Theor. Comput. Fluid Dyn.DOI 10.1007/s00162-009-0148-z

SPECIAL ISSUE ORIGINAL ARTICLE

Viatcheslav V. Meleshko

Coaxial axisymmetric vortex rings:150 years after Helmholtz

Received: 7 April 2009 / Accepted: 24 June 2009© Springer-Verlag 2009

Abstract This article addresses the fascinating 150 years history of the classical Helmholtz paper that laidthe foundation of the vortex dynamics. Among general theorems on vortex motion, this memoir contains thespecial section on circular vortex filaments and axisymmetric vortex rings, in particular. The objective of thisarticle is both to clarify some purely mathematical questions connected with the Dyson model of coaxial vortexrings 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 examplesboth of regular and chaotic motion of several vortex rings in an unbounded fluid.

Keywords Vortex rings · Helmholtz on vortex motion · Dyson model of thin coaxial vortex rings

PACS 47.32.cf · 47.15.ki · 47.15.G

In the historical record of science, the name of Helmholtz stands unique in grandeur, as a master andleader in mathematics, biology, and in physics. His admirable theory of vortex rings is one of the mostbeautiful of all beautiful pieces of mathematical work hitherto done in the dynamics of incompressiblefluids.Lord Kelvin [97, p. III]

1 Introduction

The epigraph opens a preface to the English abridged translation of the monumental three volume Germantreatise [96]. These words were written by Sir William Thomson (1824–1907), later Lord Kelvin (or, moreprecisely, Baron Kelvin of Largs) almost at the end of his long life which was so largely devoted to fluiddynamics, when he definitely could compare inputs into fluid dynamics made by many scientists of the nine-teenth century. The subject of vortex dynamics (and, in particular, a study of vortex rings) can fairly be saidto have been initiated by the seminal paper [66] of Hermann Ludwig Ferdinand Helmholtz (1821–1894)1 thatappeared 150 years ago in one of the leading scientific journals of that time (Fig. 1). Among the authors of 22papers in this volume, there are names of Jacobi, Cayley, Christoffel, Clebsch, Bjerknes, and other prominentscientists.

1 In 1882 Helmholtz was elevated by the German Emperor Wilhelm I to the ranks of the hereditary nobility and added ‘von’to his name. As this occurred later, he appears throughout this article simply as Helmholtz.

Communicated by H. Aref

V. V. Meleshko (B)Department of Theoretical and Applied Mechanics, Kiev National Taras Shevchenko University, 01601 Kiev, UkraineE-mail: meleshko@univ.kiev.ua

V. V. Meleshko

Fig. 1 Helmholtz’s (1858) paper [66]. a Title page of the journal. b First page. c Hermann Helmholtz, circa 1858. From http://www.digizeitschriften.de/resolveppn/GDZPPN002150212 and http://www-history.mcs.st-and.ac.uk/PictDisplay/Helmholtz.html

Lord Kelvin continues [97, p. IV]:

The professional career of Helmholtz was unparalleled in the history of professions. He was MilitarySurgeon in the Prussian army five years; Teacher of Anatomy in the Academy of Arts in Berlin one year;Professor of Pathology and Physiology in Königsberg six years; Professor of Anatomy in Bonn threeyears; Professor of Physiology in Heidelberg thirteen years; Professor of Physics in the University ofBerlin about twenty years till he became Director of the new ‘Physikalisch-Technische Reichsanstalt’.He occupied this post during the last years of his life, still continuing to give lectures as Professor ofPhysics.

The life and scientific work of this outstanding natural philosopher of the nineteenth century—upon hisdeath obituary notices appeared in more than 50 scientific journals all over the world—has been highly praisedby his contemporaries [49,84,95,137,179]. A man of tremendous energy and curiosity, he produced more than200 articles and books, among them classic papers on the conservation of energy and the vortex motion, a greathandbook of physiological optics, and a treatise on the physiology and psychology of sound. Besides funda-mental biography [96,97], there exists many books and articles in several languages devoted to Helmholtz,e.g., [26,27,30,41,111,141,229]. (Oddly enough that in an extensive collection [25] of 15 essays that describe,analyze, and interpret virtually all areas of Helmholtz’s work in medicine, heat and nerve physiology, colorand vision theory, physiological optics and acoustics, energy conservation, electrodynamics, mathematics,chemical thermodynamics, epistemology, philosophy of science, etc., his important results on hydrodynamicsand meteorology have been omitted. On the other hand, in comprehensive studies [87,88,186], Helmholtz’sinput in mechanics and fluid dynamics is fully addressed.) A short overview of Helmholtz’s life and scientificaccomplishments are presented on http://www-history.mcs.st-and.ac.uk/Biographies/Helmholtz.html whichalso contains 14 pictures of Helmholtz in various stages of his life, including three ones on stamps.

There exists a great number of editions in several languages (German, English, French, Italian, Russian,Slovac, Japanese) of Helmholtz scientific writings, treatises on acoustics and physiological optics, popularlectures and addresses, lectures on theoretical physics, etc. A rather complete list of these books can be found,e.g. in Deutsche Nationalbibliothek at the address http://d-nb.info/gnd/11854893X, Library of Congress at theaddress http://catalog.loc.gov/, and many other University libraries (e.g., http://katalog.ub.uni-heidelberg.de/)scattered all over the world. The digital libraries, e.g. Internet Archive at the address http://www.archive.org/,the Open Library at the address http://openlibrary.org/, the WorldCat at the address http://www.worldcat.org/,the German digital journal library at the address http://www.digizeitschriften.de/, and Max Planck Institute forthe History of Science at the address http://vlp.mpiwg-berlin.mpg.de/library/ contain a free access to scans ofbooks and some journal papers by and about Helmholtz.

Helmholtz’s motivations for taking up this new research interest remain unclear, for at that time he wasprofessor of physiology and anatomy at the University of Bonn, and the memoir appeared in the year of his

Coaxial axisymmetric vortex rings

Fig. 2 Other publications of Helmholtz’s (1858) paper [66]. a First page of English translation [67]. b Title page of the book [72].From http://www.archive.org/details/zweihydrodynamis00helmuoft. c Title page of the Russian translation [73] of the book [72]

coming to the University of Heidelberg as a professor of physiology [229]. One motivation seems to havebeen his interest in frictional phenomena, carried over from his interest in an application to organ pipes [32,p. 10], [33, p. 149], another was his growing awareness of the power of Green’s theorem in hydrodynamics[218, p. 249].

In a speech, at a banquet, on the occasion of his 70th birthday—an event that brought together 260 friendsand admirers at Kaiserhof in Berlin on November 2, 1891—Helmholtz [71] gave the following expandedaccount:2

I have been able to solve a few problems in mathematics and physics, including some that the greatmathematicians had puzzled over in vain from Euler onwards: e.g., the question of vortex motion, andthe discontinuity of motions in fluids, that of the motions of sound at the open ends of organ pipes,etc. But any pride I might have felt in my conclusions was perceptibly lessened by the fact that I knewthat the solution of these problems had always come to me as the gradual generalization of favorableexamples, by a series of fortunate conjectures, after many errors. I am fain to compare myself witha wanderer on the mountains, who, not knowing the path, climbs slowly and painfully upwards, andoften has to retrace his steps because he can go no farther—then, whether by taking thought or fromluck, discovers a new track that leads him on a little, till at length when he reaches the summit he findsto his shame that there is a royal way, by which he might have ascended, had he only had the wits tofind the right approach to it. In my works I naturally said nothing about my mistakes to the reader, butonly described the made track by which he may now reach the same heights without difficulty.

In nineteenth century, the English translation [67] made by Tait of Helmholtz’s paper [66] has been pub-lished (Fig. 2a); and there also exists a short Italian account [69] based upon a lecture by Betti at the Universityof Pisa. Later, in twentieth and even in twenty-first centuries, this paper was again published in English [75]and Russian [76].

Surprisingly [231], Helmholtz practically never directly continued his investigations of the topic of vor-tex motion established in his ground-breaking paper [66]: in the first volume of his collected writings [70],Helmholtz has only added the titles to all six sections of the paper and included three short notes related to theanimated discussion between him and French Academician Bertrand (“an acrimonious public controversy” as

2 According to the translation [97, p. 180]. There exist three other versions [32, p. 4], [33, p. 145], [74, p. 473] of translationof this fascinating passage.

V. V. Meleshko

Truesdell [217, p. 58] called it, which has even attracted attention of Maxwell, W. Thomson, and Tait [134]).However, as it was pointed in [33, p. 153], manuscript fragments No. 679 and 680 in his Nachlass, Helmholtzdiscussed various aspects of vortex motion, namely invariants, stability, and friction. Instead, he wrote anotherremarkable paper [68] on discontinuous motion of an inviscid fluid, in which he used the notion of a vortexsheet from [66] (see [32] and [33, pp. 159–166] for details). Together these two fundamental papers werepublished (Fig. 2b) as a separate book [72] in the well-known Ostwalds Klassiker der exakten Wissenschaftenseries edited by A. Wangerin, with his extensive comments and corrections of some small algebraical mistakes.In turn, the latter book was translated by S. A. Chaplygin (Fig. 2c) into Russian [73] (with a censorship approval[sic] dated 26 April 1902).

In the very beginning of his paper (Fig. 1b), Helmholtz [66, S. 25] pointed out that already Euler had men-tioned cases of fluid motion in which no velocity–potential exists, for example, the rotation of a fluid about anaxis where every element has the same angular velocity. A minute sphere of fluid may move as a whole in adefinite direction, and change its shape, all while rotating about an axis. This last motion is the distinguishingcharacteristic of vorticity.

To describe these rotations, Helmholtz introduced two new terms: Wirbellinien (vortex lines), defined aslines the direction of which coincided everywhere with the local direction of the axis of rotation of the fluid andWirbelfäden (vortex tubes), bundles of vortex lines emerging from infinitely small area elements transverse tothe rotation axes.

According to the monograph [217] which contains a rich and scholarly bibliography on the whole sub-ject, the notion of vorticity had already appeared in earlier works by d’Alembert (1749), Euler (1752–1755),Lagrange (1760), Cauchy (1827), and Stokes (1848)—in Truesdell’s view [217, p.59], “all these early worksare purely formal and somewhat mystifying”.

Helmholtz was the first to elucidate the key properties of those portions of a fluid in which vorticityoccurs in form of three important propositions. First, he proved that in such an ideal substance vortex motioncould neither be produced from irrotational flow nor be destroyed entirely by any natural forces that have apotential. Second, those fluid particles constituting a vortex line at a given moment would constitute a vortexline for all times. Thus, it was possible to speak of “the same” vortex line moving along in the fluid. Third,the product of the area of a cross section of a vortex tube and the angular velocity of the rotation at that point(called the strength of the vortex tube) was constant along the tube and in time. From this last proposition,Helmholtz concluded that vortex tubes either had to run back into themselves or else end at the boundary ofthe space in which the whole fluid was contained. Not surprising that Laue [110] called Helmholtz “den Vaterder Wirbel sätze”. (The hydrodynamical textbook tradition of the nineteenth century, e.g., [93, p. 169], [103,p. 149], [104], typically regarded Helmholtz’s third statement as a proved proposition. At the end of twentiethcentury, it appears [28, p. 27], [46, p. 313] that the assertion about the closing of vortex tubes requires furtherqualifications since topological complications like branching or aperiodic vortex lines may arise.)

The detailed account of Helmholtz’s benchmark study is provided, e.g., in [32, pp. 10–16], [46, pp. 311–316], [33, pp. 149–153], and [142, pp. 210–217]. (There even exists books [15,23] specially devoted to thisarticle!) Short descriptions of this famous memoir can be found in practically all biographies of Helmholtz andaccounts of his scientific works, e.g., [84], [95, pp. 104–105], [97, pp. 167–170], [111, pp. 225–233], [218,pp. 249–250].

