Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 47
Текст из файла (страница 47)
Most impressive is the contrast betweenthe simplicity of this experiment and the sophistication of the motivating physicomathematical analysis.254.2.4German approbation, British enthusiasm, and French suspicionHelmholtz's memoir on vortex motion is now universally regarded as the historicalfoundation of this subject. It quickly captured the attention of eminent German mathematicians and physicists. In 1 859, the Berlin mathematician Rudolph Clebsch recoveredHelmholtz's theorems by variational methods. In 1 860, Bernhard Riemann noted therelation between Helmholtz's theorems and his and Dirichlet's solutions for the problemof the rotating-fluid ellipsoid.
The Gottingen Ac�demy offered a prize for a Lagrangiandeduction of these theorems, which Hermann Hankel won in 1 8 6 1 . Helmholtz's memoirprovided the substance for two of Kirchhoff's famous lectures on mechanics.Z622Helmholtz [1858] pp. 124-7.23Jbid. pp.127-33.24Jbid. p. 1 33.25Jbid. p. 1 34.26Clebsch [1 859]; Riemann [1860]; Hankel [1861]; Kirchhoff [1 876] lectures 15, 20 . In manuscript fragments(HN, in #679 and #680), Helmholtz discussed various aspects of vortex motion, namely invariants, stability, andfriction.Fig. 4.2.Leapfrogging of two vortexrings. Photograph from Yamadaand Matsui [1978]. Courtesy of Prof.Yamada Hideo.VORTICES!55In Britain, Thomson, Maxwell, and Peter Guthrie Tait found Helmholtz's reasoninghighly congenial.
Tait's enthusiasm for Hamilton's quatemions arose when he realized, atthe end of I 858, how well suited they were to Helmholtz's decomposition offluid motion. In1867, he published a translation of Helmholtz's memoir and built a 'smoke box' with anelastic membrane on one side and a hole on the other, with which he could producespectacular shows of vortex rings and verify Helmholtz's predictions (see Fig. 4.3). 27 In aletter to Helmholtz written in January 1 867, Thomson described his friend's smoke box andhis own speculations on vortex-ring atoms, which were soon to be published in thePhilosophical magazine. In July 1 868, Tait announced to Helmholtz that Maxwell, 'oneof the most genuinely original men I have ever met', had taken up vortex motion andproved that 'two closed vortices act on one another so that the sum of the areas of theirprojections on any given plane remains constant.' He also mentioned that the forthcomingTransactions of the Royal Society of Edinburgh would be rich in papers on this subject.28The most important of these papers was a long, highly-mathematical memoir byWilliam Thomson.
There he developed Helmholtz's hints on topological aspects offluidmotion, and provided alternative proofs of his theorems based on the invariance of theintegral § v · dl over any circuit that follows the motion of the fluid. This invariance followsfrom the fact that the Lagrangian time derivative of the form v · dr, namely( )dvd1dP-(v · dr) = v · dv + - · dr = d -v2 + g · dr - 'dtdt2p(4. 1 1)is an exact differential for a perfect liquid in the gravity g. Through Stokes's theorem (ofwhich Thomson was the true discoverer), this result contains the invariance of the productw dS which is the essence of Helmholtz's laws of vortex motion. As Thomson knew, thereverse is not always true, because the volume occupied by the fluid may not be simply·Fig.
4.3.Tail's smoke box. From Tait [1 876] p. 292.27Tait to Hamilton, 7 Dec. 1858, quoted in Knott [1911] p. 127; He1mholtz [1 867]; Tait [1876] p. 292 (smokebox). A similar device had already been described in Reusch [1860]. Beautiful experiments on vortices in air and inliquids are also found in Rogers [1858]. These experimenters wereprimarily interested in the production of vorticesand were not aware of Helmholtz's predictions.28Thomson to Helmholtz, 22 Jan. 1867, HN; Thomson [1867]; Tait to Helmholtz, 28 July 1868, HN.
In a letterof2 May 1859 (HN), Thomson thanked Helmholtz for his memoir, which he had read 'with very great interest'before 'falling into the vortex of [his] winter's work'. In 1866, Maxwell set Hehnholtz's hydrodynamic theorems asa question to the Cambridge Mathematical Tripos (cf. Maxwell (1990] vol. 2, p.
241). On Thomsen's vortex atom,cf. Silliman [1963], Smith and Wise [1989] chap. 12, Kragh [2002], and the discussion in Chapter 5, pp. 191-7.!56WORLDS OF FLOWconnected. Maxwell and Tait examined the topological issues raised by Thomson, andobtained important results of the theory of knots.29Thomson had long dreamt of a world made entirely of motions in a pervasive, idealfluid. In this view, every form of energy was of kinetic origin. Every force had to be tracedto dynamical effects, as fluid pressure had been reduced to molecular collisions. Therigidity of the ether with respect to light vibrations was to be explained in terms of theinertia of small-scale motions of the primitive fluid. The permanence of atoms andmolecules was to be understood as the stability of special states of motion.
