.G., Lauder G.V.: Locomotor forces on a swimming fish: three-dimensional vortex wake dynamics quantified using digital particle image velocimetry. J. Exp. Biol. 202, 2393–2412 (1999)
[39] van Dyke M.: An Album of Fluid Motion. Parabolic, Stanford (1982)
[40] Dyson F.W.: The potential of an anchor ring. Part II. Phil. Trans. R. Soc. Lond. A 184, 1041–1106 (1893) · JFM 25.1514.01 · doi:10.1098/rsta.1893.0020
[41] Ebert H.: Hermann von Helmholtz. Wissenschafliche Verlagsgesellschaft, Stuttgart (1949)
[42] Eddington A.S.: Sir Frank Watson Dyson 1868–1939. Obit. Notices Fellows R. Soc. 3, 159–172 (1940) · doi:10.1098/rsbm.1940.0015
[43] Edser E.: General Physics for Students: A Textbook on the Fundamental Properties of Matter. Macmillan, London (1911) · JFM 42.0860.08
[44] Einstein A.: Zum hundertjährigen Gedenktag von Lord Kelvin’s Geburt. (26. Juni 1824). Naturwissenschaften 12, 601–602 (1924) · JFM 50.0587.02 · doi:10.1007/BF01506009
[45] Ellington C.P.: The aerodynamics of hovering insect flight. V. A vortex theory. Phil. Trans. R. Soc. Lond. B 305, 115–144 (1984) · doi:10.1098/rstb.1984.0053
[46] Epple M.: Topology, matter, and space, I: Topological notions in 19th-century natural philosophy. Arch. Hist. Exact Sci. 52, 297–392 (1998) · Zbl 0904.01003 · doi:10.1007/s004070050019
[47] Faber T.E.: Fluid Dynamics for Physicists. Cambridge University Press, Cambridge (1995) · Zbl 0861.76001
[48] Filippov A.T.: The Versatile Soliton. Birkhäuser, Boston (2000) · Zbl 0955.35002
[49] FitzGerald G.F.: Helmholtz memorial lecture. Nature 53, 296–298 (1896)
[50] Fleming J.A.: Waves and Ripples in Water, Air, and Aither. Young, London (1902)
[51] Fraenkel L.E.: On steady vortex rings of small cross-section in an ideal fluid. Proc. R. Soc. Lond. A 316, 29–62 (1970) · Zbl 0195.55101 · doi:10.1098/rspa.1970.0065
[52] Fraenkel L.E.: Examples of steady vortex rings of small cross-section in an ideal fluid. J. Fluid Mech. 51, 119–135 (1972) · Zbl 0231.76013 · doi:10.1017/S0022112072001107
[53] Fukumoto Y.: Higher-order asymptotic theory for the velocity field induced by an inviscid vortex ring. Fluid Dyn. Res. 30, 65–92 (2002) · Zbl 1064.76513 · doi:10.1016/S0169-5983(01)00044-2
[54] Fukumoto Y., Moffatt H.K.: Motion and expansion of a viscous vortex ring. Part 1. A higher-order asymptotic formula for the velocity. J. Fluid Mech. 417, 1–45 (2000) · Zbl 0977.76019 · doi:10.1017/S0022112000008995
[55] Glazebrook R.T.: Sir Horace Lamb 1849–1934. Obit. Notices Fellows R. Soc. 1, 375–392 (1935)
[56] Goman O.G., Karplyuk V.I., Nisht M.I.: Problem of the motion of annular vortices in an ideal fluid. Fluid Dyn. 22, 385–392 (1987) · Zbl 0648.76013 · doi:10.1007/BF01051918
[57] Gray A.: Lord Kelvin: An Account of his Scientific Life and Work. Dent, London (1908) · JFM 39.0031.05
[58] Gray A.: Notes on hydrodynamics. Phil. Mag. (Ser. 6) 28, 1–18 (1914) · JFM 45.1093.01
[59] Grinchenko V.T., Meleshko V.V., Gourjii A.A., Eisenga A.E.M., van Heijst G.J.F.: Two approaches to the analysis of the coaxial interaction of vortex rings. Int. J. Fluid Mech. Res. 30, 166–183 (2003) · doi:10.1615/InterJFluidMechRes.v30.i2.40
[60] Gröbli, W.: Spezielle Probleme über die Bewegung geradliniger paralleler Wirbelfäden. Vierteljschr. Nat. Ges. Zürich 22, 37–82, 129–168 (1877) · JFM 09.0675.01
[61] Gurzhii A.A., Konstantinov M.Yu.: Head-on collision of two coaxial vortex rings in an ideal fluid. Fluid Dyn. 24, 538–541 (1989) · doi:10.1007/BF01052414
[62] Gurzhii, A.A., Konstantinov, M.Yu.: The influence of relative sizes of coaxial vortex rings cores on the characteristics of their interaction (in Ukrainian). Dopovidi Akad. Nauk UkrSSR. Ser. A, No 3, 38–41 (1989)