In the following years, the Helmholtz theory of vortices was included in practically all classical generaltextbooks and monographs on fluid mechanics and aerodynamics. It was described and elaborated in greatdetail, in articles, in leading general and specialized encyclopedias in various countries, in major courses oftheoretical physics, in several courses of general physics, treatises on the history of mathematics and physics,and found its place in review papers, chapters or sections of popular books, and dissertations. A rather completelist of studies where the results of Helmholtz’s paper [66] were used in one way or another is given in [142,pp. 219–220].

In the present article, we will discuss only the influence made by the last section [66, Sect. 6], namely anaxisymmetric vortex flow of localized vorticity and more precisely, with one or several coaxial vortex rings inan inviscid incompressible fluid of constant density.

Saffman [182] remarks that ‘one particular motion exemplifies the whole range of problems of vortexmotion and is also a commonly known phenomenon, namely the vortex rings. < . . . > Their formation is aproblem of vortex sheet dynamics, the steady state is a problem of existence, their duration is a problem ofstability, and if there are several, we have the problem of vortex interactions’.

The topic of vortex rings is presented as a section or even a chapter in several influential textbooks onfluid mechanics and general physics by, e.g., Acheson [1], Appel [7], Basset [13], Batchelor [14], Edser [43],

Coaxial axisymmetric vortex rings

Faber [47], Joukowski [86], Kambe [90], Lamb [103,105–109], Lichtenstein [118], Milne-Thomson [145],Ogawa [154], Poincaré [160], Prandtl and Tietjens [161], Ramsay [162], Saffman [183], Sommerfeld [200],Villat [225], Wien [230], Wu, Ma, and Zhou [234], in research monographs [4,5,37,121,130,143,211], reviewarticles [8–10,78,84,125,127,128,163,168,193,198], popular books [36,48,50,129,203,223,224], and bookson the history of fluid mechanics [33,112,216]. The special attention deserves applications of this topic tolocomotion of insects, swimming of fishes, flight of birds, see, e.g., [34,38,45,83,149,164,165].

Vortex rings are easily produced experimentally by dropping drops of one liquid into another (by coinci-dence, the first observation has been made by Rogers [176] in the same year of publication of the Helmholtzpaper; later such study was performed in much more detail by J.J. Thomson and Newall [212]), or by puff-ing fluid out of a hole [35,84,119,120,139,151,152,166–168,170] (to name only a few in order to show thetime interval; many other beautiful pictures are reproduced in [14,39,184,190,232]), or by exhaling smoke[12,77,203,233]. Figures of various examples of vortex rings generators from some old papers are collected in[142, Figs. 1–4]. A nice collection of movies and many photographs of vortex rings and their interactions in anexperimental setting may be found on the web page http://serve.me.nus.edu.sg/limtt. The important issue ofmodeling of such a process which is considered in a great amount of studies is out of the scope of this article.

Searching Google for “Vortex rings” (only in English!) results in about 317,000 entries on Google Web,381,000 entries on Google Images, 56,400 entries on Google Scholar, and 3,380 entries on Google Book Search(these include not only books but also old and recent articles in many leading scientific journals). Surprisingly,the entry “Vortex rings Helmholtz” results in numbers 10,500, 5,040, 790 and 729, respectively. (As usual, notall results are relevant.)

In the following, we consider the case in which the ratio of azimuthal vorticity to cylindrical radius is con-stant within each vortex ring. Dyson [40] developed a simplified model of thin interacting coaxial vortex ringswith circular cores of small radius (for which the core dynamics can be neglected). This model has an interestingHamiltonian structure. After Dyson, this model was employed by Hicks [81] to study in detail the phenomenonof leapfrogging of two vortex rings; sometimes it is called [197] the “Dyson–Hicks model”. Much later, thismodel was rediscovered in [156], and since that time according to the Google Scholar, ISI, and Scopus databasesit has been used in more than 40 articles on the analytical and numerical studies of vortex rings interactionsin unbounded inviscid fluid [59,61–64,99,144,146,157,171–173,197,226,228], comparison of experimentalobservations of vortex rings interactions [29,155,236,237], nonintegrability and chaotic phenomena inHamiltonian systems [11,18–21,98,158,197], mixing of passive surrounding fluid [185,192,194,195,219],experimental studies of interaction of vortex rings with planes and axisymmetric rigid bodies [122,180,202,235,238], and even for the study the so-called ‘vortex sound’ [65,89,91,92,123,147,187,188,196,204–206,222].

Based upon the previous studies of the author [59,63–65,100,144] and his students [61,62,98,99], sev-eral typical cases of the vortex rings interaction are presented below. Riley et al. [171–173,226–228] discussanother interesting samples of such an interaction based upon the Dyson model.

2 Helmholtz’s (1858) memoir and its first reception by contemporary scientists

In the Introduction to his paper [66, S. 26], Helmholtz states:3

Hence it appeared to me to be of importance to investigate the species of motion for which there is novelocity-potential.

The following investigation shows that when there is a velocity-potential the elements of the fluidhave no rotation, but that there is at least a portion of the fluid elements in rotation when there is novelocity-potential.

By vortex-lines (Wirbellinien) I denote lines drawn through the fluid so as at every point to coincidewith the instantaneous axis of rotation of the corresponding fluid element.

By vortex-filaments (Wirbelfäden) I denote portions of the fluid bounded by vortex-lines drawnthrough every point of the boundary of an infinitely small closed curve.

The investigation shows that, if all the forces which act on the fluid have a potential, —-1. No element of the fluid which was not originally in rotation is made to rotate.

3 Here and in what follows, all quotations from [66] are given according to English translation [67] made by Tait. Italics arefrom the original and translated text.

V. V. Meleshko

2. The elements which at any time belong to one vortex-line, however they may be translated,remain on one vortex-line.

3. The product of the section and the angular velocity of an infinitely thin vortex-filament is con-stant throughout its whole length, and retains the same value during all displacements of thefilament. Hence vortex-filaments must either be closed curves, or must have their ends in thebounding surface of the fluid.

Helmholtz [66, S. 31] and earlier Stokes [201, p. 290] used the symbols ξ, η, ζ andω′, ω′′, ω′′′, respectively,to denote the quantities:

dv

dz− dw

dy= 2ξ = −2ω′, dw

dx− du

dz= 2η = −2ω′′, du

dy− dv

dx= 2ζ = −2ω′′′. (1)

(The notation with straight d for partial derivatives that time were opposite to the modern one ∂ .) Here u, v, ware the rectangular Cartesian components of the Eulerian velocity vector u.

Helmholtz called these quantities Rotationsgeschwindigkeiten, Stokes [201] the angular velocities of thefluid, and later Lamb [103, Art. 38], [105, Art. 31], [106, Art. 30], [107, Art. 30], [109, Art. 30] the componentangular velocities of the rotation. W. Thomson [215] and Basset [13] called these entities component rotationsand molecular rotations, respectively. For the general case of motion in which ω does not vanish, Helmholtz[66, S. 33] used the term Wirbelbewegungen, which has been later translated into English [67, p. 491] as vor-tex-motions. The name vorticity was introduced by Lamb in the fourth edition of his famous treatise [107, Art.30] (and not in the second edition [105, Art. 31], as stated by Truesdell [217, p. 58]) for the vector ω, whoseCartesian components, (ξ, η, ζ ), are given as ξ = 1

2ω′, η = 1

2ω′′, ζ = 1

2ω′′′ via expression (1). According to

Lamb [109, p. 31], this definition avoids the intrusion of an unnecessary factor 2 in a whole series of formulaerelating to vortex motion.

In the second section, Helmholtz gives proofs of his three laws of vortex motion based upon kinematicalconsiderations and ingenious transformations of the dynamical equations for incompressible, homogeneous,inviscid fluid into his now famous vorticity equations. Helmholtz derives from the Euler equations two equiv-alent systems of equations (Eqs. (3) and (3a) in the original text) in terms of velocity and rotation:

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

δξ

δt= ξ

du

dx+ η

du

dy+ ζ

du

dz,

δη

δt= ξ

dv

dx+ η

dv

dy+ ζ

dv

dz,

δζ

δt= ξ

dw

dx+ η

dw

dy+ ζ

dw

dz,

or

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

δξ

δt= ξ

du

dx+ η

dv

dx+ ζ

dw

dx,

δη

δt= ξ

du

dy+ η

dv

dy+ ζ

dw

dy,

δζ

δt= ξ

du

dz+ η

dv

dz+ ζ

dw

dz.

(2)

Here the symbol δδt in the left hand sides of (2) denotes the full derivative

δψ

δt= dψ

dt+ u

dψ

dx+ v

dψ

dy+ w

dψ

dz

in term of notation of that time. We use here the notation from the English translation [67, p. 491] instead evenmore confusing one ∂

∂t in both the original paper [66, S. 33] and subsequent reprints [70,72,73].It was pointed out in [33, p. 150] that these equations have been already known to D’Alembert and Euler.

The main innovation of Helmholtz consists in its kinematic interpretation, stated above.In the third section, Helmholtz addresses the inverse problem of finding the components of the velocity

u, v, w from the components of rotation ξ, η, ζ (up to a potential flow that covers the boundary conditions). Heindependently obtains the representations of Stokes for the classical problem of vector analysis of determininga vector field of known divergence (“hydrodynamic integrals of the first class” in his terminology) and curl(“hydrodynamic integrals of the second class”).

Determination of the velocity field for incompressible fluid leads to the Biot–Savart law of electromagne-tism, which in the present case reads that each rotating element of fluid at point (a, b, c) induces in every otherelement at point (x, y, z) a velocity with direction perpendicular to the plane through the second element thatcontains the axis of the first element. The magnitude of this induced velocity is directly proportional to thevolume of the first element, its angular velocity, and the sine of the angle between the line that joins the twoelements and the axis of rotation, and is inversely proportional to the square of the distance between the two

Coaxial axisymmetric vortex rings

elements. Helmholtz also establishes analogies between the induced velocity and the forces on magnetizedparticles.

In the fourth section, Helmholtz derives an elegant expression for the kinetic energy K —“vis viva” in histerminology—of infinite fluid with a compact distribution of vorticity within it.

In the fifth section, Helmholtz studies certain simple cases in which the rotation of the elements occursonly in a set of parallel rectilinear vortex-filaments. In particular, he considers several infinitely thin, parallelvortex-filaments each of which carries a finite, limiting value, m, of the product of the cross-sectional area andthe angular velocity. This is the now-celebrated concept of a point vortex. Helmholtz considers simple cases ofthe dynamics of such vortices. He establishes the law of conservation of the center of vorticity of an assemblyof point vortices. The discussion is phrased in terms of the “center of gravity” of the vortices (consideringtheir values of m as the analog of “masses”): ‘The centre of gravity of the vortex-filaments remains, therefore,stationary during their motions about one another, unless the sum of the masses be zero, in which case thereis no centre of gravity.’ Helmholtz also considers the reactions which two parallel straight vortex-filaments inan unbounded fluid would have exerted on one another.

Finally, in the last sixth section which has a title “Circular vortex-filaments” Helmholtz addresses theaxisymmetric motion of several circular vortex-filaments whose planes are parallel to the xy-plane, and whosecenters are on the z-axis, all motion is symmetrical about the z-axis.

