Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 31
Текст из файла (страница 31)
It was not even clearwhether the Navier-Stokes equation could be maintained. Many years elapsed before thisequation acquired the fundamental status that we now ascribe to it.VISCOSITY103The first section ofthis chapter is devoted to the hydraulic failure ofEuler's hydrodynamics, and to Girard's study of flow in capillary tubes, on which Navier relied. Section3.2describes Navier's achievements in the theory of elasticity, their transposition to fluids, andthe application to Girard's tubes.
Section3.3 discusses Cauchy's stress-strain approach to3.4 recounts Poisson'selasticity and its adaptation to a 'perfectly inelastic solid'. Sectionstruggle for rigor in the molecular approach, Cauchy's own implementation of the sameapproach, and Navier's response to Poisson's attacks. Section3.5 concerns Saint-Venant'sunique brand of applied mechanics, and his contributions to elasticity and hydraulics. The3.6, deals with Hagen's and Poiseuille's experiments on narrow-pipeFinal Section, Sectiondischarge, and their long-delayed explanation by the Navier-8tokes equation.3.1 Mathematicians' versus engineers' fluids3.1.1 Resistance and retardationAs we saw in Chapter1 , d'Alembert regarded the vanishing of fluid resistance in his theoryas a challenge for future geometers.
As a possible clue to this paradox, he evoked anasymmetry of the fluid motion (around a rear-front symmetric body) owing to 'thetenacity and the adherence of the fluid particles'. However, he did not try to formalizethis effect, presumably because he regarded the molecular interactions as too complex toyield well-defined mathematical laws at the macroscopic level. 3Euler similarly predicted zero resistance to the motion of an arbitrarily-shaped body,even before he had the fundamental equations of fluid motion.
He reasoned through aninspired, though non-rigorous, use of momentum conservation. Roughly speaking, themomentum gained by the immersed body (whatever its shape may be) in a unit of timeshould be equal to the difference of momentum fluxes across normal plane surfacessituated far ahead and far behind the body; this difference vanishes because ofthe equalityof velocity and mass flux on the two surfaces.4Euler knew of no better escape from this paradox than a partial return to EdmeMariotte's and Isaac Newton's old theories of fluid resistance.
According to these pioneersof fluid mechanics, the impact of fluid particles on the front of the immersed bodycompletely determined the resistance. Similarly, Euler cut off the rear part of the tubes offlow to which he applied his momentum balance. The true form ofthe flow and the shape ofthe rear of the body did not matter in such theories. Although their experimental inexactitude and theirad hoccharacter were already recognized in Euler's day, they remainedpopular until the beginning of the nineteenth century for the lack of any better theory.53D'Alembert [1780] p. 2 1 1 .
Cf. Saint-Venant [1887b] p. 10. The fluid resistance data used in 1 877 by the·Academic Commission for the Picardie Canal, to whlch d,Alembert belonged, were purely empirical, cf. Redondi[1997].4Euler [1745] chap. 2, prop. I, rem. 3 (French transl. pp. 3 1 6-17). Cf.
Saint-Venant [1887b] pp. 29-31 . A morerigorous reasoning would have required a cylindrical wall to limit the flow laterally, together with a proof that theworks of pressure forces on the two plane faces of the cylinder are equal and opposite. Thls cancellation resultsfrom the equality of pressures on the two faces, whlch itself derives from Bernoulli's theorem or from theconservation of live force. Compare with Saint-Venant's proof of 1 837, discussed on p. 1 34.5Cf. Saint-Venant [1887b] pp. 34--6.
On Newtons' theory, cf. ibid. pp. 15-29, G. Smith [1998], Chapter 7,pp. 265-6.WORLDS OF FLOW104The great geometers of the eighteenth century were even less concerned with thehydraulic problems of pipe and channel flow than with fluid resistance. Available knowledge in this field was mostly empirical. Since Mariotte's Traite du mouvement des eaux(1686), hydraulic engineers assumed a friction between running water and walls, proportional to the wetted perimeter, and increasing faster than the velocity of the water. Thisvelocity was taken to be roughly uniform in a given cross-section of the pipe or channel, inconformance with common observation. Claude Couplet, the engineer who designed theelaborate water system of the Versailles castle, performed the first measurements of theloss of head in long pipes of various sections.
