Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 41
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8683Saint-Venant to Savary, 13 Jan. 1 837, ibid.; Saint-Venant [1834] new version (later than 1837) of sect. 1 5;Saint-Venant [1843c].84Saint-Venant [1 843c] p. 1243 for variable e. Ibid. p. 1242n, Saint-Venant noted that Cauchy's pressuretheorems were valid to second order in the dimensions of the volume elements, 'which allows us to extend theirapplication to the case when partial irregularities of the fluid motion forces us to take faces of a certain extensionso as to have regularly varying averages.'85Saint-Venant [1843c]; Saint-Venant to Savary (ref. 82), 27 July 1834 (on Du Buat); Saint-Venant [1834] newsect.
15: 'It is experiment that should determine whether e is constant or variable.' Perhaps Saint-Venant did notbelieve in a constant e, even at the small scale, because, for the tumultuous flows observed in rivers channels andoccurring in pipes of not too small diameter, Saint-Venant believed that any irregularity of motion cascaded to asmaller and smaHer scale by 'molecular gearing'.86Boileau [1847], [1854].
Saint-Venant to Boileau, 29 Mar. 1846, Fond Saint-Venant, reproduced and discussedin Melucci [1996] pp. 65-71.WORLDS OF FLOW134Fig. 3.6.3.5.7Drawing for Saint-Venant's proof of d'Aiembert's paradox (from Saint-Venant [1887b] p. 50).Fluid resistanceIn 1 846, Saint-Venant tackled the old, difficult problem offluid resistance. He first showedthat the introduction of internal friction solved d' Alembert's paradox.
For this purpose,he borrowed from Du Buat and Poncelet the idea of placing the immersed body inside acylindrical pipe (see Fig. 3.6), from Euler the balance of momentum, and from Borda thebalance oflive forces. If the body is sufficiently far from the walls of the pipe, the action ofthe fluid on the body should be the same as for an unlimited flow.
If the body is fixed, theflow is steady, and the fluid is incompressible, then the momentum which the fluid conveysto the body in unit time is equal to the difference P0 S - P1 S between the pressures on thefaces of a column of fluid extending far before and after the body, because the momentumof the fluid column remains unchanged.
For an ideal fluid, the work (PoS - P1 S)vo ofthese pressures in unit time must vanish, because the live force of the fluid column is alsounchanged. Hence the two pressures are equal, and the fluid resistance vanishes. This is7d'Alembert's paradox, as proved by Saint-Venant. 8In a molecular fluid, the (negative) work of internal friction must be added to the workof the pressures Po and P�, or, equivalently, the live force ofnon-translatory motions mustbe taken into account. Hence the pressure falls when the fluid passes the body, and theresistance no longer vanishes. The larger the amount of non-translatory motion inducedby the body, then the higher is the resistance. When tumultuous, whirling motion occurs atthe rear of the body, the resistance largely exceeds the value it would have for a perfectlysmooth flow.
After drawing these conclusions, Saint-Venant improved· on a methodinvented by Poncelet to estimate the magnitude of the resistance and based on theassumption that the pressure P1 at the rear of the body does not differ much from thevalue that Bernoulli's law gives in the most contracted section of the flow (see Fig. 3.7). 88In summary, Saint-Venant did not accept the dichotomy between a hydrodynamicequation for ideally smooth flow on the one hand, and completely empirical retardationand resistance formulas for hydraulic engineers on the other. He sought a via media that87Saint-Venant [1846b], [!887b] pp.
45-9. In Borda [1766] p. 605, the Chevalier de Borda had derived theparadox in an even simpler manner, by applying the conservation oflive forces to a body pulled uniformly througha calm fluid. For a modern derivation, see Appendix A.88Saint-Venant [1846b] pp. 28, 72-8, 120--1 , [1887b] pp. 56-192.VISCOSITYFig. 3. 7.p.135Drawing for Poncclet's and Saint-Venant's evaluation of fluid resistance (from Saint-Venant [1887b]89).would bring theoretical constraints to bear on practical flows and yet would allow for.some experimental input. One of his strategies, later pursued by Joseph Boussinesq andsuccessfully applied to turbulent flow to this day, consisted of reinterpreting Navier'shydrodynamic equation as controlling the average, smoothed-out flow with a variableviscosity coefficient.
Another was the astute combination of momentum and energybalances with empirically-known features of the investigated flow. For hydraulics, as forelasticity, Saint-Venant was a most persevering and imaginative conciliator of fundamental and practical aims.3.6 Stokes: the pendulum3.6.1A swimming mathematicianUntil the 1 830s at least, the production of advanced mathematical physics in an engineering context remained a uniquely French phenomenon, largely depending on the creationof the Ecole Polytechnique.