Then by introducing the polar cylindrical coordinates, according to

x = χ cos ε, y = χ sin ε, z = z, a = g cos e, b = g sin e, c = c,

and taking into account that a circular vortex-filament of radius g and axial coordinate c has the single com-ponent of rotation σ(g, c) with the axis of rotation is perpendicular to direction of g and the z-axis with therectangular coordinates (ξ, η, ζ ) of angular velocity at the point (g, e, c)

ξ = −σ sin e, η = σ cos e, ζ = 0,

after some transformation, Helmholtz obtains for the radial component τ in the direction of the radius χ andaxial component w at the point (χ, ε, z), the following expressions

τ = −dψ

dz, w = dψ

dχ+ ψ

χ,

or

τχ = −d(ψχ)

dz, wχ = d(ψχ)

dχ, (3)

with expression for ψ(χ, z) given by the Eq. [(7.)], see Fig. 3, the first line. Here and in what follows theequations from the original paper [66] are enclosed into the square brackets. It should be noted that Wangerin[72] corrected some arithmetical mistakes which went unnoticed both in [70] and English translation [67].These corrections are presented in Fig. 3.

Helmholtz also uses the notion of streamline (“Strömungslinien” or “current-lines” English translation[67]) which equation is

ψχ = Const.

By considering the expression [(7.)] for ψ for a vortex filament of indefinitely small section, puttingσ(g, c)dgdc = m1 = const , Helmholtz arrives to the expression (the second line in Fig. 3) for the streamfunction of a circular vortex filament and velocity components (the third and fourth lines in Fig. 3). Theseexpressions are used now in every textbook of fluid dynamics.

Next, by considering a finite number of infinitely thin vortex filaments Helmholtz obtains several interestingidentities: besides expressions in the fifth and sixth lines in Fig. 3, he also writes

∑(mnρnτn) = 0 ,

where ρn and τn are the radius of the vortex filament n and the velocity which it receives from other filaments.

V. V. Meleshko

Fig. 3 Corrections of small errors in Sect. 6. In the left column—the original text [66], in the right column—Wangerin’s corrections[72]

Fig. 4 Vortex ring box generator. a From Tait [203, p. 292]. b From W. Thomson’s letter to Helmholtz [209, p. 514]

Finally, by considering the distributed vorticity inside a vortex ring with m = σdρdλ (ρ and λ being theradial and axial distance of the elementary circular vortex filament), Helmholtz obtains (in addition to the twolast lines in Fig. 3) the Eq. [(9.)]:

∫ ∫

σρdρ

dtdρdλ = 0,

or since the product σdρdλ = Const, as the result

1

2

∫ ∫

σρ2dρdλ = Const.

Helmholtz calls M the entire “mass” of a slice that a plane which passes through the z-axis cuts from thevortex ring

M =∫ ∫

σρdρdλ.

If R2 is the mean value of ρ2, then∫ ∫

σρ2dρdλ = M R2,

and since the integral at the left and the value of M remain unchanged in time, the value of R also remainsunchanged during the motion of the single vortex ring in an unbounded fluid.

Based upon the identity [(9b.)] (see Fig. 3, the last line) where K is the (constant) kinetic energy of the infi-nite fluid including the vortex ring, h is the constant density, after some not very strict estimates for indefinitelything vortex ring, Helmholtz arrives at the equation

2d

dt(M R2l) = − K

2πhor 2M R2l = C − K

2πht,

where l denotes the value of λ for the center of gravity of the slice section of the vortex ring. From this equationHelmholtz provides the following consequence:

Coaxial axisymmetric vortex rings

Fig. 5 Single vortex ring. a Appendix by W. Thomson to the English translation of Helmholtz paper [67]. b Self-induced forwardmotion of a vortex ring (from [203, p. 296])

Hence in a circular vortex-filament of very small section in an indefinitely extended fluid, the centre ofgravity of the section has, from the commencement, an approximately constant and very great veloc-ity parallel to the axis of the vortex-ring, and this is directed towards the side to which the fluid flowsthrough the ring. Indefinitely thin vortex-filament of finite radius would have indefinitely great velocityof translation.

A schematic picture (Fig. 5b) of self-induced motion of a vortex ring has been sketched in [203, p. 296]and then reproduced in [57, p. 172], [84, p. 42], and [142, p. 216].

Helmholtz notes several important conclusions:

We can now see generally how two ring formed vortex-filaments having the same axis would mutuallyaffect each other, since each, in addition to its proper motion, has that of its elements of fluid as producedby the other. If they have the same direction of rotation, they travel in the same direction; the foremostwidens and travels more slowly, the pursuer shrinks and travels faster, till finally, if their velocities arenot too different, it overtakes the first and penetrates it. Then the same game goes on in the oppositeorder, so that the rings pass through each other alternately.If they have equal radii and equal and opposite angular velocities, they will approach each other andwiden one another; so that finally, when they are very near each other, their velocity of approachbecomes smaller and smaller, and their rate of widening faster and faster. If they are perfectly symmet-rical, the velocity of fluid elements midway between them parallel to the axis is zero. Here, then, wemight imagine a rigid plane to be inserted, which would not disturb the motion, and so obtain the caseof a vortex-ring which encounters a fixed plane.

At the very end of his paper Helmholtz suggests a simple experiment:

In addition it may be noticed that it is easy in nature to study these motions of circular vortex-rings, bydrawing rapidly for a short space along the surface of a fluid a half-immersed circular disk, or the nearlysemicircular point of a spoon, and quickly withdrawing it. There remain in the fluid half vortex-ringswhose axis is in the free surface. The free surface forms a boundary plane of the fluid through the axis,

V. V. Meleshko

and thus there is no essential change in the motion. These vortex-rings travel on, widen when theycome to a wall, and are widened or contracted by other vortex-rings, exactly as we have deduced fromtheory.

This last problem has attracted attention of Klein [94] and his qualitative description led to discussion[2,16,17] on the possibility of vortex generation in an inviscid fluid. Quantitative calculations were performedby Taylor [208]. These studies, however, are out of topic of this article.

2.1 Some examples of receptions of Helmholtz’s paper

In Britain, a Scottish physicist Tait, a close friend of W. Thomson, read Helmholtz’s paper already in July1858 and immediately made a complete English translation for his own use. In 1866. Maxwell set Helmholtz’stheorems as a question on the Cambridge Mathematical Tripos [132, p. 242]:

If the motion of an incompressible homogeneous fluid under the action of such forces as occur in naturewe put

α = dv

dz− dw

dy, β = dw

dx− du

dz, γ = du

dy− dv

dx,

shew that

dα

dt+ u

dα

dx+ v

dα

dy+ w

dα

dz= α

du

dx+ β

du

dy+ γ

du

dz.

P, Q are adjacent particles of the fluid, such that at a given instant the projections on the axes of x, y, zof the line joining them are proportional to α, β, γ , respectively: shew that, during the subsequentmotion of the fluid, the projections of the line joining P and Q will remain proportional to α, β, γ .

On 13 November 1867, Maxwell wrote to Tait [133, p. 321]:

Dear TaitIf you have any spare copies of your translation of Helmholtz on ‘Water twists’ I should be obliged toyou if you could send me one. I set the Helmholtz dogma to the Senate house in ’66, and got it verynearly done by some men, completely as to the calculation, nearly as to the interpretation.

The Helmholtz theory of vortex motion has been essentially developed in the classical study by W. Thomson[215]. (Though read on 29 April 1867 at the meeting of the Royal Society of Edinburgh, this memoir waspublished only in 1869 after it had been ‘recast and augmented’ in 1868 and 1869.) Helmholtz and Thomsonhad met for the first time during a stay in Bad Kreuznach, a health resort in the Rhineland, in the summerof 1856, and afterwards they developed a close personal relationship. Lord Kelvin’s biographer [209, p. 512]states that Thomson had read Helmholtz’s paper [66] already in the Fall of 1858, but without the enthusiasmit later inspired. In a letter of 30 August 1859, Helmholtz reported to Thomson that he had been workingon hydrodynamic equations including friction, but did not mention his already published Wirbelbewegungenstudy. However, according to very extensive biography [199, p.417], no evidence exists that Thomson knew ofHelmholtz’s paper prior to early 1862 when Tait mentioned it to him. In his remarkable paper [215], Thomsonworked through the entire vortex theory, introduced the concept of circulation � around a closed curve C as themean value of the tangential velocity to the curve C multiplied by its length. He greatly simplified Helmholtz’sproofs of the vortex theorems and added a few theorems of his own. He showed that for any material contourmoving according to Euler’s equation for incompressible flow, the circulation is an integral of the motion, aresult known today as Kelvin’s circulation theorem This theorem was considered by Einstein [44] among themost important scientific results of Lord Kelvin.

Apparently, Lamb was the first author to treat the vortex dynamics in a separate chapter in a major textbook[103] on fluid mechanics. It had its origin, Lamb stated in his Preface ‘in a course of Lectures delivered inTrinity College, Cambridge in 1874, when the need for a treatise on the subject was strongly impressed onmy mind.’ Taylor noted [207] that ‘Lamb, in his first course of lectures on hydrodynamics broke new groundwhen he gave an account of Helmholtz’s great work on vortex motion.’ Already in 1874 Lamb lectured onHelmholtz’s memoir which became the main part of Chap. VI in [103]. Glazebrook [55, p. 377] recalls:

Coaxial axisymmetric vortex rings

And so when Lamb was put on to lecture on hydrodynamics, our coaches—I was then beginning mythird year—told us to go and hear what he had to tell us. And his lectures were a revelation. For mostof us, Besant’s Hydromechanics contained all we knew about the subject. Helmholtz was an unknownname. Some inkling of Sir William Thomson’s suggestion that certain mysterious entities known asvortex rings were the only true atoms had reached us. Lamb, in his own inimitable manner, unveiled themysteries and made the properties of a liquid in rotational motion clear to us.< . . . >When he arrivedat the lecture room, a small room, in Trinity College—I went to look at it recently for the sake of oldtimes—it was filled, but with the scholars and other honours men of the second and third years; ourcoaches had realized that Lamb would have a message for us. And he realized at once that his audiencehad not come to listen to the elements of hydrodynamics. He had read Helmholtz’s memoir in Crelle’sJournal for 1858 and Thomson’s Edinburgh paper (1867) On Vortex Motion.

Tait [203, pp. 291–293] also devised some extremely clever experiments to illustrate the vortex theoryusing smoke vortex rings in air. He used a box with a circular hole on one side and a rubber diaphragm on theopposite side. Within the box, a chemical agent (magnesium sulfate) produced a thick, white smoke. Whenstruck on the rubber diaphragm, circular smoke rings shot out of the hole (Fig. 4a). (A similar and even moresophisticated device has already been invented by Reusch [166].) Two such boxes could be placed in vari-ous positions, causing the smoke rings to interact just as Helmholtz had indicated. Tait in his lectures [203,pp. 292–298] for the audience consisted of ‘a number of my friends, mainly professional men, who wished toobtain in this way a notion of the chief advances made in Natural Philosophy since their student days’ gavethe complete verbal description of these Helmholtz’s thought experiments. Tait [203, pp. 293] also notes: ‘Ofcourse, in air, fluid friction, which depends upon diffusion, soon interferes with this state of things. But, in theexperiment, the ring (of some six or eight inches in diameter) was not sensibly altered by such causes in thefirst 20 feet of its path.’

