Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 59
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Then the condition of a given impulse must bedropped, and 'steadiness' has the ordinary meaning of constancy of the velocity field. Thecondition of a given vorticity, Thomson tells us, is the fixity of the number and intensity ofthe vortex filaments (it isnot the steadiness of the vorticity field). In more rigorous terms,this means that the distribution of vorticity at any time can be obtained from the originaldistribution by pure convection.The variation ov = w x or, with \1 · or =0,of the fluid velocity meets this condition,since it has the same effect on the vorticity distribution w as a displacement or of the fluid26particles. Therefore, the integraloT =I pv .
(wXor) d-r =must vanish for any or such that \1 · or =0.I por . (vXw) d-r(5. 1 4)This implies that\1 x (v x w) = 0.(5. 1 5)Combined with the vorticity equation (the curl of Euler's equation)at - \1 x (v x w) = 0,8w(5. 1 6)this gives the steadiness of the vorticity distribution. The fluid being incompressible, thissteadiness implies the permanence of the velocity field, as was to be proved.Thomson declared the other part of his theorem, the stability of the steady motion whenthe kinetic energy is a maximum or a minimum, to be 'obvious'. Any motion that differslittle from an energy extremum at a given time, Thomson presumably reasoned, shouldretain this property in the course of time, for its energy, being a constant, should remainclose to the extremum value.
Metaphorically speaking, a hike at a constant elevationslightly below that of a summit cannot lead very far from the summit. Thomson did notworry that the proximity of two fluid motions was not as clearly defined as the proximityZ7of two points of a mountain range.25Thomson, letters to Stokes (1872-73), ST; Thomson [1 876], [1880b] (energetic criterion); Thomson and Tait[1867] (cf. Smith and Wise [1989] chap. 1 1). For a modern interpretation of Thomsen's criterion, cf. Arnol'd[1 966], Drazin and Reid [1981], pp. 432-5.26Rigorous1y, a gradient term must be added to "' x ar in order that av be paralle1 to the walls of the container.However, this gradient term does not contribute to the variation 8T of the kinetic energy.27Thomson [1876] p.
1 1 6.WORLDS OF FLOW194At any rate, Thomson's energetic criterion helped little in determining the stability ofvortex atoms. The energy of a vortex ring turned out to be a 'minimax' (saddle point), inwhich case the energy consideration does not suffice to decide stability.28 Presumably toprepare another attack on this difficult problem, he dwelt on the simpler problem ofcylindrically-symmetric motions within a tubular container.
In this case, a simple consideration of symmetry shows that a uniform distribution of vorticity within a cylindercoaxial to the container corresponds to a maximum energy in the above sense. Similarly,a uniform distribution ofvorticity in the space comprised between the walls and a coaxialcylinder has minimum energy. These two distributions are therefore steady and stable.295.3.3Labyrinthine degradationThomson had already studied the perturbations of the former distribution, the columnarvortex, in the absence of walls. He now included a reciprocal action between the vortexvibration and a 'visco-elastic' wall.
He thus seems to have temporarily left the ideal worldof his earlier reasoning to consider what would happen to a vortex in concrete hydrodynamic experiments for which the walls of the container necessarily dissipate part of theenergy of the fluid motion. 30Thomson described how, owing to the interaction with the visco-elastic walls, 'the waves[of deformation of the surface of the vortex] of shorter length are indefinitely multipliedand exalted till their crests run out into fine laminas of liquid, and those of greater lengthare abated.' The container thus becomes filled with a very fine, but heterogeneous mixtureof rotational fluid with irrotational fluid, which Thomson called a 'vortex sponge'.31 At alater stage, the compression of the sponge leads to the minimum energy distribution forwhich the irrotational fluid is confmed in an annular space next to the wall.
