Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 65
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Accordingly, hecontented himself with proving that, for any nonzero value of the viscosity parameterand for any values of u, k, and m, convergent power-series expansions could be found forthe y dependence of the Fourier components. From this result and from the real characterof the frequencies u, he concluded that the plane Poiseuille flow was also stable. 88Lastly, Thomson dealt with the practical instability of pipe flow. In conformance withReynolds's observation that the growth of perturbations in this case depended on theirsize, he proposed that the flow was probably stable for infinitesimal perturbations (as hethought it was in two dimensions) but unstable for finite ones.
It would be so, he argued, ifthe inviscid flow with Poiseuille velocity profile was unstable, and if viscosity could onlydamp sufficiently-small perturbations. The margin of stability would then increase forhigher viscosity, as Reynolds had observed.89The instability of inviscid flow with a parabolic velocity profile clearly contradicted Rayleigh's inflection theorem. Thomson believed, however, that a 'disturbing infinity vitiate[d)[Rayleigh's] seeming proof of stability.' As Rayleigh himself noted, the stability equation( U _ I!_) (fflv -f<?-v) - d2 U v = 0k[)y2dy2(5.20)becomes singular for values of the coordinate y for which the velocity ujk of the planewave perturbation is identical to the velocity U(y) of the unperturbed flow (and U" doesnot simultaneously vanish).
At such a point, the flow is obtained by superposing a sinewave velocity pattern with a shearing motion. For an observer moving along the fluid, theflow has the 'eat's eye' outlook of Fig. 5.13, which Thomson published in 1880.90From then on, Thomson attached great importance to the disturbing infinity: 'The"awkward infinity" ', he wrote to George Darwin in August 1 880, 'threatens quite arevolution in vortex motion (in fact a revolution where nothing of the kind, nothing butthe laminar rotational movement, was even suspected before), and has been very bewildering.' Thomson believed the elliptic whirls of this flow to be the source of the turbulenceobserved by Reynolds.
Any simple perturbation of the fluid boundary necessarily contained Fourier components for which elliptic whirling would disturb the laminar flow. 91Fig. 5.13.88Thomson [1887d].Thomsen's 'eat's eye' flow pattern. From Thomson [1880c] p. 187.89Ibid. p. 335.90Thomson [1887d] p. 334; Rayleigh [1880] p. 486; Thomson [1 880c].9 1 Thomson to Darwin, 22 Aug. 1880, in Thompson [1910] p. 760. Thomson does not address the question ofthe growth of the whirls.2145.5.4WORLDS OF FLOWChallenging ThomsonRayleigh valiantly defended his stability criterion against Thomson's 'disturbing infinity':Perhaps I went too far in asserting that the motion was thoroughly stable; but it is tobe observed that if [the frequency a-] be complex, there is no 'disturbing infinity'.
Theargument, therefore, does not fail regarded as one for excluding complex values of[a-]. What happens when [a-] has a real value such that [a- - kU] vanishes at an interiorpoint, is a subject for further examination.Equation (5.20) is indeed non-singular for a complex value of a-, so that an exponentialincrease of infinitesimal perturbations and a constant sign of U" are truly incompatible.Rayleigh conceded, however, that the impossibility of an exponential increase did notrigorously establish stability.
Perhaps a less rapid increase of perturbations was stillpossible owing to the 'disturbing infinity'. Perhaps higher-order terms in the stabilityequation implied a departure from the first-order behavior. In sequels to his 1 880 study,Rayleigh provided arguments that made these escapes implausible. Modem writers onhydrodynamic stability no longer question the validity of his stability criterion.92In return to Thomson's criticism of his criterion, Rayleigh politely questioned Thornson's proofs of stability of plane viscous flow:Naturally, it is with diffidence that I hesitate to follow so great an authority, butI must confess that the argument does not appear to me demonstrative.
No attempt ismade to determine whether in free 'disturbances of the type [eiur] the imaginary partof [a-] is finite, and if so whether it is positive or negative.' If I rightly understand it,the process consists in an investigation of forced vibrations of arbitrary (real)frequency, and the conclusion depends on the tacit assumption that if these forcedvibrations can be 'expressed in periodic form, the steady motion from which they aredeviations cannot be unstable.'Rayleigh went on to show that the tacit assumption was wrong in the case of a (rigid)pendulum situated near the highest point of its orbit.