W. Thomson visited Tait in Edinburgh in mid-January 1867 and saw the smoke rings with his own eyes.On 22nd January, he wrote to Helmholtz [209, pp. 513–515]:

Just now, however, Wirbelbewegungen have displaced everything else, since a few days ago Tait showedme in Edinburgh a magnificent way of producing them. Take one side (or the lid) off a box (any oldpacking-box will serve) and cut a large hole in the opposite side. Stop the open side AB loosely witha piece of cloth, and strike the middle of the cloth with your hand. If you leave anything smokingin the box, you will see a magnificent ring shot out by every blow. A piece of burning phosphorusgives very good smoke for the purpose; but I think nitric acid with pieces of zinc thrown into it, inthe bottom of the box, and cloth wet with ammonia, or a large open dish of ammonia beside it, willanswer better (Fig. 4b). The nitrite of ammonia makes fine white clouds in the air, which, I think, willbe less pungent and disagreeable than the smoke from the phosphorus. We sometimes can make onering shoot through another, illustrating perfectly your description; when one ring passes near another,each is much disturbed, and is seen to be in a state of violent vibration for a few seconds, till it settlesagain into its circular form. The accuracy of the circular form of the whole ring, and the fineness androundness of the section, are beautifully seen. If you try it, you will easily make rings of a foot indiameter and an inch or so in section, and be able to follow them and see the constituent rotary motion.

In spite of the great popularity of Tait’s smoke box for generating vortex rings in air, the first observationof vortex rings probably corresponds with the introduction of smoking tobacco! Northrup [151, p. 211] writes:

It is not improbable that the first observer of vortex motions was Sir Walter Raleigh; if popular traditionmay be credited regarding his use of tobacco, and probably few smokers since his day have failed toobserve the curiously persistent forms of white rings of tobacco smoke which they delight to make.But some two hundred eighty years went by, after the romantic days of Raleigh and Sir Francis Drake,who made tobacco popular in England, before a scientific explanation of smoke rings was attempted.

Using the smoke-box apparatus, Dolbear [36, pp. 28–31] carefully repeated all Tait’s experiments andproduced a long list of characteristics of smoke vortex rings (see [142, p. 234] for detail). The extensive studyby Northrup [151,152] should also be mentioned here. It contains a very detailed description of a “vortexgun”, including all the parameters, together with beautiful photos of interacting vortex rings and vortex ringsinteracting with rigid obstacles, e.g., with a small silver watch chain. Modern studies with nice color picturesand video can be found at http://media.efluids.com/galleries/vortex.

On 18 February 1867, Sir William Thomson (he was recently knighted for the successful laying of theAtlantic cable in 1866) read before the Royal Society of Edinburgh, a talk entitled “Vortex Atoms” (with an

V. V. Meleshko

abstract printed next day in the local newspaper The Scotsman [209, pp. 517–519]). For the meeting, Tait alsoperfected his smoke ring experiments. Tait’s letter of 11 February 1867 captures nicely the spirit in which hewas working [46, p. 323]: ‘Have you ever tried plain air in one of your two boxes. The effect is very surprising.But eschew NO5 and Zn. The true thing is SO3+NaCl. Have the NO3 rather in excess—and the fumes arevery dense + not unpleasant. NO5 is DANGEROUS. Put your head into a ring and feel the draught.’

The talk was published as a remarkable paper [213] that led at the third part of the nineteenth century toextensive studies on the vortex atom theory and vast literature, see, e.g., [6,101,198], [142, pp. 228–233] forfurther details and references. It is worth noting that Helmholtz’s comments on the vortex atom theory werescarce [6, pp. 34–35]. On contrary, this theory received considerable attention from Kirchhoff and Maxwell.As Schuster recounts [189, p. 34]: ‘Kirchhoff, a man of cold temperament, could be roused to enthusiasm whenspeaking about it. It is a beautiful theory, he once told me, because it excludes everything else.’ Maxwell [138],in his article on the Atom written for the 1878 edition of Encyclopedia Britannica, provided a detailed descrip-tion of properties of vortices (and the vortex rings, in particular) in an inviscid fluid and strongly supported theidea of vortex atoms.

The subject of vortex motion selected by the Examiners for the Adams Prize for 1882 year was “A gen-eral investigation of the action upon each other of two closed vortices in a perfect incompressible fluid.” Inhis winning essay [211], in addition to the set subject, J.J. Thomson have discussed some points which areintimately connected with it and has applied some of the results to the vortex atom theory of matter; see, also,the review by Reynolds [169]. Part of this essay was published as an extensive memoir [210] which, in ouropinion, still deserves more close attention.

On the whole, reviewing in 1882 the progress of hydrodynamics Hicks [78, p. 63] stated: ‘During the lastforty years, without doubt, the most important addition to the theory of fluid motion has been to our knowledgeof the properties of that kind of motion where the velocities cannot be expressed by means of a potential.Helmholtz first gave us clear conceptions in his well-known paper’.

3 Development of Helmholtz’s Sect. 6 for a single vortex ring

The already cited Helmholtz’s statement on the velocity of a single circular vortex ring in an unbounded invis-cid fluid led to the vast body of results. Theoretical studies of the motion of a circular vortex ring of closedtoroidal shape with core radius a and radius R of the center line of the torus, where a � R, led to a formulafor the self-induced translational velocity Vring, directed normally to the plane of the ring:

Vring = �

4πR

(

ln8R

a− C

)

+ O

(a2

R2 ln8R

a

)

. (4)

Here � is the (constant) intensity of the vortex ring, equal to the circulation along any closed path around thevortex core, and C is a constant.

There was some disagreement in the literature concerning the value of C . The value C = 14 was given

(without proof, Fig. 5a) by W. Thomson [214] and later by Hicks [79], Basset [13], Dyson [40], and Gray[58]. (Hicks [79] has even invented, independently of Neumann who had used them in another connection,the “toroidal functions”.) This corresponds to the case where the vorticity inside the core varies directly asthe distance from the centerline of the ring. The value C = 1 was given by Lewis [116], J. J. Thomson [211],Joukovskii [85], and Lichtenstein [117] for a uniform distribution of vorticity within the core. Detailed dis-cussion of both these hypotheses and also the circular form of the core is given [117,191] and later repeatedby Fraenkel [51].

Lamb [105, p.260] provide an elegant proof of value C = 14 . This result was repeated in all further editions,

namely, [106, p.227], [107, p.233], [109, p.241]; however, it was absent in the corresponding place of the firstedition [103, Art. 143]. The proof is based upon the accurate calculation of the kinetic energy T of the fluidwith a thin circular vortex ring in it and usage relation [109, Art. 162, Eq. (6)] which coincides with (corrected)Eq. [(9 b.)] (the last line in Fig. 3) of Helmholtz [66, S. 53].4

Such a concise approach was later extended by Saffman [181] and Rott [177,178]. For a hollow vortexcore [79,159], or if one assumes the fluid inside the core is stagnant, the value C = 1

2 was obtained. A detailed

4 It was pointed out by one of the reviewers that it is worth emphasizing up front that the (corrected) Helmholtz’ Eq. [(9b.)]can be used directly to compute the ring velocity, including the O(a/R) correction. This formula, of course, supersedes Lamb’s“elegant proof”, Fraenkel’s analysis, and Saffman’s use of Lamb’s transformation to avoid Fraenkel’s difficult estimates, for Eq.[(9b.)] does not require such a transformation.

Coaxial axisymmetric vortex rings

discussion of various model is given by Donnelly [37, Chap. I]. (It is worth noting that Maxwell [135] in hisletter to W. Thomson, written well later after Thomson’s short appendix [214], uses the original Helmholtz’sequations and after some algebraical transformation arrives to the correct value of the kinetic energy of thefluid, but the omission of the second term in Eq. [(9b.)] finally leads to the erroneous value C = 7

4 in (4).)In the following, we will use the simplest of all admissible vorticity distributions (the so-called “standard

model” [37, p. 25]) in which the ratio of azimuthal vorticity to cylindrical radius is constant. For such a thinvortex ring up to the main terms the velocity is

Vring = �

4πR

(

ln8R

a− 1

4

)

, (5)

the momentum P of an unbounded fluid (or more properly, impulse) and the total kinetic energy K of the fluid(including the vortex core) for such a ring are

P = ρ�πR2, K = 1

2ρ�2 R2

(

ln8R

a− 7

4

)

. (6)

Note the relation,

Vring = ∂K

∂P, (7)

which is valid under the condition of constant volume of the vortex ring, a2 R = Const.For important particular case of a sphere in which the azimuthal vorticity varies linearly with radius, Hill

[82] discovered his famous spherical vortex, an exact solution in closed form for a steadily translating vortexring in the thick core limit. Fraenkel [51,52] and Norbury [150] provided analytical analysis and numericalsolutions for the family of a steadily propagating ring between the thin-core and Hill limits. The former authorrecovered and enlarged analytical results of Dyson [40] for steady vortex rings by applying general theory tocompute the shape of the cross-section to one order less than Dyson, and the propagation velocity to the sameorder as he did, and obtained (see [53,54] for further explanation):

P = ρπ�R2(

1 + 3

4

a2

R2

)

, K = 1

2ρ�2 R2

(

ln8R

a− 7

4+ 3

16

a2

R2 ln8R

a

)

. (8)

When a vortex ring is moving steadily through an irrotational fluid, one can distinguish three differentregions of fluid motion: (1) that of the ring which is the rotational core and which keep its identity, (2) theportion of fluid in potential motion surrounding the first, which also keeps its identity and volume and travelsuniformly through the fluid like a solid, and (3) the potential motion outside the second region which remainsat rest at infinity. The distinction between the rotational region (1) and irrotational (2) and (3) is well known.Less attention than it deserves, however, appears to have been devoted to the discussion of the relationshipbetween regions (2) and (3).

Similar to the two-dimensional case of vortex pair consisting of two rectilinear vortices of equal and oppo-site circulation which carries in its steady motion a certain oval, domain (2) (the “atmosphere” as it was calledby W. Thomson [213], the “body” [161, Art. 90], or the “vortex bubble” [202]) of surrounding fluid the vortexring also carries with it a certain body of irrotationally moving fluid in its carrier. This was already pointedout by Maxwell in a letter [135, p. 490], and later by Reynolds [167], Lodge [124], Lamb [106, p. 227], andNorthrup [151,152]. In contrast to the vortex pair, the atmosphere of the vortex ring depends upon the ratio ofits core radius and mean radius n = a/R. Different diameters of the core produce different types of the vortexring atmosphere (Fig. 6a). For a rather thick circular vortex, n > 0.0116, the carried mass of fluid looks likea solid body (Fig. 6b). For a critical value of n = 1

86 according to the Eq. (5) the velocity of translation ofthe vortex ring is equal to the velocity of fluid at its centre and we have a “figure eight”. If n < 0.0116, thevelocity of translation of the vortex ring is greater than that of the fluid through its centre, the portion of fluidwhich travels with it is always a ring and not a simply connected body (Fig. 6c). These data are due to Hicks[80] who provided a detailed numerical analysis of different core thickness. His results for four typical casesare given in Table 1.

The dividing closed line which separates the atmosphere from a surrounding potential fluid represents aseparatrix in term of the dynamical systems theory. For n > 0.0116 the vortex ring has two hyperbolic fixedpoints (more correct, the two circular lines). It is known that the stable and unstable manifolds of dynamical

V. V. Meleshko

Fig. 6 The “atmosphere” traveling with a vortex ring. a Streamlines in a moving frame. From [161, Art. 90]. b Experimentalobservation for a smoke vortex ring (from [39, Fig. 77]). c The sketch of “atmosphere” boundary for different thickness of thevortex core

Table 1 Distribution of the total kinetic energy between volumes of fluid

n = a

R

Volring

R3

Volatm

R3

K1

K(%)

K2

K(%)

K3

K(%)

0.0005 0.000005 0.56 0.3 70.4 29.30.0116 0.002 1.36 0.7 72.4 26.90.05 0.05 1.68 2.6 67.0 30.40.1 0.2 2.18 5.2 62.8 32.0

systems theory provide a powerful tool for understanding Lagrangian aspects of time-periodic flows. The peri-odic perturbation of the atmosphere boundary leads to the lobes formed by intersections of the two manifoldscan be used to quantify rates of entrainment and detrainment into the volume of fluid transported with thevortex rings, see recent papers [192,194,195] for further detail.

