Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 58
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This limit could not mean much to him: as will be seen shortly,he did not believe in the possibility of discontinuity surfaces in homogeneous fluids.12In 1 879, Rayleigh examined this very limit and derived the existence of exponentiallygrowing perturbations at any wavelength. Thus, he showed the similarity of the instabilities discovered by Helmholtz and Thomson. The modern phrase 'Kelvin-Helmholtzinstability' captures the same connection, with an unfortunate permutation of the namesof the two foundersY5.3 Vortex atoms5.3.1 Hydrodynamic analogiesEven though Thomson observed and measured waves while sailing and fishing, his maininterest in hydrodynamics derived from his belief that the ultimate substance of the worldwas a perfect liquid.
His earliest use ofhydrodynamics, in the 1 840s, was merely analogical:he developed formal analogies between electrostatics, magnetostatics, and the steadymotion of a perfect liquid, mainly for the purpose of transferring theorems from one fieldto another. His correspondence of this period contains letters to Stokes in which heenquired about the hydrodynamic results he needed. In exchange, he offered new hydrodynamic theorems that his development of the energetic aspects of electricity suggested.14One of these theorems is worth mentioning, for it played an important role in Thomson'slater discussions of kinetic stability. Consider a perfect liquid limited by a closed surfacethat moves from rest in a prescribed manner.
If the equation of this surface is F(r, t) = 0,then the condition that a fluid particle initially on this surface should remain on it reads(5. 1 1)According to a theorem by Lagrange, the motion v taken by the fluid derives from apotential cp. Now consider any other motion v' that satisfies the boundary condition at agiven instant. The kinetic energy for the latter motion differs from the former by(5.12)Partial integration of the second term givesJ p\7cp·JJ(v' - v) d-r = pcp\7 (v - v') d-r - pcp(v' - v) dS.·1 2See Helmholtz to Thomson, 3 Sept.
1868, quoted in Thompson [1910] p. 527.1 3Thomson [1 871a] p. 79; Rayleigh [1 879] pp. 365-71 .·(5.13)14See the letters of the period March...October 1 847, ST. Cf. Smith and Wise [1989] pp. 219-27, 263-75,Darrigol [2000] chap. 3.INSTABILITY191The volume integral vanishes because the fluid is incompressible. The surface integral also\1F and both motions satisfyT' - T is always positive. The energy of the motion thatthefluid takes at a given time owing to the motion impressed on its boundary is less than theenergy of any motion that satisfies the boundary condition at the same time. 15vanishes because the surface element dS is parallel tocondition(5. 1 1).Consequently,Even though in these early years Thomson constantly transposed such theorems toelectricity and magnetism, he did not yet assume a hydrodynamic nature of electricity ormagnetism. His attitude changed around1 850, after he adopted Joule's conception of heatas a kind of motion.
In this view, the elasticity of a gas results from hidden internal motion,so that an apparently potential form of energy turns out to be kinetic. Thomson and a fewother British physicists speculated, for the rest of the century, that every energy might be ofkinetic origin. The mechanical world view would thus take a seductively simple form. 1 6The kind of molecular motion that William Rankine and Thomson then contemplatedwas a whirling, fluid motion around contiguous molecules.
Gas pressure resulted from thecentrifugal force of molecular vortices. Thomson elaborated this picture to account for therotation of the polarization of light when traveling through magnetized matter, forelectromagnetic induction, and even for the rigidity of the optical ether. In privateconsiderations, he imagined an ether made of 'rotating motes' in a perfect liquid. Thegyrostatic inertia of the whirls induced by these motes provided the needed rigidity.
In1857, Thomson confided these thoughts to his friend Stokes, with an enthusiastic plea fora hydrodynamic view of nature: 17I have changed my mind greatly since my freshman's years when I thought it so muchmore satisfying to have to do with electricity than with hydrodynamics, which onlyfirst seemed at all attractive when I learned how you had fulfilled such solutions as18Fourier's by your boxes of water.
Now I think hydrodynamics is to be the root ofall physical science, and is at present second to none in the beauty of its mathematics.5.3.2A new theory of matterA year after this pronouncement, Helmholtz published his memoir on vortex motion. Inearly 1 867, Thomson saw the 'magnificent way' in which his friend Peter Guthrie Taitproduced and manipulated smoke rings.19 He gathered that Hehnholtz's theorems offereda fantastic opportunity for a theory of matter based on the perfect liquid. Instead ofrotating motes, he now considered vortex rings, and assimilated the molecules of matterwith combinations of such rings. The permanence of matter then resulted from the15Thomson [1 849].
Thomson stated two corollaries (already known to Cauchy): (i) the existence of a potentialand the boundary condition completely determine the flow at a given instant; (ii) the motion at any given time isindependent of the motion at earlier times. See also Thomson and Tait [1 867] pp. 312, 317-19.16Cf. Smith and Wise [1989] chap. 12, Stein [1981].1 7Thomson to Stokes, 20 Dec. 1857, ST. Cf. Smith and Wise [1 989] pp. 402-12, Knudsen [1971]. As noted inYamalidou [1998], the hydrodynamic view of nature implied a non-molecular idealization for the primitive fluid ofthe world.18This is an allusion to Stokes's calculation ([1 843] pp.