[63] Gurzhii, A.A., Konstantinov, M.Yu., Meleshko, V.V.: Interaction of thin coaxial vortex rings in an ideal fluid. (in Ukrainian). Dopovidi Akad. Nauk UkrSSR. Ser. A, No 4, 40–44 (1987)
[64] Gurzhii A.A., Konstantinov M.Yu., Meleshko V.V.: Interaction of coaxial vortex rings in an ideal fluid. Fluid Dyn. 23, 224–229 (1988) · doi:10.1007/BF01051891
[65] Gurzhii A.A., Meleshko V.V.: Sound emission by a system of vortex rings. Acoust. Phys. 42, 43–50 (1996)
[66] Helmholtz H.: Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen. J. Reine Angew. Math. 55, 25–55 (1858) · ERAM 055.1448cj · doi:10.1515/crll.1858.55.25
[67] Helmholtz H.: On integrals of the hydrodynamical equations, which express vortex-motion. Phil. Mag. (Ser. 4) 33, 485–510 (1867)
[68] Helmholtz H.: Ueber discontinuirliche Flüssigkeitsbewegungen. Monatsber. Akad. Wiss. Berlin 23, 215–228 (1868) · JFM 01.0341.03
[69] Helmholtz H.: Sui movimenti dei liquidi. Nuovo Cimento (Ser. 2) 1, 289–304 (1869) · doi:10.1007/BF02739634
[70] Helmholtz H.: Wissenschaftlische Abhandlungen. Band I. Barth, Leipzig (1882)
[71] von Helmholtz, H.: Autobiographisches Tischrede bei der Feier des 70. Geburtstages. In: Ansprachen und Reden, gehalten bei der am 2. November 1891 zu Ehren von Hermann von Helmholtz veranstalteten Feier, SS. 46–59. Hirschwald, Berlin (1892). (Reprinted in 1966)
[72] von Helmholtz, H.: Zwei hydrodynamische Abhandlungen. I. Ueber Wirbelbewegungen (1858). II. Ueber discontinuirliche Flüssigkeitsbewegungen (1868). In: Wangerin, A. (ed.) Ostwalds Klassiker der exakten Wissenschaften, No 79. Engelmann, Leipzig (1896). (Reprinted in 1953, 1996)
[73] Helmholtz H.: Two Studies in Hydrodynamics. (in Russian). Palas, Moscow (1902)
[74] von Helmholtz H.: An autobiographical sketch. An address delivered on the occasion of his Jubilee in Berlin, 1891. In: Kahl, R. (eds) Selected Writings of Hermann von Helmholtz, pp. 466–478. Wesleyan University Press, Middletown (1971)
[75] von Helmholtz H.: On integrals of the hydrodynamical equations, which express vortex motion. Int. J. Fusion Energy 1, 41–68 (1978)
[76] Helmholtz H.: On integrals of the hydrodynamical equations, which express vortex motion. (in Russian). Nonlinear Dyn. 2, 473–507 (2006)
[77] Hernández R.H., Cibert B., Béchet C.: Experiments with vortex rings in air. Europhys. Lett. 75, 743–749 (2006) · doi:10.1209/epl/i2006-10171-0
[78] Hicks W.M.: Report on recent progress in hydrodynamics, II. Rep. Brit. Assos. Adv. Sci. 52, 39–70 (1882)
[79] Hicks W.M.: Researches on the theory of vortex rings. Part II. Phil. Trans. R. Soc. Lond. A 176, 725–780 (1885) · JFM 17.1100.01 · doi:10.1098/rstl.1885.0015
[80] Hicks W.M.: The mass carried forward by a vortex. Phil. Mag. (ser. 6) 38, 597–612 (1919) · JFM 47.0756.03
[81] Hicks W.M.: On the mutual threading of vortex rings. Proc. R. Soc. Lond. A 102, 111–131 (1922) · JFM 49.0748.01 · doi:10.1098/rspa.1922.0075
[82] Hill M.J.M.: On a spherical vortex. Phil. Trans. R. Soc. Lond. A 185, 213–245 (1894) · JFM 25.1471.01 · doi:10.1098/rsta.1894.0006
[83] Hu D.L., Chau B., Bush J.W.M.: The hydrodynamics of water strider locomotion. Nature 424, 663–666 (2003) · doi:10.1038/nature01793
[84] Joukovskii N.E.: [Helmholtz’s] works on mechanics. (in Russian). In: Stoletov, A.G. (eds) Hermann von Helmholtz 1821–1891. Public lectures delivered at the Imperior Moscow University for the Helmholtz fund, pp. 37–52. Moscow University Press, Moscow (1892)
[85] Joukovskii N.E.: A note on the motion of vortex rings. (in Russian). Mat. Sbornik 26, 483–490 (1907)
[86] Joukowski N.: Bases théoriques de l’aéronautique. Aérodynamique: Cours professé a l’École Impériale Technique de Moscou. Gauthier-Villars, Paris (1916)