4 Interaction of coaxial vortex rings

Helmholtz’s comment on the leapfrogging motion of two coaxial equal vortex rings cited above has startedvast activity in this direction and constantly attracted the attention of scientists. However, it was extremelydifficult that time to obtain a more detailed account of such a motion. The corresponding case of two coaxialpairs of two point vortices was worked out analytically and illustrated graphically by Gröbli [60, S. 152] andlater, independently, by Hicks [81]. Also, Love [126] proposed ‘to imitate some of the circumstances of theproblem by considering the case where there are present in an infinite fluid two pairs of cylindrical vorticesof indefinitely small section, the circulations about the two vortices of each pair being equal and of oppositesign, the circulations about the four vortices being equal in absolute magnitude, and the line of symmetry forone pair coinciding with that for the other. A single pair of this kind moves parallel to the axis of symmetrywith constant velocity. Two pairs with circulations in the proper directions influence each other’s motions ina manner analogous to that exhibited by thin rings.’ He provided the complete analytical treatment of theproblem in terms of some elliptic integrals and found a condition that the motion may be periodic, the lengthof the period, and the form of the curve described by one vortex of one pair relative to the homologous vortexof the other pair. All these studies have been briefly mentioned by Lamb [105, p. 261], [106, p. 228], [107,p. 234], [109, p. 242].

Coaxial axisymmetric vortex rings

Fig. 7 Interaction of coaxial vortex rings. a Leapfrogging motion (from [9, S. 1065]). b A qualitative scheme of leapfrogging(from [200, Art. 22]). c Maxwell’s “Wheel of Life” with three vortex rings (from [136, p. 447])

Auerbach [9, S. 1065] gave a quantitative illustration (Fig. 7a) of leapfrogging by two vortex rings, how-ever, the details of calculations remain unclear,5 while Sommerfeld [200, Art. 22] provided the qualitativedescription of the process (Fig. 7b). Experimentally, the most clear picture of visualization of the leapfrogginginteractions was obtained much later [236,237]. These pictures were reproduced in many studies, e.g., [32],[33, p. 154], [39, Fig. 79], [194]. Lim [119] gives another nice experimental example of the vortex ringsleapfrogging.

Much earlier Maxwell has even created the so-called “Wheel of Life” to understand the behavior of threeidentical coaxial vortex rings (Fig. 7c) and on 6 October 1868, he wrote to W. Thomson [136, pp. 446–447]:

H2 [Hermann Helmholtz]’s 3 rings do as the 2 rings in his own paper that is those in front expand andgo slower those behind contract and when small go faster and thread through the others. I drew 3 tomake the motion more slow and visible not that I have solved the case of 3 rings more than to get arough notion about this case and to make the sum of the three areas const. I have made them fat whensmall and thin when big.

4.1 Dyson’s model of interaction of coaxial vortex rings

At the very end of large memoir consisting of two parts (the second part was communicated by ProfessorJ. J. Thomson, F. R. S., received March 16, read April 20, 1893) and mainly devoted to a gravitational potentialof an anchor ring whose cross-section is elliptic being in connection with Saturn, Dyson [40, pp. 1091–1106]6

considered two problems of vortex rings motion in an inviscid fluid. These problems deal with a steady motionof a single vortex ring in an infinite fluid, and of several thin coaxial vortex rings. The first problem has beenreferred to by Lamb already in [105, Art. 163] and later briefly in [109, Art. 166] with further reference to Love[127]. Surprisingly, the second problem has not been mentioned at all. Thus, the Dyson model for interactionof several coaxial vortex rings remain unnoticed till the last quarter of twentieth century. Below we outline themain points of the model (with correction some misprints in the original paper [40, Eqs. (90), (97), (106)]).

In short, the essence of Dyson model is that each circular vortex ring moves with its self-induced axialvelocity (which preserves its ring radius) and in addition has a velocity due to interaction with another vor-tex rings considered as circular vortex filaments. It is supposed that a thin coaxial vortex ring (V R)i withi = 1, . . . , N of mean radius Ri and position Z i along axial z-direction in the cylindrical polar coordinatesystem (r, φ, z) (Fig. 8a) has always the circular cross-section of radius ai which remains small comparing to

5 This picture was repeated later [10, S. 139] without any change, in spite of detailed study by Hicks [81]. It seems this picturebeing qualitatively correct is quantitatively wrong, for the small vortex ring initially has much bigger self-induced velocity thanthe large one; see below.

6 Sir Frank Watson Dyson (1868–1939) is mainly known as an astronomer, for he occupied the post of Astronomer Royal atGreenwich since 1910 till 1933. Among other things, he introduced the ‘Six Pips’ in 1924 and organized in 1919, two expeditionsto study the Sun eclipse—one from Greenwich to Sobral in Brazil, and one from Cambridge, under Sir Arthur Eddington, toPrinciple in West Africa. The successful results of these expeditions confirmed Einstein’s prediction of the deflection of starlightin the Sun’s gravitational field. Dyson died on 25 May 1939 during the homeward voyage from Australia, where he had beenvisiting one of his daughters, and was buried at sea. Eddington notes [42, p. 160]: ‘He seems to have been most interested in theapplication of his results to hydrodynamics (motion of vortex rings); but gentle persuasion from the Isaac Newton Electors, who“while approving his programme, think it important that, as far as possible, he should pay attention to the astronomical applicationof his method”, [Dyson in 1892 was awarded the Isaac Newton Studentship for research in astronomy] led him to deal also witha ring of strongly elliptical section, for which his formulae were well-suited, and so to the theory of Saturn’s rings.’

V. V. Meleshko

Fig. 8 The Dyson model for coaxial vortex rings. a Geometry of the problem. b A vortex ring approaching a rigid plane. From[40]

Ri. The volume of each ring remains constant in time,

2π2a2i Ri = Consti, i = 1, . . . , N . (9)

The flow of incompressible fluid is the axisymmetric one with radial u(r, z) and axial w(r, z) componentsof the velocity vector and the single scalar component ω(r, z) = ∂u

∂z − ∂w∂r of the vorticity vector along the

φ-axis which can be expressed via the stream function �(r, z) as

u = −1

r

∂�

∂z, w = 1

r

∂�

∂r, ω = −1

r

(∂2�

∂z2 + ∂2�

∂r2 − 1

r

∂�

∂r

)

.

Suppose that for each vortex ring the single scalar component of the vorticity vector along the φ-axisvaries proportional over r, ωi = Mir , with Mi is constant. Then the Helmholtz vorticity equation written inthe modern notation [14, Art. 7.1]

D(ω/r)

Dt= 0

is satisfied identically inside each vortex ring.The stream-function �(r, z) satisfies the equation

∂2�

∂z2 + ∂2�

∂r2 − 1

r

∂�

∂r= 0 outside the rings,

∂2�

∂z2 + ∂2�

∂r2 − 1

r

∂�

∂r+ Mir

2 = 0 inside the ring (V R)i, (10)

while � and its normal derivative d�dn

are continuous everywhere.The value �(r ′, z′) at any point (r ′, z′) outside the rings is given by

�(r ′, z′) = r ′N∑

i=1

Mi

4π

∫ ∫ ∫r2 cosφ dz dr dφ

√[(z′ − z)2 + r ′2 − 2rr ′ cosφ + r2

] , (11)

where the integrals are taken throughout the volume of each ring.Inside the ring (V R)i the solution for �(r ′, z′) is more complicated. Dyson [40, Eq. (87)] presents it in a

symbolic form, which has got an admiration [51,53,54] even in recent time. After some algebra and expansioninto Taylor series on ai he comes to the expression for the stream-function at a point (Ri−ai cosχ, Z i+ai sin χ)with 0 ≤ χ ≤ 2π , on the surface of the ring (V R)i

�(Ri − ai cosχ, Z i + ai sin χ) = �i

2π

[

Ri

(

ln8Ri

ai− 2

)

− 1

2

(

ln8Ri

ai− 1

4

)

ai cosχ + . . .

]

+N∑

j=1

′ �j

2π

[

Iij + ∂ Iij

∂Z iai sin χ − ∂ Iij

∂Riai cosχ + . . .

]

, (12)

Coaxial axisymmetric vortex rings

where symbol ′ means the omission of the term in the sum with j = i, �i = πMia2i Ri = Consti is the intensity

of the vortex ring (V R)i (or its circulation, according to W. Thomson; here we use the notation � instead ofthe original m, remember that m = �/2), and

Iij =∫ π

0

Ri Rj cosφdφ√[(Z i − Z j)2 + R2

i − 2Ri Rj cosφ + R2j

] =√

Ri Rj

[(2

kij− kij

)

K(kij)− 2

kijE(kij)

]

(13)

k2ij = 4Ri Rj

(Ri + Rj)2 + (Z i − Z j)2.

Here K(k),E(k) are complete elliptic integrals of the first and second kind, respectively.On the other hand, let the ring (V R)i moves forward with velocity Z i and its radius increases with the

velocity Ri. The radius of the cross-section will change, but since a2i Ri = Consti and ai = − 1

2aiRi

Ri, therefore

(ai/Ri being small), ai is negligible compared with Ri. Calculating the normal velocity of any point on thering’s surface and remembering the meaning of the stream function, we obtain at the surface of the vortex ring(V R)i

�(Ri − ai cosχ, Z i + ai sin χ) = Ci − Ri Z iai cosχ − Ri Riai sin χ + · · · . (14)

Comparing this with the value already found for � at the surface of the ring, we obtain

Ri Z i = �i

4π

(

ln8Ri

ai− 1

4

)

+N∑

j=1

′ �j

2π

∂ Iij

∂Ri,

Ri Ri = −N∑

j=1

′ �j

2π

∂ Iij

∂Z i, i = 1, . . . , N . (15)

If we introduce the notation,

U =N∑

i=1

N∑

j=1

′�i�j

2πIij, (16)

then the system of equations (15) can be written as

�i Ri Z i = �2i

4π

(

ln8Ri

ai− 1

4

)

+ ∂U

∂Ri,

�i Ri Ri = − ∂U

∂Z i, i = 1, . . . , N . (17)

The system of equations (17) together with (9) and initial conditions

Ri(0) = R(0)i , Z i(0) = Z (0)i , ai(0) = n(0)i R(0)i , n(0)i � 1 i = 1, . . . , N (18)

represent a nonlinear system of ordinary differential equations for determining the trajectory (Ri(t), Z i(t)) ofthe i-th vortex ring.

Remarkably, the system (17) has two independent first integrals:

N∑

i=1

�i R2i ≡ P

πρ= Const, (19)

and

N∑

i=1

�2i

4πRi

(

ln8Ri

ai− 7

4

)

+ U ≡ T

2πρ= Const . (20)

V. V. Meleshko

These integrals are the equations expressing the constancy of the linear momentum P and of the kinetic energyT of the infinite fluid and can be obtained in the following manner.

Since U in (16) is a function of only Z i − Z j then from summation on i from 1 to N of the second lineof (17)

N∑

i=1

∂U

∂Z i= 0, ⇒

N∑

i=1

�i Ri Ri = 0.

