Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 30
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Computing dy/dx from the previous expressions for x and y, we obtaindX = -X tan a(2. 125)daThe integral X = a cos a of this equation then givesx= a cos a(2 - cos2 a),y=a sin a cos2 a,(2.126)which are the same as eqns (2. 1 19) with r = tan a.153The extreme simplicity of this derivation strikingly illustrates the transformation ofmathematical physics announced in the introduction to this chapter.
In 1775, Laplacealready knew the equations of hydrodynamics that are needed to formulate the ship-waveproblem mathematically. Had he dared to approach this problem, he would probably have1 5"This reasoning is from Lighthill [1957] pp. [21-2, [1978] pp. 269-79. See also Billingham and King [2000] pp.99-105. Thomson ([1887fl pp. 425-7) gives this geometrical construction of the characteristic angle, without thephysical interpretation.153The form X = a cos rx of the constant-phase condition also derives from </> = wt - kd (witht = 2X/V, d = Xcosrx, V cos a = wfk = .,fijk), which leads to </> = gXfV2 coso:.100WORLDS OF FLOWfallen into the same error as in the waves-by-emersion problem, for he did not know howto synthesize local perturbations from sinusoidal ones.
Some forty years later, Poisson andCauchy could have written the multiple integral that yields the water disturbance behindthe ship. But they lacked efficient means to evaluate this integral. Ninety years later,Thomson succeeded in this task thanks to 'the principle of interference'. Through therelated intuition of wave groups, he even suggested a way to circumvent the integral andreason in geometric terms.This story exemplifies a symbiotic evolution of mathematical analysis and physical interpretation in the nineteenth century. The need to solve the differential equations of physicsproblems such as the propagation of heat inspired new mathematical tools such as Fourieranalysis.
In turn, the application of these tools to a broad array of physical phenomenaprovided them with physical interpretations that suggested more efficient ways of handling them. From raw, algebraic procedures for combining and transforming mathematicalexpressions, they became genuine physico-mathematical tools. Whereas in their moreprimitive guise they often generated impenetrable integrals, in their mature form theyrevealed the behavior of the integrals.This evolution largely explains the success of nineteenth-century theorists in dealingwith complex wave patterns in the linear approximation. That Stokes, Boussinesq, andRayleigh could also solve an important class of nonlinear problems depended on anotherquality, namely, their ability to develop methods of approximation that combined twodifferent small parameters, the slope and the elevation of the waves.
In both cases, acentury elapsed between the basic formulation of the problem in Lagrange's memoir of1781 and a fairly complete mastery of the observed wave behaviors. Although this mayseem a long time, it is less than what was needed for a fragmentary answer to otherhydrodynamic questions.With hindsight, there are three peculiarities of water-wave motion that make it moreeasily amenable to mathematical analysis than other forms of fluid motion. Firstly, it canbe studied with reasonable accuracy without taking into account the small viscosity ofwater. Secondly, in the same approximation it can be regarded as irrotational (except forGerstner's waves) and therefore admits a harmonic velocity potential.
Thirdly, it is stableand non-turbulent, except in the limit of breaking waves. We will now leave this relativelysimple domain and enter more troubled waters.3VISCOSITYM. Navier himself only gives his starting principle as a hypothesis that can beverified solely by experiment. If, however, the ordinary formulas of hydrodynamics resist analysis so strongly, what should we expect from new, farmore complicated formulas?1 (Antoine Coumot, 1828)As far as I can see, there is today no reason not to regard the hydrodynamicequations [ofNavier and Stokes] as the exact expression of the laws that rule themotions of real fluids. 2 (Hermann Helmholtz, 1 873)In the early nineteenth century, the rational fluid mechanics of d'Alembert, Euler, andLagrange remained irrelevant to the mundane problems of pipe flow and ship resistance.Engineers had their own empirical formulas, and mathematicians their own paper theoryof perfectly unresisted flow.
A similar contrast existed in the case of elasticity: the formulasestablished by mathematicians for the flexion of prisms were oflittle help in evaluating thelimits of rupture in physical constructions. In the 1 820s and 1 830s, a new breed of Frenchengineer-mathematicians trained at the Ecole Polytechnique, mainly Navier, Cauchy, andSaint-Venant, struggled to fill this gap between theory and practice. As a preliminary steptoward a more realistic theory of elasticity, in 1 821 Navier announced the general equations of equilibrium and motion for an (isotropic, one-constant) elastic body. Transposinghis reasoning to fluids, he soon obtained a new hydrodynamic equation for viscous flow,namely the Navier-Stokes equation.Navier'slattertheoryreceivedlittlecontemporaryattention.