60-8) of the inertial moments of boxes filled with perfectliquid and his subsequent experimental verification of the results by measuring the torsional oscillations ofsuspended boxes of this kind. Cf. Chapter 3, p. 136.1 9Thomson to Helmholtz, 22 Jan. 1 867, quoted in Thompson [1910] p. 513.192WORLDS OF FLOWconservation of vorticity.
The chemical identity of atoms became a topology of mutuallyembracing or self-knotted rings. Molecular collisions appeared to be a purely kinetic effectresulting from the mutual convection of two vortices by their velocity fields. In a long,highly mathematical memoir, Thomson developed the energy and momentum aspects ofthe vortex motions required by this new theory of matter.20The most basic property of matter being stability, Thomson faced the question ofthe stability of vortex rings. Helmholtz's theorems only implied the permanence of theindividual vortex filaments of which the rings were made.
They did not exclude significantchanges in the shape and arrangement of these filaments when subjected to externalvelocity perturbations. Thomson had no proof of stability, except in the case of acolumnar vortex, that is, a circular-cylindric vortex of uniform vorticity. He showed thata periodic deformation of the surface of the column propagated itself along and aroundthe vortex with a constant amplitude.
An extrapolation of this behavior to thin vortexrings did not seem too adventurous to him. Moreover, Tait's smoke-ring experimentsindicated stability as long as viscous diffusion did not hide the ideal behavior.21During the next ten years, Thomson had no decisive progress to report on his vortextheory of matter. The simplest, non-trivial problem he could imagine, that of a cylindrically-symmetric distribution of vorticity within a cylindrical container, proved to be quitedifficult. In1 872/73,he exchanged long letters with Stokes on this question, with nodefinite conclusion.22 Thomson's arguments were complex, elliptic, and non-rigorous.As he admitted to Stokes, 'This is an extremely difficult subject to write upon.' Abenevolent and perspicacious Stokes had trouble guessing what his friend was hintingat.
I have fared no better. 23A stimulus came in1878from Alfred Mayer's experiments on floating magnets. TheAmerican professor had shown that certain symmetric arrangements of the magnets weremechanically stable. Realizing that the theoretical stability criterion was similar to that ofa system of vortex columns, Thomson exulted: 'Mr Mayer's beautiful experiments bring usvery near an experimental solution of a problem which has for years been before meunsolved-of vital importance in the theory of vortex atoms: to find the greatest numberof bars which a vortex mouse-mill can have.' Thomson claimed to be able to prove thesteadiness and stability of simple regular configurations, mathematically in the triangleand square cases, and experimentally in the pentagon case.24These considerations only shed light on the stability of a mutual arrangement ofvorticeswith respect to a disturbance of this arrangement, and not on their individual stability.They may have prompted Thomson's decision to complete his earlier, mostly unpublished20Thomson [1867], [1869].
Cf. Silliman [1963], Smith and Wise [1989] pp. 417-25, Kragh [2002]. See alsoChapter 4, pp. 1 54-5.21 Thomson [1867] p. 4, [1 880a]. As John Hinch told me, the relevance of the latter observation is questionable;the smoke rings may not indicate the actual distribution ofvorticity, because the diffusivity ofvorticity is muchmore efficient than that of smoke particles.22-rhomson to Stokes, 1 9 Dec. 1 872, 1-2, 8, 1 1 , 21-22 Jan. 1 873; Stokes to Thomson, 6, 18, 20 Jan.
1873, ST.Cf. Smith and Wise [1989] pp. 431-8.23Thomson to Stokes, 19 Dec. 1 872, ST.''Thomson [1878] p. 135. The subject was further discussed by Alfred Green hill in 1 878, J. J. Thomson in 1883,and William Hicks in 1882. Cf. Love [1901] pp. 122-5, Kragh [2002].INSTABILITY193considerations on the stability of cylindrical vortices. In harmony with the energy-basedprogram developed in his and Tait'sTreatise on natural philosophy, Thomson formulatedan energetic criterion of stability. In problems of statics, stable equilibrium corresponds toa minimum of the potential energy.
In any theory that reduces statics to kinetics, thereshould be a similar criterion for the stability of motion. For the motion of a perfect liquidwith the vorticity andimpulse given, the kinetic energy is stationary, then the motion is steady. If it is a (local)minimum or maximum, then the motion is not only steady but stable.25of unlimited extension, Thomson stated the following theorem. If,Some thinking is necessary to understand what Thomson had in mind, since he did notcare to provide a proof. For simplicity, I only consider the case of a fluid confined in a rigidcontainer with no particular symmetry.
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