[87] Jungnickel C., McCormmach R.: Intellectual Mastery of Nature: Theoretical Physics from Ohm to Einstein, vol 1. The Torch of Mathematics 1800–1870. University of Chicago Press, Chicago (1990) · Zbl 1181.01033
[88] Jungnickel C., McCormmach R.: Intellectual Mastery of Nature: Theoretical Physics from Ohm to Einstein, vol. 2. The Now Mighty Theoretical Physics 1870–1925. University of Chicago Press, Chicago (1990) · Zbl 1181.01033
[89] Kambe T.: Acoustic emissions by vortex motions. J. Fluid Mech. 173, 643–666 (1986) · Zbl 0624.76110 · doi:10.1017/S0022112086001301
[90] Kambe T.: Elementary Fluid Mechanics. World Scientific, Singapore (2007) · Zbl 1118.76001
[91] Kambe T., Minota T.: Sound radiation from vortex systems. J. Sound Vibr. 74, 61–72 (1981) · Zbl 0452.76063 · doi:10.1016/0022-460X(81)90491-0
[92] Kambe T., Minota T.: Acoustic wave radiated from head-on collision of two vortex rings. Proc. R. Soc. Lond. A 386, 277–308 (1983) · doi:10.1098/rspa.1983.0037
[93] Kirchhoff G.: Vorlesungen über Matematische Physik: Mechanik. Teubner, Leipzig (1876)
[94] Klein F.: Über die Bildung von Wirbeln in reibungslosen Flüssigkeiten. Z. Math. Phys. 58, 259–262 (1910) · JFM 41.0819.02
[95] Koenigsberger L.: The investigations of Hermann von Helmholtz on the fundamental principles of mathematics and mechanics. Smithsonian Inst. Annu. Rep. 51, 93–123 (1896)
[96] Koenigsberger L.: Hermann von Helmholtz. Vieweg, Braunschweig (1902)
[97] Koenigsberger L.: Hermann von Helmholtz. (With a preface by Lord Kelvin). Clarendon, Oxford (1906) · JFM 37.0015.02
[98] Konstantinov M.Yu.: Chaotic phenomena in the interaction of vortex rings. Phys. Fluids 6, 1752–1767 (1994) · Zbl 0829.76015 · doi:10.1063/1.868237
[99] Konstantinov M.Yu.: Numerical investigation of the interaction of coaxial vortex rings. Int. J. Num. Meth. Heat Fluid Flow 7, 120–140 (1997) · Zbl 0973.76592 · doi:10.1108/09615539710163220
[100] Konstantinov M.Yu., Meleshko V.V.: Motion of vortex ring generated in an ideal fluid near solid walls. Fluid Mech. Soviet Res. 20(3), 1–6 (1991) · Zbl 0741.76010
[101] Kragh H.: The vortex atom: A Victorian theory of everything. Centaurus 44, 32–114 (2002) · doi:10.1034/j.1600-0498.2002.440102.x
[102] Krueger P.S., Moslemi A.A., Nichols J.T., Bartol I.K., Stewart W.J.: Vortex rings in bio-inspired and biological jet propulsion. Adv. Sci. Tech. 58, 237–246 (2008) · doi:10.4028/www.scientific.net/AST.58.237
[103] Lamb H.: A Treatise on the Mathematical Theory of the Motion of Fluids. Cambridge University Press, Cambridge (1879)
[104] Lamb H.: The motion of fluids. Nature 22, 145 (1880) · doi:10.1038/022145a0
[105] Lamb H.: Hydrodynamics. 2nd edn. Cambridge University Press, Cambridge (1895) · JFM 26.0868.02
[106] Lamb H.: Hydrodynamics. 3nd edn. Cambridge University Press, Cambridge (1906) · JFM 36.0817.07
[107] Lamb H.: Hydrodynamics. 4nd edn. Cambridge University Press, Cambridge (1916) · Zbl 0828.01012
[108] Lamb H.: Hydrodynamics. 5nd edn. Cambridge University Press, Cambridge (1924) · JFM 50.0567.01
[109] Lamb H.: Hydrodynamics. 6nd edn. Cambridge University Press, Cambridge (1932) · JFM 58.1298.04
[110] von Laue M.: Zum 50. Todestage von Hermann v. Helmholtz (8. September 1944.). Naturwissenschaften 32, 206–207 (1944) · doi:10.1007/BF01475256
[111] Lebedinskii A.V., Frankfurt U.I., Frenk A.M.: Helmholtz. (in Russian). Nauka, Moscow (1966)
[112] Levy E.: The Science of Water: The Foundation of Modern Hydraulics. American Society of Civil Engineers, New York (1995)
[113] Levy H., Forsdyke A.G.: The stability of an infinite system of circular vortices. Proc. R. Soc. Lond. A 114, 594–604 (1927) · JFM 53.0788.01 · doi:10.1098/rspa.1927.0061