Whether he correctly interpretedThomson's intentions is questionable. He was right, however, to judge Thomson's reasoning incomplete.93The Irish mathematician William Orr clearly identified the gaps in 1907. Consider firstThomson's proof of stability of plane Poiseuille flow. This proof assumes that a superposition ofharmonic solutions (with respect to t, x, andz) that satisfies the boundary conditionsis sufficient to reproduce any initial value of the velocity perturbation. This does not need tobe tme, because the boundary conditions might restrict the harmonic solutions too much.Thomson's proofalso fails in the case ofplane Couette flow.
The forced solution in this proofdoes not need to vanish for t = 0, even though it is forced to vanish on the boundaries of thefluid for any negative time. Indeed, the boundary conditions completely determinethe Fourier-type solution, thus leaving no room for a further restriction ofthe initial motion.Consequently, the complete solution may not have the requested initial value.9492Rayleigh [1892] p. 380, [1887], [1895].
Cf. Drazin and Reid [1981] pp. 126-47.93Rayleigh (1892] p. 582. Yet, in 1895 Rayleigh (unwisely) endorsed Thomson's 'special solution' for disturbances of the plane Couette flow.940rr (1907]. This paper also contains an unconvincing interpretation of Rayleigh's criticism of 1 892.INSTABILITY5.5.5215Rayleigh 's paradoxThomson himself had become aware of the weakness of his reasoning, as appears in aletter he wrote to Stokes in December 1 898: 'Several papers of mine in Phi!. Mag. about1887 touch inconclusively on this question [of the stabilizing effect of viscosity].' Yet hestill believed that the instability observed by Reynolds depended on the instability of theparabolic velocity profile at zero viscosity.
In contrast, Rayleigh never really doubted thetruth of his inflection theorem, which forbade this sort of instability. This led him toenunciate the basic paradox of pipe flow:If[my criterion] is applied to a fluid of infinitely small viscosity, how are we to explainthe observed instability which occurs with moderate viscosities? It seems very unlikelythat the first effect of increasing viscosity should be to introduce an instability notpreviously existent, while, as observation shows, a large viscosity makes for stability.Rayleigh offered a few suggestions to explain this discrepancy.
Firstly, irregularities of thewall surface could play a role. Secondly, instability could occur forfinite disturbances evenwhen the Rayleigh criterion gave stability. Thirdly, the three-dimensional case of Reynolds's experiments could qualitatively differ from the two-dimensional case studied byRayleigh and Thomson. Fourthly, Rayleigh wrote, 'it is possible that, after all, theinvestigation in which viscosity is altogether ignored is inapplicable to the limiting caseof a viscous fluid when the viscosity is supposed infinitely small.
'95The main purpose of Rayleigh's paper was to exclude the third possibility by extendinghis stability criterion to cylindrically-symmetric flow. In retrospect, his short comments onthe fourth conjecture are most interesting: 96There is more to be said in favour of this view than would at first be supposed. In thecalculated motion there is a finite slip at the walls [when viscosity is ignored], and thisis inconsistent with even the smallest viscosity. And further, there are kindredproblems relating to the behaviour of a viscous fluid in contact with fluid walls forwhich it can actually be proved that certain features of the motion which could notenter into the solution, were the viscosity ignored from the first, are neverthelessindependent of the magnitude of viscosity, and therefore not to be eliminated bysupposing the viscosity to be infinitely small.Rayleigh had in mind the explanation he had given in 1883 of an acoustic anomalydiscovered by Savart in 1820 and studied by Faraday in 1831, namely that, when a platesprayed with light powder is set into vibration, the powder gathers at the antinodes of themotion, whereas Chladni's earlier experiments with sand gave the expected nodal figures.Faraday traced this anomaly to the action of currents of air, rising from the plate at theantinodes, and falling back at the nodes.
97In his confirming calculation, Rayleigh assumed a plane monochromatic standing wavefor the motion of the plate and solved the Navier-Stokes equation for the fluid motionabove the plate perturbatively, taking the nonlinear (v · 'V)v tenn as the perturbation. The95Thomson to Stokes, 27 Dec. 1898; Rayleigh [1 892] pp.
576-7.96Rayleigh [1 892] p. 577.97Cf. Rayleigh [1 883a] pp. 239-40; [1 896] vol. I, pp. 367-<l. Rayleigh also explained the air currents observedby Vincenz Dvofak in 1876 in Kundt's tubes.WORLDS OF FLOW216l � T � 1 '==: T� 7i"0nodeFig. 5.14.loop1fnode�?floopMotion of air near a vibrating plate. From Rayleigh [1883a] p. 250.resulting motion is confined near to the plate in a layer of thickness (v/!) 1 12, where v is thekinematic viscosity andfis the frequency of the oscillations.
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