TheNavier-Stokes equationwas rediscovered or rederived at least four times, by Cauchy in 1 823, by Poisson in 1 829, bySaint-Venant in 1837, and by Stokes in 1 845. Each new discoverer either ignored or denigrated his predecessors' contribution. Each had his own way tojustify the equation, althoughthey all exploited the analogy between elasticity and viscous flow. Eachjudged differently thekind ofmotion and the nature ofthe system to which it applied. The comparison between thevarious derivations of this equation-or of the equations of motion of an elastic bodybrings forth important characteristics of mathematical physics in the period 1820-1 850.A basic methodological and ontological issue was the recourse to molecular reasoning.Historians have often perceived an opposition between Laplacian molecular physics onthe one hand, and macroscopic continuum physics on the other, with Poisson being thechampion of the former physics, and Fourier the champion of the latter.
Closer studies ofFourier's heat theory have shown that the opposition pertains more to the British readingof this work than to its actual content. Fourier actually combined molecular intuitions1 Cournot [1 828] p. 13.2Helmholtz [1 873] p. 158.102WORLDS OF FLOWwith more phenomenological reasoning. Viscous-fluid and elastic-body theorists similarlyhybridized molecular and continuum physics. Be they engineers or mathematicians, theyall agreed that the properties of real, concrete bodies required the existence of noncontiguous molecules. However, they differed considerably over the extent to whichtheir derivations materially involved molecular assumptions.At one extreme was Poisson, who insisted on the necessity of discrete sums overmolecules. At the other extreme was Cauchy, who combined infmitesimal geometry andspatial symmetry arguments to defme strains and stresses and to derive equations ofmotion without referring to molecules.
Yet the opposition was not radical. Poisson reliedon Cauchy's stress concept, and Cauchy eventually provided his own molecular derivations. Others compromised between the molecular and the molar approach. Navierstarted with molecular forces, but quickly jumped to the macroscopic lev!!l by consideringvirtual works. Saint-Venant insisted that a clear definition of the concept of stress couldonly be molecular, but nevertheless provided a purely macroscopic derivation of theNavier-Stokes equation. Stokes obtained the general form of the stresses in a fluid by aCauchy type of argument, but he justified the linearity of the stresses with respect todeformations by reasoning on hard-sphere molecules.These methodological differences largely explain why Navier's successors ignoredor criticized his derivation of the Navier--Stokes equation.
His short cuts from the molecular to the macroscopic levels seemed arbitrary or even contradictory. Cauchy and Poissonsimply ignored Navier's contribution to fluid dynamics. Saint-Venant and Stokes bothgave credit to Navier for the equation, but believed an alternative derivation to be necessary. To this day, Navier's contribution has been constantly belittled, even though hisapproach was far more consistent than a superficial reading may suggest.This wide spectrum of methodological attitudes, both in fluid mechanics and in elasticity theory, corresponds to different views of mathematical rigor and different degrees ofconcern with engineering problems. Navier's way of injecting physical intuition intomathematical derivations was alien to Cauchy and Poisson, who were the least involvedin engineering and the most versed in higher mathematics.
Yet many engineers judgedNavier's approach too mathematical and too idealized. Personal ambitions and prioritycontroversies enhanced, and at times even determined, the disagreements. Acutely awareof these tensions, Saint-Venant developed innovative strategies that combined the demands of mathematical rigor and practical usefulness.The many fathers of the Navier--Stokes equation also differed in the types of applicationthey envisioned. Navier and Saint-Venant had pipe and channel flow in mihd.
Cauchy'sand Poisson's interests were more philosophical than practical. Cauchy did not evenintend the equation to be applied to real fluids; he derived it for 'perfectly inelastic solids',and noted its identity with Fourier's heat equation in the limiting case of slow motion.Stokes was motivated by British geodesic measurements that required aerodynamic corrections to pendulum oscillations.To Navier's disappointment, his equation worked well only for slow, regular motions,as occurs around pendulums and within capillary tubes. In most hydraulic cases, thereseemed to be no alternative to the empirical approach of engineers.
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