The main British contributors to elasticity theory and hydrodynamics in this period had little or no connection with engineering. Typically, they wereastronomers like Airy and Challis, or mathematicians like Green and Kelland. Their workon elasticity was subordinate to their interest in the new wave optics, and the aspectsof hydrodynamics that captured their attention tended to be wave and tide theory.A Cambridge-trained mathematician, and the first Wrangler and Smith prize winner(1841), George Gabriel Stokes was not much closer to the world of engineers.
He nonetheless was a keen observer of nature, a first-rate swimmer, and a naturally giftedexperimenter. He was quick to note the gaps between idealized theories and real processes,and sometimes eager to fill them.89During the two decades preceding Stokes's student years, British mathematical physicshad undergone deep reforms that eliminated archaic Newtonian methods in favor of thenewer French ones. While Fourier's theory of heat and Fresnel's theory of light were most89Cf. Stokes [1846a], Parkinson [1976], Wilson [1987].WORLDS OF FLOW136admired for their daring novelty, the hydrodynamics of Euler and Lagrange provided thesimplest illustration of the necessary mathematics of partial differential equations.
Thefamous Cambridge coach William Hopkins made it a basic part of the Tripos examination, and persuaded Stokes to choose it as his first research topic. In his first papers,published in the early 1 840s, Stokes already noted discrepancies between real and idealflows and suggested a few remedies, including the introduction of viscosity.903.6.2The pendulumStokes's interest in imperfect fluidity derived from the pendulum experiments performedby Edward Sabine in 1 829. This artillery officer had led a number of geodesic projects, oneof which, in 1821122, dealt with the pendulum determination of the figure of the Earth. In1 828, the German astronomer Friedrich Bessel published a memoir on the seconds'pendulum that brought pendulum studies, and quantitative experiment in general, to anunprecedented level of sophistication. Bessel not only improved experimental proceduresand data analysis, but he also brought new theoretical insights into the various effects thataltered the ideal pendulum motion.
Most importantly, he was the first to take into accountthe inertia of the air moved by the pendulum. His study played a paradigmatic role indefining a Konigsberg style of physics. It also induced further experimental and theoreticalpendulum studies in Britain and France.91While investigating Bessel's inertial effect, Captain Sabine found that the mass correction of a pendulum oscillating in hydrogen was much higher than the density ratio betweenhydrogen and air would imply. Sabine suggested that gas viscosity could be responsible forthis anomaly. The remark prompted Stokes to study the way viscosity affected fluidmotion. His first strategy, implemented in a memoir of 1 843, was to study special casesof perfect-fluid motion in order to appreciate departures from reality:92The only way by which to estimate the extent to which the imperfect fluidity of fluidsmay modify the laws of their motion, without making any hypothesis on the molecular constitution of fluids, appears to be, to calculate according to the hypothesisof perfect fluidity some cases of fluid motion, which are of such a nature as to becapable of being accurately compared with experiment.900n the transformation of British physics, cf.
Smith and Wise [1989] chap. 6. On Hopkins's role, cf. Wilson[1987] p. 132. Henry Moseley's hydrodynamic treatise [1830], written for the students of Cambridge Universityunder Challis's advice, marked a transition between older Newtonian methods and Euler's hydrodynamics: it onlyintroduced the fundamental equations (in integral form) at a very late stage, and based most reasoning on preEulerian techniques such as Bernoulli's Jaw or d'Alembert's principle; it gave a Newtonian treatment of fluidresistance, ignored d'Alembert's paradox, and failed to mention Navier's equations of fluid motion.910n Stokes and pendulums, cf. Stokes [1850b] pp.
1-7. On Sabine, cf. Reingold [1975] pp. 49-53. On Bessel'swork, cf. Olesko [1991] pp. 67-73. On pendulum studies in general, cf. Wolf [1889]. Bessel's inertial effect wasalready known to Du Buat [1786] vol. 3, in a hydraulic contexi. Poisson [1832] gave the theoretical value of theinertial mass correction for a sphere as half of the mass of the displaced fluid, in conformance with Du Buat's resultfor water.92Sabine [1829], commentary to his eighth experiment; Stokes [1850b] p. 2 (Sabine); Stokes [1843] pp.
17-18(quote). Stokes assumed that the motion started from rest, which implies the existence of a velocity potential for aperfect liquid. Stokes hoped that this property would still hold approximately for the small oscillations of a realfluid (ibid. p. 30; this turned out to be wrong in the pendulum case).137VISCOSITYAmong his cases of motion Stokes included oscillating spheres and cylinders that couldrepresent the bulb and the suspending thread of a pendulum. In the spherical case, hefound the mass correction to be equal to half the mass ofthe fluid expelled by the sphere, inconformance with a calculation made by Poisson in 1831, but only five-ninths of Bessel'sexperimental result of 1828.
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