[114] Levy H., Forsdyke A.G.: The vibrations of an infinite system of vortex rings. Proc. R. Soc. Lond. A 116, 352–379 (1927) · JFM 53.0788.02 · doi:10.1098/rspa.1927.0140
[115] Lewis T.C.: On the images of vortices in a spherical vessel. Q. J. Pure Appl. Math. 16, 338–347 (1879) · JFM 11.0679.02
[116] Lewis T.C.: Some cases of vortex motion. Mess. Math. 9, 93–95 (1880) · JFM 11.0681.02
[117] Lichtenstein, L.: Über einige Existenzprobleme der Hydrodynamik homogener, unzusammendrückbarer, reibungsloser Flüssigkeiten und die Helmholtzschen Wirbelsätze, Math. Z. 23, 89–154, 310–316 (1925) · JFM 51.0658.01
[118] Lichtenstein L.: Grundlagen der Hydromechanik. Springer, Berlin (1929) · JFM 55.1124.01
[119] Lim T.T.: A note on the leapfrogging between two coaxial vortex rings at low Reynolds numbers. Phys. Fluids 9, 239–241 (1997) · doi:10.1063/1.869160
[120] Lim T.T., Nickels T.B.: Instability and reconnection in the head-on collision of two vortex rings. Nature 357, 225–227 (1992) · doi:10.1038/357225a0
[121] Lim T.T., Nickels T.B.: Vortex rings. In: Green, S.I. (eds) Fluid Vortices, pp. 95–153. Kluwer, Dordrecht (1995)
[122] Lim T.T., Nickels T.B., Chong M.S.: A note on the cause of rebound in the head-on collision of a vortex ring with a wall. Exp. Fluids 12, 41–48 (1991) · doi:10.1007/BF00226564
[123] Liow Y.S.K., Thompson M.C., Hourigan K.: Sound generated by a pair of axisymmetric coaxial vortex rings. AIAA J. 43, 326–336 (2005) · doi:10.2514/1.5797
[124] Lodge O.J.: The stream-lines of moving vortex-rings. Phil. Mag.(ser. 5) 20, 67–70 (1885)
[125] Love A.E.H.: On recent English researches in vortex motion. Math. Ann. 30, 326–344 (1887) · JFM 19.1046.01 · doi:10.1007/BF01443950
[126] Love A.E.H.: On the motion of paired vortices with a common axis. Proc. Lond. Math. Soc. 25, 185–194 (1894) · JFM 25.1468.01 · doi:10.1112/plms/s1-25.1.185
[127] Love A.E.H.: Hydrodynamik: Theoretische Ausführungen. In: Klein, F., Müller, C. (eds) Encyklopädie der Mathematischen Wissenschaften, Band. IV/3, SS, pp. 84–147. Teubner, Leipzig (1901) · JFM 32.0753.03
[128] Love A.E.H.: Développements d’hydrodynamique. In: Molk, J., Appell, P. (eds) Encyclopédie des Sciences Mathématiques, vol. IV/5, pp. 102–208. Gauthier-Villars, Paris (1913)
[129] Lugt H.: Vortex Flow in Nature and Technology. Wiley, New York (1983)
[130] Lugt H.: Introduction to Vortex Theory. Vortex Flow Press, Potomac (1996) · Zbl 0893.76014
[131] Marshall J.S.: The flow induced by periodic vortex rings wrapped around a columnar vortex core. J. Fluid Mech. 345, 1–30 (1997) · Zbl 0895.76029 · doi:10.1017/S0022112097005739
[132] Maxwell J.C.: Manuscript fragments on the stability of fluid motion [a question set for the Cambridge Mathematical Tripos in 1866]. In: Harman, P.M. (eds) The Scientific Letters and Papers of James Clerk Maxwell, vol. 2, pp. 241–244. Cambridge University Press, Cambridge (1995)
[133] Maxwell J.C.: Letter to Peter Guthrie Tait, 13 November 1867. In: Harman, P.M. (eds) The Scientific Letters and Papers of James Clerk Maxwell, vol. 2, pp. 321–322. Cambridge University Press, Cambridge (1995)
[134] Maxwell J.C.: Letter to Peter Guthrie Tait, 18 July 1868. In: Harman, P.M. (eds) The Scientific Letters and Papers of James Clerk Maxwell, vol 2, pp. 391–394. Cambridge University Press, Cambridge (1995)
[135] Maxwell J.C.: Letter to William Thomson, 18 July 1868. In: Harman, P.M. (eds) The Scientific Letters and Papers of James Clerk Maxwell, vol. 2, pp. 398–403. Cambridge University Press, Cambridge (1995)
[136] Maxwell J.C.: Letter to William Thomson, 6 October 1868. In: Harman, P.M. (eds) The Scientific Letters and Papers of James Clerk Maxwell, vol. 2, pp. 446–448. Cambridge University Press, Cambridge (1995)