Some fifty years later, Charles Bossut, aJesuit who taught mathematics at the engineering school of Mezieres, performed moreprecise and extensive measurements of the same kind. 6So did his contemporary Pierre Du Buat, an engineer with much experience in canal andharbor development, and the author of a very influential hydraulic treatise. Du Buat'ssuperiority rested on a sound mechanical interpretation of his measurements.
He was thefirst, in print, to give the condition for steady flow by balancing the pressure gradient (inthe case of a horizontal pipe) or the paraiiel component of fluid weight (in the case of anopen channel) with the retarding frictional force. He took into account the loss of head atthe entrance of pipes (due to the sudden increase in velocity), whose neglect had flawed hispredecessors' results for short pipes. Lastly, he proved that fluid friction, unlike solidfriction, did not depend on pressure. 7Bossut found the retarding force to be proportional to the square of the velocity, and DuBuat found it to increase somewhat slower than that with velocity. Until the mid-nineteenthcentury, German and French retardation formulas were usuaily based on the data accumulated by Couplet, Bossut, and Du Buat. In 1804, the Directeur of the Ecole des Pants etChaussees, Gaspard de Prony, provided the most popular formula, which made the frictionproportional to the sum of a quadratic and a smaii linear term.
The inspiration for this formcame from Coulomb's study of fluid coherence, to be discussed shortly. 83 . 1 .2Fluid coherenceFor Du Buat's predecessors, the relevant friction occurred between the fluid and the wailsof the tube or channel. In contrast, Du Buat mentioned that viscosity was needed to checkthe acceleration of internal fluid filaments. He observed that the average fluid velocityused in the retardation formulas was only imaginary, that the real flow velocity increasedwith the distance from the wails, and even vanished at the wails in the case of a very smailflux.
The molecular mechanism he suggested for the resistance implied the•adherence offluid molecules to the wails, so that the retardation truly depended on internal fluidprocesses. Specificaily, Du Buat imagined that the adhering fluid layer impeded themotion of the rest of the fluid, partly as a consequence of molecular cohesion, and mostly6Mariotte [1686] part 5, discourse I. Cf. Saint-Venaut [1887b] pp.
39-40, Rouse and Ince [1957] pp. 1 14(Couplet), 126-1l (Bossut).7Du Buat [1786], vol. 1, pp. xvii, 14-15, 40. Cf. Saint-Venant [1866], Rouse aud Ince [1957] pp. 129-34. In 1 775,Antoine Chezy had already given the condition of steady motion in an unpublished report for the Yvette Caual (cf.ibid. pp. 1 1 7-20). More will be said on Bossut and Du Buat in Chapter 6, pp. 221-2.8Cf. Rouse and Ince [1957] pp.
141-43. In 1803, Girard had used a non-homogenous v + if formula, alsoinspired by Coulomb.105VISCOSITYbecause of the granular structure of this layer. This structure implied a 'gearing' oftraveling-along molecules(engrenage des molecules),through which they lost a fractionof their momentum proportional to their average velocity, at a rate itself proportional tothis velocity.
Whence came the quadratic behavior of the resistance.9In1 800, the military engineer Charles Coulomb used his celebrated torsional-balancetechnique to study the 'coherence of fluids and the laws of their resistance in very slowmotion'. The experiments consisted of measuring the damping of the torsional oscillationsof a disk suspended by a wire through its center and immersed in various fluids.
Theirinterpretation depended on Coulomb's intuition that the coherence of fluid moleculesimplied a friction proportional to the velocity, and that surface irregularities implied aninertial retardation proportional to the square of the velocity. In conformance with thisview, Coulomb found that the quadratic component depended only on density and thatthe total friction became linear for small velocities. From his further observation thatgreasing or sanding the disk did not alter the linear component, he concluded:1 0The part of the resistance which we found to be proportional to the velocity is due tothe mutual adherence of the molecules, not to the adherence of these molecules withthe surface of the body.
Indeed, whatever be the nature of the plane, it is strewn withan infinite number of irregularities wherein fluid molecules take permanent residence.Although Du Buat's and Coulomb's emphasis on internal fluid friction or viscosity wasexceptional in their day, the notion was far from new. Newton had made it the cause of thevortices induced by the rotation of an immersed cylinder, and he had even provided aderivation (later considered to be flawed) of the velocity field around the cylinder. Heassumed (in conformance with later views) that the friction between two consecutive,coaxial layers of the fluid was proportional to their velocity difference. After a centuryduring which this issue was virtually ignored, in1799 the Italian hydraulic engineerGiovanni Battista Venturi offered experiments that displayed important effects of internalfluid friction.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