Therefore, after integration, we obtain the equation (19).Again, multiply the first line in (17) by Ri and the second line by −Z i and then add, therefore

N∑

i=1

�2i

4π

(

ln8Ri

ai− 1

4

)

Ri +N∑

i=1

(∂U

∂RiRi + ∂U

∂Z iZ i

)

= 0. (21)

Remembering (9) we arrive at (20).For any two vortex rings, the invariants (19) and (20) are sufficient for complete integrability of the system

(17).Dyson also writes the system of equations (17) in terms of the kinetic energy T

�i Ri Z i = 1

2πρ

∂T

∂Ri, −�i Ri Ri = 1

2πρ

∂T

∂Z i, i = 1, . . . , N , (22)

which can be transformed into the Hamiltonian system

qi = ∂H

∂pi, pi = −∂H

∂qi, i = 1, . . . , N (23)

with the canonical variables qi = �i Z i, pi = Z2i , and the Hamiltonian H = T/πρ. This representation has

been independently obtained in [24,153,174,175,220].We dimensioned all parameters in respect to the parameters of the first vortex ring, namely �1 and R(0)1 ;

the time is dimensioned by �1/(R(0)1 )2. Time integration of the system (17), with the associated initial con-

ditions (18) and condition (9) was performed using the fourth order Runge–Kutta scheme with a variablestep. The accuracy of the computed trajectories has been thoroughly investigated by integrating the system ofequations with various initial step sizes and local error values. The initial step �t = 0.005 with maximumstep �tmax = 0.001 and a local error 10−13 were sufficiently demeaned to ensure proper accuracy. Doubleprecision computations were found necessary. Besides, the invariants (19) and (20) are used to additionallycontrol the accuracy of simulations.

To understand the various rich possibilities, it is worth stated that there are four dimensionless parameters(including an initial separation distance) for the two vortex rings case and eight dimensionless parameters forthe three vortex rings case.

4.2 Interaction of two vortex rings

For the classical Helmholtz’s case of mutual threading, the initial condition at t = 0 is that the two identicalvortex rings of strength �1 =�2 = 1.0, and radii R(0)1 = R(0)2 = 1.0 with n(0)1 = n(0)2 = 0.01 are located at some

distance Z0 = Z (0)1 − Z (0)2 > 0. The interaction of the vortices in this configuration is that of the well-knowngame of leapfrogging. In course of the interaction, the radius of the rear vortex ring initially decreases, whilethe radius of the front vortex ring increases. The interaction causes the translational speed of the rear vortex tobecome larger than the speed of the front one. Therefore, the rear vortex gradually catches up with the frontone and threads through it.

The invariants (19) and (20) provides the possibility to construct the phase space for such a dynamicalsystem: the ring radius R1 against the vortex separation Z1 − Z2. Such phase trajectories were obtained inseveral studies [22,56,91].

Figure 9a illustrates the classical Helmholtz leapfrogging interaction and presents the trajectories of ringsfor this case of interaction with R(0)2 = 1.0, Z (0)2 = 1.0, n(0)1 = n(0)2 = 0.01. The trajectory of the first ring is

Coaxial axisymmetric vortex rings

Fig. 9 Interaction of two coaxial vortex rings with positive circulations, the Dyson model. The trajectories of the centroids formutual threading (a) and slip-through (c) motion. a Helmholtz vortex rings �1 = 1, �2 = 1, R(0)1 = 1.0, R(0)2 = 1.0, Z (0)1 =0.0, Z (0)2 = 1.0. b Admissible initial values of R(0)2 and Z (0)2 − Z (0)1 for the mutual treading of vortex rings for several values of

�2. c �1 = 1, �2 = 2, R(0)1 = 1.0, R(0)2 = 1.0, Z (0)1 = −2.0, Z (0)2 = 0.0

shown by solid line, the trajectory of the second ring is indicated by dashed line. Vortex positions via equalmoment �t = 2.0 is denoted by filled circles. Directions of vortex motion are indicated by arrows.

Dyson [40, Art. 46] proved the following theorem concerning the possibility of permanent mutual threadingof two equal vortex rings:

If κ√

2 sin θ0 and κ√

2 cos θ0 be the radii of two coaxial vortex rings of equal strength and volumewhen the rings are in the same plane (θ0 being< π/4, so that κ

√2 sin θ0 is the radius of the inner ring):

then these two rings will continue to thread one another in turns, or will separate to an infinite distanceaccording as θ0 is > or < β, where β is determined by the equation

sin β√2

[

ln8

n− 7

4+ 3

2ln(

√2 sin β)

]

+ cosβ√2

[

ln8

n− 7

4+ 3

2ln(

√2 cosβ)

]

+ 1√2

∫ π

0

sin 2β cosφdφ√(1 − sin 2β cosφ)

= ln8

n− 7

4. (24)

Dyson calculated the value of β for several values of n = a/R and found that roughly speaking, when n isbetween 0.125 and 0.01, the initial ratio R(0)1 /R(0)2 with Z (0)1 = Z (0)2 = 0 for the leapfrogging motion shouldstay between 1

3 and 25 . Addressing now to Auerbach’s picture (Fig. 7a), we can see that this ratio is out of the

defined interval, and, consequently, these two vortex rings can not perform the leapfrogging motion.Hicks [81] proved that the mutual threading of two vortex rings of arbitrary circulation is possible (inde-

pendent of initial separation Z (0)1 − Z (0)2 ) only if their self-induced velocities at infinity are equal. In other case,either second vortex ring never reaches first vortex ring or if it were slight greater, it would ultimately reachit, and, since the trajectories are symmetrical, passes over to infinity at another side. This condition providesa rather complicated analytical formula [64], which is hardly necessary to write down in full. Instead, Fig. 9bshows the domain of admissible initial parameters for mutual threading for several typical values of �2. Thisfigure generalizes the results of Dyson [40] and Hicks [81].

The example of a single interaction of two vortex rings with �2 �= 1 is presented in Fig. 9c. Here �2 =0.5, R(0)2 = 1.0, Z (0)2 = 1.0, n(0)1 = n(0)2 = 0.01. After a single interaction, the distance between vortex ringsindefinitely increases.

An isolated single vortex ring in a unbounded space moves stationary with a self-induced velocity. Itappears that the system of two vortex rings has also a steady regime of motion. To find the parameters for thisregime, it is necessary to satisfy three conditions

R1 = 0, R2 = 0, Z1 − Z2 = 0.

V. V. Meleshko

Fig. 10 Interaction of two coaxial vortex rings with opposite circulations, the Dyson model. The trajectories of the centroids fordirect scattering (a) and mutual trapping (c) motion. a �1 = 1, �2 = −0.6, R(0)1 = 1.0, R(0)2 = 0.7, Z (0)1 = 0.0, Z (0)2 = 5.0. b

Domains for several types of interactions. c �1 = 1, �2 = −0.6, R(0)1 = 1.0, R(0)2 = 0.85, Z (0)1 = 0.0, Z (0)2 = 0.0

This provides a complicated relation between the intensity �2 of the second vortex and its (then constant)radius R2

�2 =ln

(8

a1− 1

4

)

− 2

Rmax

[

K(k2)− E(k2)+ 2(1 − R2)

RminE(k2)

]

1

R2

(

ln8R2

a2− 1

4

)

− 2

Rmax

[

K(k12)− E(k2)+ 2R2(R2 − 1)

RminE(k2)

] . (25)

with R2max = (1 + R2)

2, R2min = (1 − R2)

2, k22 = 4R2/(1 + R2)

2.In most cases the value �2 is negative except for the region of small radii for very thin rings.If vortex rings have different sign of circulation, �2 < 0, then their self-induced velocities have opposite

directions. If �2 �= −1 or R(0)2 �= 1, then a symmetry in vortex rings motions is absent. Vortex rings moveone through another. We point out two examples of vortex interaction in this case. Let the first vortex ring,which moves in the direction of positive z, has greater own energy in comparison with the second ring. Thiscase corresponds to the following initial conditions: �2 = −0.6, R(0)2 = 0.7, Z (0)1 = 0.0, Z (0)2 = 5.0, n(0)1 =n(0)2 = 0.01. The results of calculations are shown in Fig. 10a. It is seen that ring 1 increases its own radius andpasses through another ring. Detailed analysis shows that vortices have analogous interaction in cases whenvalues of initial radii have a small difference.

Vortex interaction follows another scenario for initial conditions, which are defined by large enough differ-ence in radii of vortex rings. The trajectories of vortices for case �2 = −0.6, R(0)2 = 0.5, Z (0)1 = 0.0, Z (0)2 =5.0, n(0)1 = n(0)2 = 0.01 are similar to ones in Fig. 10a, but here the smaller vortex ring does not changeconsiderably its radius and once passes through the first vortex ring.

Both cases for �2 < 0 are characterized by a single interaction. Two vortex rings after the interactioncontinue their motion in the directions of their self-induced velocities.

The character of trajectories of vortex interaction is determined by parameters of rings in the moment whenvortex rings have the same axial position, Z1 = Z2. A limiting case, when one type of interaction is changedby another, is defined by values of ring radii, when their self-induced velocities are equal for given �2. Thislimiting regime of motion exists for arbitrary values �2. Each state is characterized by the defined values ofradii of rings at Z1 = Z2.

Approaching of vortices to the stationary motion of rings determines the type of their further interaction.For example, if rings have radii less than values corresponded to the stationary motion, their radial velocities atZ1 = Z2 change the signs. As a result, the second vortex ring passes through the first ring. If vortices achievethe plane Z1 = Z2 parameters in a neighborhood of the stationary regime, the radial components of vortexvelocities do not change their sign. Consequently, we have an interaction, which is characterized by pass ofthe second vortex ring above the first one.

Figure 10b presents the results of calculations for different values of intensities of the second vortical ring�2 = −0.2,−0.4,−0.6,−0.8, and −1.0. Regions of initial parameters for each possible type of interaction at

Coaxial axisymmetric vortex rings

�2 = −0.6 are shaded in different way. The region A is characterized by pass of the first vortex ring throughsecond one, the region B corresponds to the motion of the second ring through the first vortex ring. Investi-gation shows that there is a third closed region C of parameters (except the case �2 = −1). The boundary ofthis region crosses the vertical axis at two points, the bottom point determines the values of the second ringsfor the stationary motion of vortices.

Our results show that the region C corresponds to periodic interaction of the two vortex rings with inten-sities of opposite signs. The example of this motion is shown in Fig. 10c for the following initial parametersof the second vortex ring: �2 = −0.6, R(0)2 = 0.85, Z (0)1 = Z (0)2 = 0.0, n(0)1 = n(0)2 = 0.01. Vortex rings,participating in an leapfrogging interaction, have a certain displacement along the axis of symmetry. Vorticesdo not withdraw at large distances during the interaction in comparison with periodic motion for rings withthe same signs of intensities.

Thus, all possible situations of interactions of two coaxial vortex rings as they move in the same direction,or in opposite directions, are: leapfrogging or mutual threading (Fig. 9a), slip-through motion (Fig. 9c), directscattering (Fig. 10a), and mutual trapping (Fig. 10c).

4.3 Interaction of a vortex ring with a rigid sphere

Dyson [40, Art. 45] also considered the case of a single vortex ring passing over a fixed rigid sphere of radiusk. He applies the method of images and states following Lewis [115] that if �2, R2, a2, Z2 are the intensity,the radius, the core radius, and the distance along Z -axis, respectively, then an imaginary coaxial vortex ringwith parameters �1, R1, a1, Z1, where

�1 = −�2

√R1

R2, a1 = a2

R1

R2,

√(R2

1 + Z21

)√(R2

2 + Z22

) = k2, (26)

together with the original one provides the zero normal velocity at the surface of the sphere.The equation of energy (20) written as

�22

2R2

(

ln8R2

a2− 7

4

)

+ �22 R2

2R1R1

(

ln8R1

a1− 7

4

)

− �22

√R2

R1I12 = �2

2 R∞(

ln8R∞a∞

− 7

4

)

, (27)

where R∞ and a∞ are the values of R2 and a2, respectively, when the vortex ring and the sphere are a longway apart, together with relations (26) provides an implicit equation [40, Eq. (118)] for the path of the vortexring from infinity up to the sphere. Dyson provided some numerics for several typical cases. Additional resultscan be found, e.g., in [100,158,180,238]. This model can be generalized on the case of an oscillating spherewith periodically shedding coaxial vortex rings [202].