Helmholtz'stheorems, applied to vortex rings, and Thomson's further topological considerationsseemed to offer a rigorous basis for developing this program.30Yet, in the middle of writing his memoir on vortex motion, Thomson went through afew weeks of despair: 'It is a pity', he then wrote to Tait, 'that H2 [Hermann Helmholtz inMaxwell's notation] is all wrong and that we all dragged so deep in the mud after him.' Thecause of this lament was a criticism published by the eminent mathematician and academician Joseph Bertrand in the Comptes rendus of the French Academy of Sciences.31Bertrand rejected Helmholtz's interpretation of (1/2)\7 x v as the rotation velocity ofthe elements of the fluid.
Many cases of motion, Bertrand showed, could be reduced tothree dilations of the fluid elements along three oblique axes. Although, intuitively, suchmotions involve no rotation, the corresponding \7 x v vanishes only when the three axesare orthogonal. From this remark, Bertrand concluded that the integrability of v dl couldnot be identified with the absence of rotation. The consequences were devastating: 'Despite his very deep knowledge of mathematics, the author has committed a slight inadvertence at the beginning of his memoir that mars all his results by making him attach a quiteexcessive importance to the integrability condition [\7 x v = 0].'32Helmholtz promptly replied that the rule according to which one decomposes a complexmotion into simpler ones was to some extent arbitrary.
The decomposition of the motionof a fluid element into three orthogonal dilations and a rotation (also a global translation)is one possibility; that into three oblique dilations (with real or imaginary axes) is another.Although the former choice seems to contradict the geometrical intuition of a rotation, it isthe only one suited to fluid dynamics, because the angular momentum of the elements offluid is determined by a rotation defmed in this sense. Helmholtz added that his usage of'rotation' was not new and could be found in Kirchhoff's memoir on vibrating plates.33The source of the conflict is clear.
On one side, Helmholtz and Kirchhoff (also Stokesand Thomson) adjusted their geometrical and kinematic concepts to the needs of dynamics. On the other, Bertrand refused to let physical arguments control his geometricalintuition. His first reaction to Helmholtz's rebuttal was to give a 'decisive example' offluid motion that allegedly contradicted Helmholtz's definition of rotation: the fluidmoves uniformly in planes parallel to the Oxy plane, with a velocity increasing linearly·29Thomson [1869]. On Thomson's, Tait's, and Maxwell's contributions to topology, cf. Epple [1998].
On theorigin of Stokes's theorem, see SMPP 5, pp. 320-1.30Cf. Smith and Wise [1989] chap. 12.3 1Thomson to Tait, !I July 1868, quoted by Harman in Maxwell [1990] vol. 2, 399n; Bertrand [! 868a].32Bertrand [1868a] p. 1227.33Helmholtz [1868a]. Helmholtz referred to Stokes's earlier analysis in his next reply to Bertrand.VORTICES157with the coordinate z. According to Helmholtz's definition, this case involves a constantrotation of the fluid elements, against Bertrand's intuition of their motion. 34More came to irritate Bertrand.
In a note to the Comptes rendus, Saint-Venant sidedwith Helmholtz and referred to Cauchy's earlier interpretation of (1/2)\7 x v as the'average rotation' of the fluid particles. In reaction, Bertrand insisted that Helmholtz'smemoir contained false theorems as well as an aberrant definition of rotation. Forexample, he denied that the velocity field could be determined from the vorticity field inHelmholtz's manner, because the potential cp of eqn (4.6) indirectly depended on thevorticity, through the boundary conditions (for example, the tangential component of\7cp along the fixed walls of the fluid must be opposed to the tangential componentof \7 x A, which depends on the vorticity).35This criticism was slightly more embarrassing to Helmholtz, because he had not discussed the nature of the \i'cp contribution to the velocity field.
In his reply, he claimed thathe had been aware that this contribution in general depended on the vorticity, but hadnevertheless ignored it because the harmonic character of the function cp severely restrictedits form. In particular, it vanished whenever the fluid mass could be regarded as infinite(and the fluid motion did not extend to infinity). All the special cases of motion treated inhis paper were of that kind or could be brought back to it. 36In this second note, Helmholtz's tone was far less deferential than in the first; he accusedBertrand of disfiguring his theorems, and used mildly ironic phrases such as 'J'invite monsavant critique a se rappeler que . .
. '. To make things worse, in his journal Les mondes,the xenophilic Abbot Moigno ridiculed Bertrand's attitude: 'While M. Bertrand persistsin his inconsiderate criticism, he now wraps it in so many polite words and insistent,eloquent praises that any intelligent reader can conclude that he is certainly wrong.'In his final note, Bertrand protested his sincerity and intellectual honesty. He accusedMoigno of giving a poor idea of French manners and Helmholtz of believing him. Inthe next issue of Les mondes, Moigno retorted: 'We have perfectly felt the coups de griffewhich [M. Bertrand) gave us in his bad mood. If he maintains his jest in the Comptesrendus, we will have no pain to prove that of the great Academician and the humbleabbot, the most serious is not the one he thinks.' More diplomatically, Helmholtzapologized for having used phrases that could lead to misinterpretations of his trueintentions. 37Bertrand's attack was only a minor and ephemeral threat to Helmholtz's theory.Thomson quickly regained his faith in it.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