[137] Maxwell J.C.: Hermann Ludwig Ferdinand Helmholtz. Nature 15, 389–391 (1877) · doi:10.1038/015389a0
[138] Maxwell J.C.: Atom. Encyclopædia Britannica. 9th edn. 3, 36–48 (1878)
[139] Maxworthy T.: Some expermental studies of vortex rings. J. Fluid Mech. 81, 465–495 (1977) · doi:10.1017/S0022112077002171
[140] Maxworthy T.: Comments on ”Preliminary study of mutual slip-through of a pair of vortices”. Phys. Fluids 22, 200 (1979) · doi:10.1063/1.862467
[141] McKendrick J.G.: Hermann Ludwig Ferdinand von Helmholtz. Longmans, Green & Co, New York (1899) (Reprinted in 2007, 2008)
[142] Meleshko V.V., Aref H.: A bibliography of vortex dynamics 1858–1956. Adv. Appl. Mech. 41, 197–292 (2007) · doi:10.1016/S0065-2156(07)41003-1
[143] Meleshko V.V., Konstantinov M.Yu.: Dynamics of Vortex Structures. (in Russian). Naukova Dumka, Kiev (1993)
[144] Meleshko V.V., Konstantinov M.Yu., Gurzhii A.A.: Ordered and chaotic movement in the dynamics of three coaxial vortex rings. J. Math. Sci. 68, 711–714 (1994) · doi:10.1007/BF01249412
[145] Milne-Thomson L.M.: Theoretical Aerodynamics. Macmillan, New York (1966) · Zbl 0029.28204
[146] Minota T., Nishida M., Lee M.G.: Head-on collision of two compressible vortex rings. Fluid Dyn. Res. 22, 43–60 (1998) · doi:10.1016/S0169-5983(97)00025-7
[147] Miyazaki T., Kambe T.: Axisymmetrical problem of vortex sound with solid surfaces. Phys. Fluids 29, 4006–4015 (1986) · Zbl 0616.76092 · doi:10.1063/1.865741
[148] Moffatt K.: Vortex dynamics: The legasy of Helmholtz and Kelvin. In: Borisov, A.V., Kozlov, V.V., Mamaev, I.S., Sokolovskiy, M.A. (eds) IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence Proceedings of the IUTAM Symposium held in Moscow, 25–30 August 2006, pp. 1–10. Springer, Dordrecht (2008)
[149] Mohseni K.: A formulation for calculating the translational velocity of a vortex ring or pair. Bioisp. Biomim. 1, S57–S64 (2006) · doi:10.1088/1748-3182/1/4/S08
[150] Norbury J.: A family of steady vortex rings. J. Fluid Mech. 57, 417–431 (1973) · Zbl 0254.76018 · doi:10.1017/S0022112073001266
[151] Northrup, E.F. An experimental study of vortex motions in liquids. J. Franklin Inst. 172, 211–226, 345–368 (1911)
[152] Northrup E.F.: A photographic study of vortex rings in liquids. Nature 88, 463–468 (1912) · doi:10.1038/088463b0
[153] Novikov E.A.: Hamiltonian description of axisymmetric vortex flows and the system of vortex rings. Phys. Fluids 28, 2921–2922 (1985) · Zbl 0576.76024 · doi:10.1063/1.865213
[154] Ogawa A.: Vortex Flow. CRC Press, Boca Raton (1993) · Zbl 0801.35125
[155] Oshima Y.: Head-on collision of two vortex rings. J. Phys. Soc. Jpn. 44, 328–331 (1978) · doi:10.1143/JPSJ.44.328
[156] Oshima Y., Kambe T., Asaka S.: Interaction of two vortex rings moving along a common axis of symmetry. J. Phys. Soc. Jpn. 38, 1159–1166 (1975) · doi:10.1143/JPSJ.38.1159
[157] Oshima Y., Noguchi T., Oshima K.: Numerical study of interaction of two vortex rings. Fluid Dyn. Res. 1, 215–227 (1986) · doi:10.1016/0169-5983(87)90006-2
[158] Pedrizzetti G., Novikov E.A.: Instability and chaos in axisymmetric vortex-body interactions. Fluid Dyn. Res. 12, 129–151 (1993) · doi:10.1016/0169-5983(93)90018-6
[159] Pocklington H.C.: The complete system of the periods of a hollow vortex ring. Phil. Trans. R. Soc. Lond. A 186, 603–619 (1895) · JFM 26.0875.01 · doi:10.1098/rsta.1895.0017
[160] Poincaré H.: Théorie des tourbillons. Carré, Paris (1893)
[161] Prandtl L., Tietjens O.G.: Fundamentals of Hydro- and Aeromechanics. Dover, New York (1957) · Zbl 0078.39603
