Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 60
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176-7, this process only occurs if the canister offers no resistance torotation (so that the angular momentum of the fluid is constant).3"Thomson to Stokes, 1 872, ST, pp. 378-9.35Thomson [1 889] p. 202.36Ibid. p. 204. Cf. Hunt [1991]p.102.37Stokes to Thomson, 8 Jan. 1873; Thomson to Stokes, 27 Dec. 1898, ST.WORLDS OF FLOW196I now believe that this is the fate of vortex rings, and of every kind of irrotational[rotational?] motion (with or without finite slips anywhere) in a limited portion of aninviscid mass of fluid, which is at rest at great distances from the moving parts. Thisputs me in mind of a thirty-year-old letter of yours with a drawing in black and redink suggesting instability of the motion of a columnar vortex, which I did not thenbelieve.
I must see if l can find the letter.According to Thomson's later recollections, he became aware of the instability of vortexrings in unpublished work of 1 887:38It now seems to me certain that if any motion be given within a fmite portion of aninfinite incompressible liquid originally at rest, its fate is necessarily dissipation toinfinite distances with infinitely small velocities everywhere; while the total kineticenergy remains constant. After many years of failure to prove that the motion in theordinary Helmholtz circular ring is stable, I came to the conclusion that it is essentially unstable, and that its fate must be to become dissipated as now described.I came to this conclusion by extensions not hitherto published of the considerationsdescribed in a short paper entitled: 'On the stability of steady and periodic fluidmotion', in thePhi/.
Mag. for May 1887.In this short paper, Thomson proved that the energy of any vortex motion of a fluidconfined within deformable walls could be increased indefinitely by doing work on thewalls in a systematic manner. More relevantly, he announced that the energy of the motionwould gradually vanish if the walls were viscously elastic. It is not clear, however, why thisresult would have been more threatening to vortex atoms than the degradation of a vortexcolumn surrounded by viscously-elastic walls already was. 39Another paper of the same year seems more relevant. Therein Thomson considered thesymmetric arrangement of vortex rings represented in Fig. 5.3 as a possible model of arigid ether. He worried:Fig. 5.3.()..()•cr=i==-..{)..(}....()r•()l{)0T..i()..i()..1()nf!tnf•() f•..Williarn Thomsen's arrangement ofvortex rings as a tentative model of the optical ether.
The arrowsrepresent the axes of the rings, and the black and white dots their intersections with the plane of the fignre.From Thomson [1887e] p. 317.38Thomson [1905] pp. 370n-37ln.39Thomson [1 887b].197INSTABILITYIt is exceedingly doubtful, so far as I can judge after much anxious considerationfrom time to time during these last twenty years, whether the configuration represented [in Fig. 5.3] or any other symmetrical arrangement, is stable when the rigidityof the ideal partitions enclosing each ring separately is annulled through space . . .The symmetric motion is unstable, and the rings shuffle themselves into perpetuallyvarying relative positions, with average homogeneousness, like the ultimate moleculesof a homogeneous liquid.This instability threatened not only the vortex theory of ether-on which Thomsonpronounced 'the Scottish verdict ofnot proven'-butalso any attempt at explainingchemical valence by symmetric arrangements of vortex rings.
After twenty years of brooding, Thomsen's hope for a grand theory of ether and matter was turning into disbelief.405.4 The Thomson-Stokes debate5.4.1Conflicting idealsWhen, in1857, Thomson was contemplating an ether made of a perfect liquid and rotatingmotes, his friend Stokes warned him about the instability of the motion of a perfect liquidaround a solid body.41 Thomson confidently replied: 'Instability, or a tendency to run toeddies, or any kind of dissipation of energy, is impossible in a perfect fluid.' As he hadlearned from Stokes ten years earlier and as Cauchy had proved in1827,the motion ofsolids through a perfect liquid completely determines the fluid motion if the solids andfluid are originally at rest.
Following Lagrange's theorem, the latter motion is irrotationaland devoid of eddying. Following Thomson's theorem of 1849, it is the motion that has atevery instant the minimum energy compatible with the boundary conditions. Thomsonbelieved these two results to imply stability.42Stokes disagreed. He insisted: 'I have always inclined to the belief that the motion of aperfect incompressible liquid, primitively at rest, about a solid which continually progressed, was unstable. ' The theorems of Lagrange, Cauchy, and Poisson, he argued, onlyhold 'on theassumption ofcontinuity, and I have always been rather inclined to believe thatsurfaces of discontinuity would be formed in the fluid.' The formation of such surfaceswould imply a loss ofvis viva in the wake of the solid and thus induce a finite resistance to itsmotion.