4.4 Interaction of three vortex rings

Three coaxial vortex rings need for the determination of all possible situations at least six independent determi-native parameters (if the parameters of all rings are divided by the parameters of one ring). An accurate proofof the nonintegrability of the system (17) in general case, is given in [11,20,21]. It is based on the phenomenonof splitting of the separatrices of the Hamiltonian system (23), which leads to the absence of additional analyticintegrals.

At the same time, the trajectories of the vortex rings become very sensitive to possible changes in the initialconditions. Figure 11, the simplest example of such sensitiveness is shown. Here the trajectories of motion forthe simplest case of initial conditions for three identical vortex rings are slightly different in the Z coordinate.One can conclude, that it is impossible to predict the process of the interaction even with such a little changeof one of the co-ordinates.

Essential changes in values of circulations of the three vortex rings lead to splitting the system into twoleapfrogging vortex ring and a single one moving either faster or slowly the two others (Fig. 12).

Additional results for another initial positions of three vortex rings can be found, e.g., in [98,99,144].

V. V. Meleshko

Fig. 11 Interaction of three coaxial vortex rings, the Dyson model. The trajectories of the centroids.�1 = 1, �2 = 1, �3 = 1, R(0)1

= 1.0, R(0)2 = 1.0, R(0)3 = 1.0. a Z (0)1 = 0.0, Z (0)2 = 1.00, Z (0)3 = 2.00. b Z (0)1 = 0.0, Z (0)2 = 1.01, Z (0)3 = 2.00. c Z (0)1 = 0.0,

Z (0)2 = 1.00, Z (0)3 = 2.01

Fig. 12 Interaction of three coaxial vortex rings, the Dyson model. The trajectories of the centroids. R(0)1 = 1.0, R(0)2 = 1.0,

R(0)3 = 1.0, Z (0)1 = 0.0, Z (0)2 = 1.0, Z (0)3 = 2.0. a �1 = 2, �2 = 1, �3 = 1. b �1 = 1, �2 = 2, �3 = 1. c �1 = 1, �2 = 1, �3 = 2

4.4.1 Fluid transport by two interacting vortex rings

The problem of motion and interaction of three vortex rings has one important application. It concerns withLagrangian analysis of fluid transport in leapfrogging motion of two vortex rings. This issue is naturally relatedto stable and unstable manifolds—the separatrix of an atmosphere of a “fat” vortex ring (Fig. 6c), the hetero-clinic tangle of these manifolds under periodic perturbation, and the process of vortex ring fluid entrainmentand detrainment. Recent papers [192,194] contain a solid background of this subject together with the historyof lobe dynamics associated to coaxial vortex ring interaction.

The main idea in study of motion of two vortex ring atmosphere is to consider their boundaries as lines(in the cross-section plane) consisting of some passive fluid particles. Each particle can be considered as athird “vortex ring” with zero circulation. The contour tracking algorithms or more simple a collection of pas-sive particles along the boundary of each atmosphere provides reasonable correspondence with well-knownexperimental photographs [236] of two equal vortex ring interaction (Fig. 13). In spite of earlier discussion[140,237], this comparison supports the conclusion [194] ‘that the observed experimental pattern may be duemerely to complex motion of tracer in irrotational or weakly vortical fluid, with vortex cores behaving in asimple, non-deforming and almost classical manner. It is only the tracer that appears to deform and roll uparound the leading vortex.’

Additional experimental and numerical observations of the Lagrangian transport and a quantitative mea-sures for estimation of the entrainment and detrainment of the surrounding fluid under vortex rings interactioncan be found in [31,99,102,185,219].

Coaxial axisymmetric vortex rings

Fig. 13 Leapfrogging of two identical vortex rings. a Photographs are from [236]. b Contour tracking of the atmospheres, theDyson model

4.5 Interaction of N > 3 vortex rings

The chaotic behavior of that system is observed [98,99] more clearly. On the other hand, it was reported [3]about an effect of localization discrete vortex rings into groups of large eddies consisting of coaxial vortexrings in jet flows under harmonic forcing. Similar phenomena was observed [131] in more complicated caseof coaxial vortex rings with a rectilinear line vortex along the z-axis.

The infinite train of identical vortex rings evenly spaced along the Z -axis propagates with a constant veloc-ity [221]. The interesting question of a (linear) stability and vibration of such an infinite system was addressedin [113]. The main conclusion is that the arrangement is definitely longitudinally unstable for all values of theratio of the distances apart the rings to their radii. Another paper [114] deals with possible nonaxisymmetricwaves on the vortex ring. In particular, ‘it was found that for any given ratio of radius of ring section to radiusof ring there exists a critical ratio of ring spacing to radius, separating the region of stable oscillation fromthat of instability, a result in some respects closely analogous to that found by Kármán for the stability of twoinfinite parallel rows of rectilinear vortices.’ In our opinion, these remarkable papers still deserve attention.The Dyson model might appear in great help.

5 Conclusion

In 1981 in his influential paper for the special issue devoted to the 25th anniversary of the Journal of FluidMechanics Saffman [182, p. 57] wrote:

There can be no doubt that in any list of the most important papers in fluid mechanics, a prominentplace will be held by Helmholtz’s great paper ‘Ueber Integrale der hydrodynamischen Gleichungen,welche den Wirbelbewegungen entsprechen’ whose appearance in 1858 laid the foundations of thestudy of rotational or vortex motion and together with the subsequent papers of Lord Kelvin providesthe basis for the understanding and description of rotational fluid motion under conditions such thatthe direct effect of viscosity is not important. A measure of the quality and significance of the paperis that, although it is 123 years old, it is as good and clear an exposition as any, and is old-fashionedonly in the use of Cartesian notation (like Lamb’s 1932 text) rather than the modern vector or tensornotation.

V. V. Meleshko

Looking back upon the 150 years since publication of Helmholtz’s great paper [66], we see that these wordsremain true, and the study of the dynamics of vortex rings and, generally, the legacy of Helmholtz and Kelvin[148] is still an active and important topic in fluid mechanics research.

Acknowledgements My best thanks are due to Hassan Aref for various discussions on fascination of vortex dynamics during along period of mutual work starting in August 1990 at the IUTAM Symposium on Stirring and Mixing (La Jolla, USA), for sharingmy interest in the historical part of any scientific research, and for his hospitality and care during my attendance in October 2008of the IUTAM Symposium on 150 Years of Vortex Dynamics (Copenhagen, Denmark). This visit was funded by Kiev NationalTaras Shevchenko University (Ukraine). I am grateful to G.J.F. van Heijst (Eindhoven, The Netherlands), L. Zannetti (Turin,Italy), A. Leonard (Pasadena, United States), T.S. Krasnopolskaya, A.A. Gourjii (both from Kiev, Ukraine) for their unfailingkindness in coming to my aid with various suggestions and advice.

References

1. Acheson, D.J.: Elementary Fluid Dynamics. Clarendon, Oxford (1990)2. Ackeret, J.: Über die Bildung von Wirbeln in reibungslosen Flüssigkeit. Z. Angew. Math. Mech. 15, 3–4 (1935)3. Acton, E.: A modelling of large eddies in an axisymmetric jet. J. Fluid Mech. 98, 1–31 (1980)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)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)8. Auerbach, D.: Some open questions on the flow of circular vortex rings. Fluid Dyn. Res. 3, 209–213 (1988)9. Auerbach, F.: Wirbelbewegung. In: Winkelmann, A. (ed.) Handbuch der Physik, Band 1, SS. 1047–1074. Barth, Leipzig

(1908)10. Auerbach, F.: Wirbelbewegung. In: Auerbach, F., Hort, W. (eds) Handbuch der Physikalischen und Technischen Mechanik,

Band 5, SS. 115–156. Barth, Leipzig (1927)11. Bagrets, A.A., Bagrets, D.A.: Nonintegrability of two problems in vortex dynamics. Chaos 7, 368–375 (1997)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)14. Batchelor, G.K.: An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge (1967)15. Bauer, G.: Die Helmholtzsche Wirbeltheorie für Ingenieure Bearbeitet. Oldenbourg, München-Berlin (1919)16. Betz, A.: Wirbelbildung in idealen Flüssigkeiten und Helmholtzscher Wirbelsatz. Z. Angew. Math. Mech. 10,

413–415 (1930)17. Betz, A.: Wie entsteht ein Wirbel in einer wenig zähen Flüssigkeit?. Naturwissenschaften 37, 193–195 (1950)18. Blackmore, D., Brøns, M., Goullet, A.: A coaxial vortex ring model for vortex breakdown. Physica D 237, 2817–2844 (2008)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)20. Blackmore, D., Knio, O.: KAM theory analysis of the dynamics of three coaxial vortex rings. Physica D 140, 321–348 (2000)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)22. Boyarintzev, V.I., Levchenko, E.S., Savin, A.S.: Motion of two vortex rings. Fluid Dyn. 20, 818–819 (1985)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)25. Cahan, D. (ed.): Hermann von Helmholtz and Foundations of Nineteenth-Century Science. University of California

Press, Berkeley (1993)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)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)29. Chu, C.-C., Wang, C.-T., Chang, C.-C., Chang, R.-Y., Chang, W.-T.: Head-on collision of two coaxial vortex rings: exper-

iment and computation. J. Fluid Mech. 296, 39–71 (1995)30. Cromby, A.C.: Helmholtz. Sci. Amer. 198, 94–102 (1958)31. Dabiri, J.O., Gharib, M.: Fluid entrainment by isolated vortex rings. J. Fluid Mech. 511, 311–331 (2004)32. Darrigol, O.: From organ pipes to atmospheric motion: Helmholtz on fluid mechanics. Hist. Stud. Phys. Sci. 29, 1–51 (1998)33. Darrigol, O.: Worlds of Flow: A History of Hydrodynamics from the Bernoullis to Prandtl. Oxford University

Press, Oxford (2005)34. Dickinson, M.: How to walk on water. Nature 424, 621–622 (2003)35. Didden, N.: On the formation of vortex rings: rolling-up and production of circulation. Z. Angew. Math. Phys. 30,

101–116 (1979)36. Dolbear, A.E.: Modes of Motion or Mechanical Conceptions of Physical Phenomena. Lee and Shepard, Boston (1897)37. Donnelly, R.J.: Quantized Vortices in Helium II. Cambridge University Press, Cambridge (1991)

Coaxial axisymmetric vortex rings

38. Drucker, E.G., Lauder, G.V.: Locomotor forces on a swimming fish: three-dimensional vortex wake dynamics quantifiedusing 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)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)43. Edser, E.: General Physics for Students: A Textbook on the Fundamental Properties of Matter. Macmillan, London (1911)44. Einstein, A.: Zum hundertjährigen Gedenktag von Lord Kelvin’s Geburt. (26. Juni 1824). Naturwissenschaften 12,

601–602 (1924)45. Ellington, C.P.: The aerodynamics of hovering insect flight. V. A vortex theory. Phil. Trans. R. Soc. Lond. B 305,

115–144 (1984)46. Epple, M.: Topology, matter, and space, I: Topological notions in 19th-century natural philosophy. Arch. Hist. Exact

Sci. 52, 297–392 (1998)47. Faber, T.E.: Fluid Dynamics for Physicists. Cambridge University Press, Cambridge (1995)48. Filippov, A.T.: The Versatile Soliton. Birkhäuser, Boston (2000)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)52. Fraenkel, L.E.: Examples of steady vortex rings of small cross-section in an ideal fluid. J. Fluid Mech. 51, 119–135 (1972)53. Fukumoto, Y.: Higher-order asymptotic theory for the velocity field induced by an inviscid vortex ring. Fluid Dyn.