[162] Ramsay A.S.: A Treatise of Hydromechanics. Part II. Hydrodynamics. Bell, London (1913)
[163] Rayleigh L.: Fluid motions. Proc. R. Instn. Gt. Britain 21, 70–83 (1914) · JFM 12.0711.02
[164] Rayner J.M.V.: A vortex theory of animal flight. Part 1. The vortex wake of a hoyering animal. J. Fluid Mech. 91, 697–730 (1979) · Zbl 0436.76019 · doi:10.1017/S0022112079000410
[165] Rayner J.M.V.: A vortex theory of animal flight. Part 2. The forward flight of birds. J. Fluid Mech. 91, 731–763 (1979) · Zbl 0436.76020 · doi:10.1017/S0022112079000422
[166] Reusch E.: Ueber Ringbildung in Flüssigkeiten. Ann. Phys. Chem. (Ser. 2) 110, 309–316 (1860) · doi:10.1002/andp.18601860611
[167] Reynolds O.: On the resistance encountered by vortex rings and the relation between vortex rings and the stream-lines of a disc. Nature 14, 477–479 (1876)
[168] Reynolds O.: On vortex motion. Proc. R. Instn. Gt. Britain 8, 272–279 (1877)
[169] Reynolds O.: Vortex rings. [The Motion of Vortex Rings. By J.J. Thomson (London: Macmillan and Co., 1883)]. Nature 29, 193–195 (1883) · doi:10.1038/029193a0
[170] Reynolds O.: Study of fluid motion by means of coloured bands. Nature 50, 161–164 (1894) · doi:10.1038/050161a0
[171] Riley N.: On the behaviour of pairs of vortex rings. Q. J. Mech. Appl. Math. 46, 521–539 (1993) · Zbl 0783.76027 · doi:10.1093/qjmam/46.3.521
[172] Riley N.: The fascination of vortex rings. Appl. Sci. Res. 58, 169–189 (1998) · Zbl 0912.76012 · doi:10.1023/A:1000723416667
[173] Riley N., Stevens D.P.: A note on leapfrogging vortex rings. Fluid Dyn. Res. 11, 235–244 (1993) · doi:10.1016/0169-5983(93)90114-P
[174] Roberts P.H.: A Hamiltonian theory for weakly interacting vortices. Mathematika 19, 169–179 (1972) · Zbl 0256.76015 · doi:10.1112/S0025579300005611
[175] Roberts P.H., Donnelly R.J.: Dynamics of vortex rings. Phys. Lett. A 31, 137–138 (1970) · doi:10.1016/0375-9601(70)90193-3
[176] Rogers W.B.: On the formation of rotating rings by air and liquids under certain conditions of discharge. Am. J. Sci. (Ser. 2) 26, 246–258 (1858)
[177] Rott N.: Vortex drift: a historical survey. In: Fung, K.-Y. (eds) Symposium on Aerodynamics & Aeroacoustics, Tuscon, Arizona, 1–2 March 1993, pp. 173–186. World Scientific, Singapore (1994)
[178] Rott N., Cantwell B.: Vortex drift, I. Dynamic interpretation. Phys. Fluids 5, 1443–1450 (1993) · Zbl 0804.76034 · doi:10.1063/1.858581
[179] Rücker, A.W.: The physical work of von Helmholtz. Nature 51, 472–475, 493–495 (1895) · JFM 26.0025.02
[180] Ryu K.W., Lee D.J.: Interaction between a vortex ring and a rigid sphere. Eur. J. Mech. B-Fluids 16, 645–664 (1997) · Zbl 0902.76022
[181] Saffman P.G.: The velocity of viscous vortex rings. Stud. Appl. Math. 49, 371–380 (1970) · Zbl 0224.76032
[182] Saffman P.G.: Dynamics of vorticity. J. Fluid Mech. 106, 49–58 (1981) · Zbl 0465.76020 · doi:10.1017/S0022112081001511
[183] Saffman P.G.: Vortex Dynamics. Cambridge University Press, Cambridge (1992) · Zbl 0777.76004
[184] Samimy G.K., Breuer K.S., Leal L.G., Steen P.H.: A Galery of Fluid Motion. Cambridge University Press, Cambridge (2003)
[185] Sau R., Manesh K.: Passive scalar mixing in vortex rings. J. Fluid Mech. 582, 449–461 (2007) · Zbl 1177.76106 · doi:10.1017/S0022112007006349
[186] Scheel H. (eds): Gedanke von Helmholtz über schöpferische Impulse und über das Zusammenwirken Verschiedener Wissenschafttszweige. sAkademie, Berlin (1972) · Zbl 0258.00020
[187] Schram C., Hirschberg A.: Application of vortex sound theory to vortex-pairing noise: sensitivity to errors in flow data. J. Sound Vibr. 266, 1079–1098 (2003) · doi:10.1016/S0022-460X(02)01630-9
[188] Schram C., Hirschberg A., Verzicco R.: Sound produced by vortex pairing: prediction based on particle image velocimetry. AIAA J. 42, 2234–2244 (2004) · doi:10.2514/1.13570