A few yearslater, George Francis FitzGerald and Thomson himself based a reputed theory of the etheron the intermediate vortex-sponge state. 32Some aspects of the dissipative evolution of a columnar vortex are relatively easy tounderstand. According to Helmholtz's vortex theorems, the rotational and irrotationalparts of the fluid (which have, respectively, the vorticity w of the original vortex column andzero vorticity) behave like two incompressible, immiscible fluids. Since the original configuration is that of maximum energy, the dissipative interaction with the visco-elastic wallleads to a lesser-energy configuration for which portions of the rotational fluid are closer tothe walls. As the w-fluid is incompressible, this evolution implies a corrugation of the vortexsurface. As Thomson proved in his study of columnar vortex vibrations, the corrugationrotates at a frequency that grows linearly with its inverse wavelength (and linearly with the28Thomson [1876] p.
124. For a given vorticity and a given impulse, the energy of a thin vortex ring (with quasicircular cross�section) is decreased by making its cross-section oval; it is increased by making the ring thicker inone place than in another.29Thomson (1880b] p. 173.31'Thomson to Stokes, 19 Dec. 1872, ST; Thomson (1880b] pp. 176-SO."Thomson [1880b] p. 177. In his correspondence of 1872, Thomson imagined a different process of 'labyrinthine' and 'spiraling' penetration of the rotational fluid into the irrotational fluid.32Cf. Hunt [1991] pp. 96-1 04. FitzGera1d first wrote on the vortex-sponge ether in FitzGera1d [1885]. Thomsonfirst wrote on this topic in Thomson [1887e].
See Chapter 6, pp. 242-3.INSTABILITY195vorticity w). Since the energy-damping effect of the walls is proportional to the frequency oftheir perturbation, the energy of the smaller corrugation waves diminishes faster. As forthese special waves (unlike sea waves) a smaller energy corresponds to a higher amplitude,the shorter waves must grow until they reach the angular shape that implies frothing andmixing with the irrotational fluid.33 On the latter point, Thomson probably reasoned byanalogy with the finite-height sea-wave problem, which he had been discussing with Stokes.Thomson expected a similar degradation to occur for any vortex in the presence ofvisco-elastic matter. 'An imperfectly elastic solid', he noted in1 872,'is slow but surepoison to a vortex.
The minutest portion of such matter, would destroy all the atoms ofany finite universe.' Yet Thomson did not regard this peculiar instability as a threat to hisvortex theory of matter. Visco-elastic walls did not exist at the scale of his ideal world fluid:all matter, including container walls, was made of vortices in the same fluid. In Thomson'simagination, the interactions of a dense crowd of vortices only resembled visco-elasticdegradation to the extent needed to explain the condensation of a gas on the walls of itscontainer.345.3.4DelusionFor a few more years, Thomson contented himself with the observed stability of smokerings and with the demonstrated stability of the columnar vortex.
By1 889,however, heencountered difficulties that ruined his hope of a vortex theory of matter. This is attestedby a letter he wrote to the vortex-sponge enthusiast FitzGerald: 'I have quite confirmedone thing I was going to write to you (in continuation with my letter of October26), viz.that rotational vortex cores must be absolutely discarded, and we must have nothing butirrotational revolution around vacuous cores.' He adduced the following reason: 'Steadymotion, with crossing lines of vortex columns, is impossible with rotational cores, but ispossible with vacuous cores and purely irrotational circulations around them.'35Crossing lines of vortex columns occurred in FitzGerald's and Thomson's vortex ether.They were also a limiting case of the mutually-embracing vortex rings that Thomsoncontemplated in his theory of matter.
Their unsteadiness was therefore doubly problematic. Thomson was pessimistic: 'I do not see much hope for chemistry and electromagnetism.' Although vacuous-core vortices with zero vorticity still remained possible, Thomsonwas much less eager to speculate on vortex atoms than he had been earlier. In subsequentletters, he tried to persuade FitzGerald to abandon the vortex ether. 36Considerations of stability also played a role in Thomson's renunciation. Since1867, hisfriend Stokes had been warning him about possible instabilities: 'I confess', Stokes wrotein January1873,'I am skeptical about the stability of many of the motions which youappear to contemplate.' In a letter to Stokes of December1 898,Thomson described thefrittering and diffusion of an annular vortex, with the comment:37"According to Thomson [ 1880b] pp.
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