Res. 30, 65–92 (2002)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)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)57. Gray, A.: Lord Kelvin: An Account of his Scientific Life and Work. Dent, London (1908)58. Gray, A.: Notes on hydrodynamics. Phil. Mag. (Ser. 6) 28, 1–18 (1914)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)60. Gröbli, W.: Spezielle Probleme über die Bewegung geradliniger paralleler Wirbelfäden. Vierteljschr. Nat. Ges. Zürich 22,

37–82, 129–168 (1877)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)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)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)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)69. Helmholtz, H.: Sui movimenti dei liquidi. Nuovo Cimento (Ser. 2) 1, 289–304 (1869)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 discontinu-irliche 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. (ed.) 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)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)80. Hicks, W.M.: The mass carried forward by a vortex. Phil. Mag. (Ser. 6) 38, 597–612 (1919)81. Hicks, W.M.: On the mutual threading of vortex rings. Proc. R. Soc. Lond. A 102, 111–131 (1922)82. Hill, M.J.M.: On a spherical vortex. Phil. Trans. R. Soc. Lond. A 185, 213–245 (1894)83. Hu, D.L., Chau, B., Bush, J.W.M.: The hydrodynamics of water strider locomotion. Nature 424, 663–666 (2003)

V. V. Meleshko

84. Joukovskii, N.E.: [Helmholtz’s] works on mechanics. (in Russian). In: Stoletov, A.G. (ed.) Hermann von Helmholtz 1821–1891. Public lectures delivered at the Imperior Moscow University for the Helmholtz fund, pp. 37–52. Moscow UniversityPress, 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)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)89. Kambe, T.: Acoustic emissions by vortex motions. J. Fluid Mech. 173, 643–666 (1986)90. Kambe, T.: Elementary Fluid Mechanics. World Scientific, Singapore (2007)91. Kambe, T., Minota, T.: Sound radiation from vortex systems. J. Sound Vibr. 74, 61–72 (1981)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)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)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)98. Konstantinov, M.Yu.: Chaotic phenomena in the interaction of vortex rings. Phys. Fluids 6, 1752–1767 (1994)99. Konstantinov, M.Yu.: Numerical investigation of the interaction of coaxial vortex rings. Int. J. Num. Meth. Heat Fluid

Flow 7, 120–140 (1997)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)101. Kragh, H.: The vortex atom: A Victorian theory of everything. Centaurus 44, 32–114 (2002)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)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)105. Lamb, H.: Hydrodynamics. 2nd edn. Cambridge University Press, Cambridge (1895)106. Lamb, H.: Hydrodynamics. 3rd edn. Cambridge University Press, Cambridge (1906)107. Lamb, H.: Hydrodynamics. 4th edn. Cambridge University Press, Cambridge (1916)108. Lamb, H.: Hydrodynamics. 5th edn. Cambridge University Press, Cambridge (1924)109. Lamb, H.: Hydrodynamics. 6th edn. Cambridge University Press, Cambridge (1932)110. von Laue, M.: Zum 50. Todestage von Hermann v. Helmholtz (8. September 1944.). Naturwissenschaften 32, 206–

207 (1944)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)114. Levy, H., Forsdyke, A.G.: The vibrations of an infinite system of vortex rings. Proc. R. Soc. Lond. A 116, 352–379 (1927)115. Lewis, T.C.: On the images of vortices in a spherical vessel. Q. J. Pure Appl. Math. 16, 338–347 (1879)116. Lewis, T.C.: Some cases of vortex motion. Mess. Math. 9, 93–95 (1880)117. Lichtenstein, L.: Über einige Existenzprobleme der Hydrodynamik homogener, unzusammendrückbarer, reibungsloser

Flüssigkeiten und die Helmholtzschen Wirbelsätze, Math. Z. 23, 89–154, 310–316 (1925)118. Lichtenstein, L.: Grundlagen der Hydromechanik. Springer, Berlin (1929)119. Lim, T.T.: A note on the leapfrogging between two coaxial vortex rings at low Reynolds numbers. Phys. Fluids 9,

239–241 (1997)120. Lim, T.T., Nickels, T.B.: Instability and reconnection in the head-on collision of two vortex rings. Nature 357,

225–227 (1992)121. Lim, T.T., Nickels, T.B. : Vortex rings. In: Green, S.I. (ed.) 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)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)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)126. Love, A.E.H.: On the motion of paired vortices with a common axis. Proc. Lond. Math. Soc. 25, 185–194 (1894)127. Love, A.E.H.: Hydrodynamik: Theoretische Ausführungen. In: Klein, F., Müller, C. (eds) Enzyklopädie der Mathematis-

chen Wissenschaften, Band. IV/3, SS. 84–147. Teubner, Leipzig (1901)128. Love, A.E.H.: Développements d’hydrodynamique. In: Molk, J., Appell, P. (eds) Enzyclopédie des Sciences Mathéma-

tiques, 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)131. Marshall, J.S.: The flow induced by periodic vortex rings wrapped around a columnar vortex core. J. Fluid Mech. 345,

1–30 (1997)

Coaxial axisymmetric vortex rings

132. Maxwell, J.C.: Manuscript fragments on the stability of fluid motion [a question set for the Cambridge MathematicalTripos in 1866]. In: Harman, P.M. (ed.) 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. (ed.) The Scientific Letters and Papersof 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. (ed.) The Scientific Letters and Papers of JamesClerk 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. (ed.) The Scientific Letters and Papers of JamesClerk 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. (ed.) The Scientific Letters and Papers ofJames 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)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)140. Maxworthy, T.: Comments on “Preliminary study of mutual slip-through of a pair of vortices”. Phys. Fluids 22, 200 (1979)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)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)145. Milne-Thomson, L.M. : Theoretical Aerodynamics. Macmillan, New York (1966)146. Minota, T., Nishida, M., Lee, M.G.: Head-on collision of two compressible vortex rings. Fluid Dyn. Res. 22, 43–60 (1998)147. Miyazaki, T., Kambe, T.: Axisymmetrical problem of vortex sound with solid surfaces. Phys. Fluids 29, 4006–4015 (1986)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. Proceedingsof 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)

150. Norbury, J.: A family of steady vortex rings. J. Fluid Mech. 57, 417–431 (1973)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)153. Novikov, E.A.: Hamiltonian description of axisymmetric vortex flows and the system of vortex rings. Phys. Fluids 28,

2921–2922 (1985)154. Ogawa, A.: Vortex Flow. CRC Press, Boca Raton (1993)155. Oshima, Y.: Head-on collision of two vortex rings. J. Phys. Soc. Jpn. 44, 328–331 (1978)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)157. Oshima, Y., Noguchi, T., Oshima, K.: Numerical study of interaction of two vortex rings. Fluid Dyn. Res. 1, 215–227 (1986)158. Pedrizzetti, G., Novikov, E.A.: Instability and chaos in axisymmetric vortex-body interactions. Fluid Dyn. Res. 12,

129–151 (1993)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)160. Poincaré, H.: Théorie des tourbillons. Carré, Paris (1893)161. Prandtl, L., Tietjens, O.G.: Fundamentals of Hydro- and Aeromechanics. Dover, New York (1957)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)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)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)166. Reusch, E.: Ueber Ringbildung in Flüssigkeiten. Ann. Phys. Chem. (Ser. 2) 110, 309–316 (1860)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)170. Reynolds, O.: Study of fluid motion by means of coloured bands. Nature 50, 161–164 (1894)171. Riley, N.: On the behaviour of pairs of vortex rings. Q. J. Mech. Appl. Math. 46, 521–539 (1993)172. Riley, N.: The fascination of vortex rings. Appl. Sci. Res. 58, 169–189 (1998)173. Riley, N., Stevens, D.P.: A note on leapfrogging vortex rings. Fluid Dyn. Res. 11, 235–244 (1993)174. Roberts, P.H.: A Hamiltonian theory for weakly interacting vortices. Mathematika 19, 169–179 (1972)175. Roberts, P.H., Donnelly, R.J.: Dynamics of vortex rings. Phys. Lett. A 31, 137–138 (1970)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. (ed.) 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)179. Rücker, A.W.: The physical work of von Helmholtz. Nature 51, 472–475, 493–495 (1895)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)

V. V. Meleshko

181. Saffman, P.G.: The velocity of viscous vortex rings. Stud. Appl. Math. 49, 371–380 (1970)182. Saffman, P.G.: Dynamics of vorticity. J. Fluid Mech. 106, 49–58 (1981)183. Saffman, P.G.: Vortex Dynamics. Cambridge University Press, Cambridge (1992)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)186. Scheel, H. (ed.): Gedanke von Helmholtz über schöpferische Impulse und über das Zusammenwirken Verschiedener

Wissenschafttszweige. Akademie, Berlin (1972)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)188. Schram, C., Hirschberg, A., Verzicco, R.: Sound produced by vortex pairing: prediction based on particle image veloci-

metry. AIAA J. 42, 2234–2244 (2004)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)193. Shariff, K., Leonard, A.: Vortex rings. Ann. Rev. Fluid Mech. 24, 235–279 (1992)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)195. Shariff, K., Leonard, A., Ferziger, J.H.: A contour dynamics algorithm for axisymmetric flow. J. Comput. Phys. 227,

9044–9062 (2008)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)197. Shashikanth, B.N., Marsden, J.E.: Leapfrogging vortex rings: Hamiltonian structure, geometric phases and discrete reduc-

tion. Fluid Dyn. Res. 33, 333–356 (2003)198. Silliman, R.H.: William Thomson: Smoke rings and nineteenth-century atomism. Isis 54, 461–474 (1963)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)201. Stokes, G.G.: On the theories of the internal friction of fluids in motion, and the equilibrium and motion of elastic sol-

ids. 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)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)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)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)207. Taylor, G.I.: Sir Horace Lamb, F.R.S. Nature 135, 255–257 (1935)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)209. Thompson, S.P.: The Life of William Thomson, Baron Kelvin of Largs, vol. I. Macmillan, London (1910)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)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)213. Thomson, W.: On vortex atoms. Phil. Mag. (Ser. 4) 34, 15–24 (1867)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)216. Tokaty, G.A.: A History and Philosophy of Fluid Mechanics. Dover, New York (1994)217. Truesdell, C: The Kinematics of Vorticity. Indiana University Press, Bloomington (1954)218. Turner, R.S.: Helmholtz, Hermann von. In: Gillispie, C.C. (ed.), 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)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)223. de Villamil, R.: ABC of Hydrodynamics. Spon, London (1912)

Coaxial axisymmetric vortex rings

224. de Villamil, R.: Motion of Liquids. Spon, London (1914)225. Villat, H.: Leçons sur la théorie des tourbillons. Gauthier-Villars, Paris (1930)226. Wakelin, S.L., Riley, N.: Vortex rings interactions II. Inviscid models. Q. J. Mech. Appl. Math. 49, 287–309 (1996)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)228. Weidman, P.D., Riley, N.: Vortex ring pairs: numerical simulation and experiment. J. Fluid Mech. 257, 311–337 (1993)229. Werner, F.: Hermann Helmholtz’ Heidelberger Jahre (1858–1871). Springer, Berlin (1997)230. Wien, W.: Lehrbuch der Hydrodynamik. Hirzel, Leipzig (1900)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)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)236. Yamada, H., Matsui, T.: Preliminary study of mutual slip-through of a pair of vortices. Phys. Fluids 21, 292–294 (1978)237. Yamada, H., Matsui, T.: Mutual slip-through of a pair of vortex rings. Phys. Fluids 22, 1245–1249 (1979)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)