[189] Schuster A.: The Progress of Physics During 33 years, 1875–1908. Cambridge University Press, Cambridge (1911) (Reprinted in 1975)
[190] Schwenk T.: Sensitive Chaos. 2nd edn. Steiner, London (1996)
[191] Sen, N.R.: On circular vortex rings of finite section in incompressible fluids. Bull. Calcutta Math. Soc. 13, 117–140 (1922/1923)
[192] Shadden S.C., Dabiri J.O., Marsden J.E.: Lagrangian analysis of fluid transport in empirical vortex ring flows. Phys. Fluids 18, 047105-1–047105-11 (2006) · Zbl 1185.76700 · doi:10.1063/1.2189885
[193] Shariff K., Leonard A.: Vortex rings. Ann. Rev. Fluid Mech. 24, 235–279 (1992) · Zbl 0743.76024 · doi:10.1146/annurev.fl.24.010192.001315
[194] Shariff K., Leonard A., Ferziger J.H.: Dynamical systems analysis of fluid transport in time-periodic vortex ring flows. Phys. Fluids 18, 047104-1–047104-11 (2006) · Zbl 1185.76701 · doi:10.1063/1.2189867
[195] Shariff K., Leonard A., Ferziger J.H.: A contour dynamics algorithm for axisymmetric flow. J. Comput. Phys. 227, 9044–9062 (2008) · Zbl 1146.76041 · doi:10.1016/j.jcp.2007.10.005
[196] Shariff K., Leonard A., Zabusky N.J., Ferziger J.H.: Acoustics and dynamics of coaxial interacting vortex rings. Fluid Dyn. Res. 3, 337–343 (1988) · doi:10.1016/0169-5983(88)90088-3
[197] Shashikanth B.N., Marsden J.E.: Leapfrogging vortex rings: Hamiltonian structure, geometric phases and discrete reduction. Fluid Dyn. Res. 33, 333–356 (2003) · Zbl 1032.76537 · doi:10.1016/j.fluiddyn.2003.05.001
[198] Silliman R.H.: William Thomson: Smoke rings and nineteenth-century atomism. Isis 54, 461–474 (1963) · doi:10.1086/349764
[199] Smith C., Wise M.N.: Energy and Empire: A biographical Study of Lord Kelvin. Cambridge University Press, Cambridge (1989)
[200] Sommerfeld A.: Lectures on Theoretical Physics., vol. 2. Mechanics of Deformable Bodies. Academic, New York (1950) · Zbl 0038.37107
[201] Stokes G.G.: On the theories of the internal friction of fluids in motion, and the equilibrium and motion of elastic solids. Trans. Cambr. Phil. Soc. 8, 287–319 (1845)
[202] Sullivan I.S., Niemela J.J., Hershberger R.E., Bolster D., Donnelly R.J.: Dynamics of thin vortex rings. J. Fluid Mech. 609, 319–347 (2008) · Zbl 1147.76011 · doi:10.1017/S0022112008002292
[203] Tait P.G.: Lectures on Some Recent Advances in Physical Science, with a Special Lecture on Force. 2nd edn. Macmillan, London (1876)
[204] Tang S.K., Ko N.W.M.: Sound generation by a vortex ring collision. J. Acoust. Soc. Am. 98, 3418–3427 (1995) · doi:10.1121/1.413793
[205] Tang S.K., Ko N.W.M.: On sound generated by the interaction of two inviscid vortex rings moving in the same direction. J. Sound Vibr. 187, 287–310 (1996) · doi:10.1006/jsvi.1995.0522
[206] Tang S.K., Ko N.W.M.: Basic sound generation mechanisms in inviscid vortex interactions at low Mach number. J. Sound Vibr. 262, 87–115 (2003) · doi:10.1016/S0022-460X(02)01029-5
[207] Taylor G.I.: Sir Horace Lamb, F.R.S. Nature 135, 255–257 (1935) · JFM 61.0955.02 · doi:10.1038/135255a0
[208] Taylor G.I.: Formation of a vortex ring by giving an impulse to a circular disk and then dissolving it away. J. Appl. Phys. 24, 104 (1953) · doi:10.1063/1.1721114
[209] Thompson S.P.: The Life of William Thomson, Baron Kelvin of Largs, vol. I. Macmillan, London (1910) · JFM 41.0019.01
[210] Thomson J.J.: On the vibrations of a vortex ring, and the action of two vortex rings upon each other. Phil. Trans. R. Soc. Lond. A 173, 493–521 (1882) · JFM 14.0776.01 · doi:10.1098/rstl.1882.0010
[211] Thomson J.J.: A Treatise on the Motion of Vortex Rings An Essay to Which the Adams Prize was Adjudged in 1882, in the University of Cambridge. Macmillan, London (1883) (Reprinted in 1968)
