Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 42
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Stokes also transposed Thomsen's method of electricalimages to show that a rigid wall placed near the oscillating sphere modified the masscorrection. Lastly, he addressed the most evident contradiction with observation, namely,that a perfect fluid does not have any more damping effect on oscillatory motion than itwould have on a uniform translational motion. 93Stokes evoked three possible causes of the observed resistance, namely, fluid friction,discontinuous flow, and instability leading to a turbulent wake. As he did not yet feel readyto explore any of these options by means of theory, he looked for further experimentalresults. He was not himself planning pendulum measurements, presumably because therequired apparatus and protocol were too complex for his taste; he usually favoredexperiments that could be performed with the minimum equipment and time consumption.For testing the departure of real fluids from perfect ones, he judged that the moments ofinertia of water-filled boxes offered a better opportunity.
Unfortunately, the experimentshe soon performed with suspended water boxes could only confirm the perfect-fluid theory.They were not accurate enough to show any effect of imperfect fluidity. 943.6.3FluidfrictionHaving exhausted the possibilities of his first strategy for studying the imperfection offluids, Stokes tried another approach.
In 1 845, he sought to include internal fluid frictionin the fundamental equations of hydrodynamics. To Du Buat's arguments for the existence of internal friction he added pendulum damping and a typically British observation:'The subsidence of the motion in a cup of tea which has been stirred may be mentionedas a familiar instance of friction, or, which is the same, of a deviation from the law ofnormal pressure.' From Cauchy he borrowed the notion of transverse pressure, as wellas the general idea of combining symmetry arguments and the geometry of infinitesimaldeformations.9 5Stokes's first step was the decomposition of the rate of changefluid segment dr into a symmetric and an antisymmetric part:1o;vjdx; of an infinitesimalIo;vjdx; = 2 (o;vj + Ojv;)dx; + 2 (o;vj - Ojv;)dx;.(3.
32)Then he showed that the antisymmetric part corresponded to a rotation of the vector dr,and the symmetric part to the superposition of three dilations (or contractions) along threeorthogonal axes. Thata small deformationO;Uj - OjU; represents the rotation of an element of a continuum foruwas known to Cauchy. No one, however, had explicitly given.93Stokes [1 843] pp.
36, 38-49, 53; Poisson [1832]. Stokes made his calculation in the incompressible case,knowing from Poisson that the effects of compressibility were negligible in the pendulum problem.94Ibid. pp. 60-8; Stokes [1846b] p. 196. On Stokes' experimental style, cf. Liveing [1907].95Stokes [1 849a] pp. 75-6; [! 848a] p. 3 (cup of tea). Stokes ([1849a] p. 1 1 8) refers to Cauchy as follows: 'Themethod which I have employed is different from [Cauchy's], although in some respects it much resembles it.'WORLDS OF FLOW138Stokes's decomposition and its geometrical interpretation. Cauchy and other theorists ofelasticity directly studied the quadratic form (112)egdx;dxj that gives the change in thesquared length of the segment dr.96Stokes then required, as Cauchy had done, the principal axes of pressure to be identicalwith those of deformation.
He decomposed the three principal dilations into anisotropic dilation and three 'shifting motions' along the diagonals of these axes. To theisotropic dilation he associated an isotropic normal pressure, and to each shift a paralleltransverse pressure. In order to get the complete pressure system, he superposed these fourcomponents and transformed the result back to the original system of axes. So far,Stokes's procedure was similar to Saint-Venant's, except that Saint-Venant dealt directlywith slides in the original system of axes and did not require any superposition of principalpressures nor any transformation of axes.97The analogy with Saint-Venant-whose connunication Stokes was probably unawareof-ends here.
Stokes wanted the pressures to depend linearly on the instantaneousdeformations. He justified this linearity (including the above-mentioned superposition),as well as the zero value he chose for the pressure implied by an isotropic compression, bymeans of a somewhat obscure model of 'smooth molecules acting by contact'. His previousapproach to the imperfect fluid had been deliberately non-molecular. The new, internalfriction approach was explicitly molecular.
Undoubtedly Stokes grew to be an overcautious physicist who avoided microphysical speculation as much as he could. Yet, no morethan his French predecessors could he conceive of internal friction without transversemolecular actions.983.6.4Elastic bodies, ether, andpipesStokes's reasoning of course led to the Navier-stokes equation, since this is the onlyhydrodynamic equation that is compatible with local isotropy and a linear dependencebetween stress and distortion rate. After reading Poisson's memoir of 1 829, which proceeded from the equations of elastic bodies to those of real fluids, Stokes tried the reversecourse and transposed his hydrodynamic reasoning to elastic bodies.