A surface of discontinuity, he told Thomson, is surely formed when fluid passesfrom one vessel to another through a small opening (see Fig.5.1 ) , which implies the instability of the irrotational, spreading-out motion. Similarly, Stokes went on, the spreadingout motion behind a moving sphere (see Fig. 5.4) should be unstable. Stokes was only repeating the considerations he had used in 1 842/43 to reconcile perfect- and real-fluid behaviors. 4340Thomson [1887e] pp. 3 1 8, 320.41This is inferred from the letter from Thomson to Stokes of! 7 June 1857, ST: 'I think the instability you speak,of cannot exist in a perfect . . .
liquid.42Thomson to Stokes, 23 Dec. 1 857. Presumably, Thomson believed that a slightly-perturbed motion wouldremain close to the original motion because its energy would remain close to that of the minimum-energy solution.However, this is only true in a closed system for which there is no external energy input. As Stokes later argued,such an input may feed the perturbation.43Stokes to Thomson, 12-13 Feb. 1858, ST.198WORLDS OF FLOWFig. 5.4.The spreading motion of a fluid behind a sphere.
From Stokes to Thomson, 13 Feb. 1 858, ST.In general, Stokes drew his ideas on the stability of perfect-liquid motion from thebehavior of real fluids with small viscosity, typically water. In 1 880, while preparing thefirst volume of his collected papers, he reflected on the nature of the zero-viscosity limit.His remark of 1 849 on the discontinuity surface from an edge, he then noted, depended onthe double idealization of a strictly inviscid fluid and an infinitely-sharp edge:44A perfect fluid is an ideal abstraction, representing something that does not exist innature.
All actual fluids are more or less viscous, and we arrive at the conception of aperfect fluid by starting with fluids such as we find them, and then in imaginationmaking abstraction of the viscosity. Similarly, any edge we can mechanically form ismore or less rounded off, but we have no difficulty in conceiving of an edge perfectlysharp.Stokes then considered the flow for a finite viscosity f.L and a finite curvature radius a ofthe edge, and argued that the limit of this flow when a and f.L reached zero depended on theorder in which the two limits were taken.
If the limit f.L -> 0 is taken first, then the resultingflow is continuous and irrotational, and it obviously remains so in the limit a -> 0. If thelimit a -> 0 is taken first, then the resulting flow is that of a viscous fluid passing aninfinitely-sharp edge. The viscous stress is easily seen to imply the formation of a trail ofvorticity from the edge. In the limit f.L -> 0 this trail becomes infinitely narrow, and a vortexsheet or discontinuity surface is formed. Stokes believed the latter double limit to be theonly one of physical interest, because the result of the former was unstable in the sense thatan infinitely-small viscous stress was sufficient to turn it into a widely different motion.45Stokes returned to his idea of the double limit in several letters.46 In 1894, it led him to aninstructive comment on the nature of his disagreement with Thomson: 'Your speculations44SMPP 1, pp.
31 1-12.45Ibid. As Thomson later pointed out, in this alleged instability there is an apparent contradiction between tbevanishing work of the viscous stress and the finite energy difference between the two compared motions. Stokesreplied witb a metaphor (27 Oct. 1 894, STJ: 'Suppose there is a railway AB which at B branches off towards C andtowards D. Suppose a train travels without stopping along AB and onwards.
Will you admit that the muscularexertion of the pointsman at B is the merest trifle of the work required to propel the train along BC or CD? NowI look on viscosity in the neighborhood of a sharp, though not absolutely sharp, edge as performing the part of thepointsman at B.'461 Nov. 1 894, 22-23 Nov. 1898, 14 Feb. 1 899, ST. In this last letter, Stokes considers the state of things at timet from the commencement of motion and at distance r from an edge, and argues that the limit t --). 0 gives the'mike' (minimum kinetic energy) solution, whereas the limit r --+ 0 gives a discontinuity surface.INSTABILITY199about vortex atoms led you to approach the limit in the first way [/.L -> 0 first]; my ideas,derived from what one sees in an actual fluid, led me to approach it in the other way [a0first].' Indeed, Thomson's reflections on stability mostly occurred in the context of histheory of ether and matter.
He was therefore prejudiced in favor of stability, and generallyexpected important qualitative differences between real- and perfect-fluid behavior.-->5.4.2Careless vortices, goring, and dead waterIn 1 887, Thomson publicly rejected the possibility of surfaces of discontinuity, arguingthat they could never be formed by any natural action. In his opinion, continuity ofvelocity was always obtained when two portions of fluid where brought into contact. Henow agreed with Stokes and Helmholtz that the flow around a solid obstacle was unstablewhen the velocity exceeded a certain value, but denied that this instability had anything todo with surfaces of discontinuity.
For a perfect liquid, the determining effect was theseparation of the fluid from the solid surface.47In the case of flow around a sphere, Thomson described the instability as follows. Thefluid separates at the equator when the asymptotic velocity V of the fluid exceeds the valuefor which the pressure at the equator becomes negative Ci p V2 according to Bernoulli's lawapplied to the irrotational solution of Euler's equation).48 A careless vortex is formed, asindicated in Thomson's drawing (see Fig.
5.5). This vortex grows until it separates fromthe sphere and follows the flow. The whole process repeats itself indefinitely and results ina 'violently disturbed motion'.49Stokes did not comment on this cavitational instability, which was known to occur onthe edges of swiftly-moving immersed solids, for instance ship propellers. He did, however,Fig. 5.5.The formation of a careless vortex H near the equator G of a sphere inunersed in a moving liquid.From Thomson [1887a] p. 151.47Thomson [1887a].49Ibid. p. 149.480n negative pressure.
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