Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 44
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1 11In order to determine the viscosity coefficients of liquids, Helmholtz asked his studentGustav von Piotrowski to measure the damping of the oscillations of a hollow metallicsphere filled with liquid and suspended by a torsion-resisting wire. Helmholtz integratedthe Navier-Stokes equation so as to extract the viscosity coefficient from these measurements and also from Poiseuille's older experiments on capillary tubes.
The two valuesdisagreed, unless a finite slip of the fluid occurred on the walls of the metallic sphere. Whenhe learned about this analysis, Stokes told Thomson that he inclined against the slip, butdid not exclude it. 11 2This episode shows that, as late as 1 860, the Navier-Stokes equation did not yet belong tothe physicist's standard toolbox. It could still be rediscovered. The boundary condition,which is crucial in judging consequences for fluid resistance and flow retardation, was stilla matter of discussion. Nearly twenty years elapsed before Hor,ace Lamb judged theNavier-Stokes equation and Stokes's boundary condition to be worth a chapter a treatiseon hydrodynamics. This evolution rested on the few successes met in the ideal circumstances of slow or small-scale motion, and on the confirmation of the equation byMaxwell's kinetic theory of gases in 1 866.
Until Reynolds's and Boussinesq's studies ofturbulent flow in the 1 880s, described in Chapter 7, the equation remained completely113irrelevant to hydraulics.Thus, the mere introduction of viscous terms in the equations of motion did notsuffice to explain the flows most commonly encountered in natural and artificial circumstances. This failure long confmed the Navier-Stokes equation to the department ofphysico-mathematical curiosities, despite the air of necessity that its multiple molecularand non-molecular derivations gave it. As we will see in the following two chapters, a fewhydrodynamicists left this equation aside and speculated that much of the true behavior ofslightly-viscous fluids such ·as air and water could be understood without leaving theperfect-liquid context.1 1 1 Helmholtz to Thomson, 30 Aug. 1859, Kelvin Collection, Cambridge University Library; Thomson toHelmholtz, 6 Oct.
1859, HN. Cf. Darrigol [1998], and Chapter 4, pp. 148, 158-9.112Helmholtz and Piotrowski [1860] pp. 1 95-214 (calculations in the spherical case), 215-17 (calculation forthe Poiseuille flow); Stokes to Thomson, 22 Feb. [1862], in Wilson [1990]. Helmholtz was aware of Girard'smeasurements (Helmholtz and Piotrowski [1860] pp. 217-19), which he unfortunately trusted, but not ofHagen's.113Lamb [1 879] chap. 9. The verification of the consequences ofMaxwell's kinetic theory by viscous dampingexperiments required new, improved solutions of the Navier-Stokes equation, cf. Hicks [1 882] pp.
61-70.4VORTICESI have been able to solve a few problems of mathematical physics on which thegreatest mathematicians since Euler have struggled in vain . . . But the pride I couldhave felt over the final results . . . was considerably diminished by the fact that Iknew well how the solutions had almost always come to me: by gradual generalization of favorable examples, through a succession of felicitous ideas after manyfalse trails. I should compare myself to a mountain climber who, without knowingthe way, hikes up slowly and laboriously, often must return because he cannot gofurther, then, by reflection or by chance, discovers new trails that take him a littlefurther, and who, when he finally reaches his aim, to his shame discovers a royalroad on which he could have trodden up if he had been clever enough to find theright beginning.
Naturally, in my publications I have not told the reader about thefalse trails and I have ouly described the smooth road by which he can now reachthe summit without any effort.1 (Hermann Helmholtz,1891)One way of addressing the practical failures of Euler's fluid mechanics was to introduceviscosity into the fundamental equations. This approach, described in the previous chapter, only helped in cases of Iaminar flow, such as the loss of head in capillary tubes or thedamping of pendulum oscillations.