From the 'principleof superposition of small quantities', he derived the linearity of the stress-strain relation.He then exploited isotropy in the principal-axis system to introduce two elastic constants,one for the shifts, and the other for isotropic compression.Stokes thus retrieved the two-constant stress system that Cauchy had obtained forisotropic elastic bodies in his non-molecular theory of 1828. He imputed Poisson's single96Stokes [1849a] pp.80--4; Cauchy [1841] p. 321 (cf. Dugas [1950] pp. 402--6).
Stokes's reasoning did not seem22 Jan. 1862, in Larmor [1907] vol. I, pp. 156-159. Larmor'stoo clear to Saint-Venant; see his letter to Stokes,comment, 'The practical British method of development in mathematical physics, by fusing analysis with directphysical perception or intuition, still occasionally present similar difficulties to minds trained in a more formalmathematical discipline', does not seem to apply well to Saint-Venant, although it certainly applies to thecontinental perception of Larmor's own work.97Stokes [1 849a] pp.83-4.98Ibid. pp. 84--6.
Cf. Yamalidou [1998]. Stokes mentioned Saint-Venant's proof in his [1 846a] pp. 183-4, withthe observation: 'This method does not reqnire the consideration of ultimate molecules at all.' Stokes's modelimplies a zero trace for the viscous stress tensor, so that his equation includes the term (p,/3)\1(\1 · v) (besides thep.Av term) in the case of a compressible fluid.VISCOSITY139constant result to the assumption that the sphere of action of a given molecule containedmany other molecules-which only shows that he had not read the memoir in whichCauchy proved this assumption to be unnecessary. More pertinently, Stokes argued thatsoft solids such as India rubber or jelly required two elastic constants, for they had a muchsmaller resistance to shifts than to compression.
He also suggested that the optical ethermight correspond to the case of infi�ite resistance to compression, for which longitudinalwaves no longer exist. In summary, Stokes had both down-to-earth and ethereal reasons torequire two elastic constants instead of one. With George Green, whose works he praised,he inaugurated the British preference for the multi-constant theory.99Stokes's immediate purpose was, however, a study of the role of internal friction in fluidresistance and flow retardation.
Here boundary conditions are essential. When, in 1 845,Stokes read his memoir on fluid friction, he was already inclined to assume a vanishingrelative velocity at a rigid wall. He worried, however, about the resulting pipe retardationlaw, which contradicted Bossut's and Du Buat's results. Navier's and Poisson's conditionthat the tangential pressure at the wall should be proportional to the slip did not work anybetter, except for a very small velocity, in which case the measured retardation becameproportional to the velocity.
Girard's measurements, as interpreted by Navier, seemed torequire a finite slip in this case, although Du Buat had found a zero velocity near the wallsof a very reduced flow. In this perplexing situation, Stokes refrained from publishingdischarge calculations. He only gave the parabolic velocity profile for cylindrical pipeswith zero velocity at the walls. 1003.6.5Back to the pendulumIn the pendulum case Stokes knew the retardation to be proportional to velocity, inconformance with both the Navier-Poisson boundary condition and the zero-slip condition. He also knew, from a certain James South, that a tiny piece of gold leaf attachedperpendicularly to the surface of a pendulum's globe remained perpendicular duringoscillation.
This observation, together with Du Buat's and Coulomb's small-velocityresults, brought him to try the analytically simpler zero-slip condition. The success ofthis choice required justification. In his major memoir of 1 850 on the pendulum, Stokesargued that it was 'extremely improbable' that the forces called into play by an infinitesimal internal shear and by a fmite wall shear would be of the same order of magnitude, asthey should be for the dynamical equilibrium of the layer of fluid next to the wall. 101Neglecting the quadratic (v · \7)v terms in the Navier-Stokes equation, Stokes found anexact analytical solution for an oscillating sphere representing the globe of the pendulum,and a power-series solution for an oscillating cylinder representing the suspending threadof the pendulum.
The results explained Sabine's mass-correction anomaly, and permitteda close fit with Francis Baily's extensive experiments of 1832. Ironically, Stokes obtainedthis impressive agreement with a wrong value for the viscosity coefficient. The explanationof this oddity is that his data analysis depended on the assumption that viscosity is99Stokes [1849a] sects 3-4.100/bid. pp. [93-9]; Stokes [1846a] l86.
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