[212] Thomson J.J., Newall H.F.: On the formation of vortex rings by drops falling into liquids, and some allied phenomena. Proc. R. Soc. Lond. 39, 417–436 (1885) · doi:10.1098/rspl.1885.0034
[213] Thomson W.: On vortex atoms. Phil. Mag. (Ser. 4) 34, 15–24 (1867) · JFM 15.0767.01
[214] Thomson W.: [The translatory velocity of a circular vortex ring]. Phil. Mag. (Ser. 4) 34, 511–512 (1867)
[215] Thomson W.: On vortex motion. Trans. R. Soc. Edinburgh 25, 217–260 (1869) · JFM 02.0405.01
[216] Tokaty G.A.: A History and Philosophy of Fluid Mechanics. Dover, New York (1994) · Zbl 0853.76001
[217] Truesdell C: The Kinematics of Vorticity. Indiana University Press, Bloomington (1954) · Zbl 0056.18606
[218] Turner R.S.: Helmholtz, Hermann von. In: Gillispie, C.C. (eds) Dictionary of Scientific Biography, vol VI, pp. 241–253. Scribner’s Sons, New York (1972)
[219] Uchiyama T., Yagami H.: Numerical simulation for the collision between a vortex ring and solid particles. Powder Tech. 188, 73–80 (2008) · doi:10.1016/j.powtec.2008.03.015
[220] Vasil’ev N.S.: Reduction of the equations of motion of coaxial vortex rings to canonical form. (in Russian). Zap. Mat. Otd. Novoross. Obshch. Estest. 21, 1–12 (1913)
[221] Vasil’ev N.S.: On the motion of an infinite row of coaxial circular vortex rings with the same initial radii. (in Russian). Zap. Fiz.-Mat. Fak. Imp. Novoross. Univ. 10, 1–44 (1914)
[222] Verzicco R., Iafati A., Ricardi G., Fatica M.: Analysis of the sound generated by the pairing of two axisymmetric co-rotating vortex rings. J. Sound Vibr. 200, 347–358 (1997) · Zbl 1235.76170 · doi:10.1006/jsvi.1996.0714
[223] de Villamil R.: ABC of Hydrodynamics. Spon, London (1912)
[224] de Villamil R.: Motion of Liquids. Spon, London (1914) · JFM 45.1106.04
[225] Villat H.: Leçons sur la théorie des tourbillons. Gauthier-Villars, Paris (1930) · JFM 56.1247.14
[226] Wakelin S.L., Riley N.: Vortex rings interactions II. Inviscid models. Q. J. Mech. Appl. Math. 49, 287–309 (1996) · Zbl 0960.76513 · doi:10.1093/qjmam/49.2.287
[227] Wakelin S.L., Riley N.: On the formation and propagation of vortex rings and pairs of vortex rings. J. Fluid Mech. 332, 121–139 (1997) · Zbl 0892.76012
[228] Weidman P.D., Riley N.: Vortex ring pairs: numerical simulation and experiment. J. Fluid Mech. 257, 311–337 (1993) · doi:10.1017/S002211209300309X
[229] Werner F.: Hermann Helmholtz’ Heidelberger Jahre (1858–1871). Springer, Berlin (1997)
[230] Wien W.: Lehrbuch der Hydrodynamik. Hirzel, Leipzig (1900) · JFM 31.0717.04
[231] Wien, W.: Hydrodynamische Untersuchungen von H. v. Helmholtz. Sber. Preuss. Akad. Wiss. 716–736 (1904)
[232] Wilkens F., Jacobi M., Schwenk W.: Understanding Water. 2nd edn. Floris Books, Edinburgh (2005)
[233] Wood R.W.: Vortex rings. Nature 63, 418–420 (1901) · doi:10.1038/063418c0
[234] Wu Y.-Z., Ma H.-Y., Zhou M.-D.: Vorticity and Vortex Dynamics. Springer, Berlin (2006)
[235] Yamada H., Konsaka T., Yamabe H., Matsui T.: Flow field produced by a vortex ring near a plane wall. J. Phys. Soc. Jpn. 51, 1663–1670 (1982) · doi:10.1143/JPSJ.51.1663
[236] Yamada H., Matsui T.: Preliminary study of mutual slip-through of a pair of vortices. Phys. Fluids 21, 292–294 (1978) · doi:10.1063/1.862206
[237] Yamada H., Matsui T.: Mutual slip-through of a pair of vortex rings. Phys. Fluids 22, 1245–1249 (1979) · doi:10.1063/1.862739
[238] Ye Q., Chu C., He Y.: Coaxial interactions of two vortex rings or of a ring with a body. Acta Mech. Sinica 11, 219–228 (1995) · Zbl 0854.76022 · doi:10.1007/BF02487725
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