In the 1 860s, Herrnann Helmholtz invented anotherapproach based on vortex-like solutions of Euler's equations.Helmholtz arrived at this idea while studying a specific problem of acoustics, thesounding of organ pipes. In his efforts to improve the theory of this instrument, he cameto consider the internal friction of the air and its damping effect. As he was unaware of theNavier-Stokes equation, he began by analyzing the solutions ofEuler's equation for whichinternal friction would play a role. This is the source of his famous memoir of 1 858 onvortex motion.In this study, Helmholtz included the simple case of a 'vortex sheet', that is, a continuous alignment of rectilinear vortices, and found it to be equivalent to a tangentialdiscontinuity of the fluid velocity across the sheet. He later appealed to such discontinuousmotions to explain another mystery of organ pipes, namely, the production of an alternating motion by a continuous stream of air through the mouth of the pipe.
In I 868, hedescribed the general properties of surfaces of discontinuity, the most essential one beingtheir instability, whereby any protuberance of the surface tends to grow and to unrollspirally, as shown in Fig. 4. 1 .Helrnholtz reached these notions by focusing on the difficulties o f a concrete applicationof Euler's equations to the specific system of organ pipes. By analogy, he believed that theneglection of surfaces of discontinuities or similar structures explained the failure of many1Helmholtz [1891] p. 14.WORLDS OF FLOW146Fig. 4.1.Spiral unrolling of a protuberance on a surface of discontinuity. Courtesy of Greg Lawrence (inFernando [1991] p.
475).other applications of theoretical fluid mechanics. The first three sections of this chapter,on acoustics, vortex motion, and vortex sheets, recount the emergence of the methods andconcepts that justified this conviction. Section 4.4 documents Helmholtz's interest inmeteorology and his understanding of cyclonic vortices. Section 4.5 shows how, inspiredby a singular observation in the Swiss sky, he came to apply discontinuity surfaces to thegeneral circulation of the atmosphere and to the theory of storms, thus foreshadowingsome central notions of modem meteorology. As is explained in the final section, Section4.6, he predicted atmospheric waves resulting from the instability of such surfaces, anddevoted much time and effort to the analogous waves induced by wind blowing overwater.4.1Sound the organDuring his studies at the University of Berlin, Helmholtz read widely in physics, as is clearfrom the erudition displayed in the memoir of 1 847 on the conservation of force.
Afterobtaining the Konigsberg chair of physiology, he specialized in the study of perception, atthe intersection of his interests in physics, physiology, aesthetics, and philosophy. Although his first research in this field concerned vision, in the mid-1850s he began a parallelstudy of the perception of sound. Acoustics was then a developing branch of physics, andan ideal subject for someone who loved both music and mathematics. Here is Helmholtz'seloquent statement of his motivation:2I have always been attracted by this wonderful, highly interesting mystery: It isprecisely in the doctrine of tones, in the physical and technical foundations ofmusic, which of all arts appears to be the most immaterial, fleeting, and delicatesource of incalculable and indescribable impressions on our mind, that the science ofthe purest and most consistent thought, mathematics, has proved so fruitful.4.1.1From acoustics to mathematicsThe earliest trace of Helmholtz's interest in acoustics is a review of works 'concerningtheoretical acoustics' that Helmholtz wrote for the Fortschritte der Physik of 1 848 and1849.
They all dealt with the physics of sound, that is, the first of the three components2He1mho1tz [1857] pp. . 121-2. On Helmholtz's biography, cf. Koenigsberger [1902]. On his interest in tbeperception of sound, cf. Voge1 [1993], Hatfield [1993], Hiebert and Hiebert [1994].VORTICES147that Helmholtz distinguished in the study of sensations, namely physical, physiological,and psychological. 3In particular, Helmholtz criticized Guillaume Wertheim's measurements of the velocityof sound with organ pipes. Following earlier theories by Daniel Bernoulli, Euler, andLagrange, Wertheim assumed stationary air waves in the pipes, with a velocity node at thebottom of the pipe and an anti-node at the opening.
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