Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 4
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If the fluid isinstead regarded as a true continuum, then an unwarranted extension of the law toinfinitesimal elements of mass is needed.To one who has these difficulties in mind, the canonical derivation of Euler's equationsseems largely illusory. It rests on axioms that need further justification, possibly through akinetic-molecular theory of matter, or through the empirical success of their consequences. Accordingly, one should expect the historical genesis of Euler's equations tohave been a difficult, roundabout process. Indeed, seventeen years elapsed between DanielBernoulli's first attempt at applying a general dynamic principle to fluid motion andLeonhard Euler's strikingly modern derivation of the equations named after him.Another fact makes the history of early hydrodynamics even more intricate and interesting: the basic physico-mathematical tools of the modern derivation ofEuler's equationswere not originally available.
In the early eighteenth century, there was no concept of adimensional quantity, no practice of writing vector equations (even in the so-caiiedCartesian form), no concept of a velocity field, and no calculus of partial differentialequations. The idea of founding a domain of physics on a system of general equationsrather than on a system of general principles expressed in words did not exist.Any historical investigation of the origins of hydrodynamics requires atabula rasa ofquite a few familiar notions of today's physicist. The purpose of the present chapter is toshow how these notions graduaily emerged together with modern hydrodynamics; it is notto determine who the main founder of this new science was. Excessive concern withpriority questions leads to misinterpretations of the goals and concepts of the actors,and it harbors the myth of sudden, individual discovery.
In contrast, the present chapterdescribes a long, multifaceted process in which fluid motion was graduaily subjected togeneral dynamics, with a concomitant evolution of the principles of dynamics and withZextensions of the classes of investigated flows.The first attempt at applying a general dynamical principle to fluid motion occurred inDaniel Bernoulli'sHydrodynamica of I 738. The principle was the conservation of live forces,expressed in terms ofHuygens' pendulum paradigm. The main problem of fluid motion wasefflux, or the parailel-slice flow through an opening on a vessel.
The result was a geometricalexpression of laws that implicitly contained the one-dimensional version of Euler's equations. This approach did not require the concept of internal pressure. Daniel Bernoullinonetheless extended the concept of wall pressure to moving fluids, and derived 'Bernoulli'slaw' for this pressure. These inaugural achievements are described in Section 1. I.'SectioninI 742.I .2 is devoted to the Hydraulica that Daniel's father Johann Bernouiii publishedThe standard problem was stiii parailel-slice efflux, approached through the2In the history found in his Mechanique analitique [1788] pp. 436-7, Lagrange made d'Alembert the founder ofhydrodynamics.
He did not even mention Euler's name, although he no doubt appreciated his contributions (thesecond edition([18 1 1 115] vol. 2, p. 271) has the sentence:'It is to Euler that we owe the ftrst general formulas forthe motion of fluids, founded on the laws of their equilibrium, and presented with the simple and luminousnotation of partial differentials.' In disagreement, Truesdell([1954] p.CJOtvn) writes: 'It seems that much of whatd'Alembert is commonly credited with having done is taken from the simple and clear attributions of Lagrange,for I have searched for it in vain in d' Alembert's own works.' GrimbergTruesdell's reading of d'Alembert.[1998] has identified gaps and flaws in3THE DYNAMICAL EQUATIONSpendulum analogy.
The dynamical principle was now Newton's second law, together witha rule for replacing the gravities and the accelerations of the various parts of the systemwith equivalent gravities and accelerations acting on one part only. Again, this method didnot require the concept of internal pressure. Johann Bernoulli nonetheless defmed theinternal pressure as some sort of contact force between successive slices of fluid, and gaveits value in a generalization of Bernoulli's law to non-permanent flow. A key point of hissuccess was his awareness of two contributions to the acceleration of a fluid slice: thevelocity variation per time unit at a given height (our 8vtime unit due to the change of section (ourv8vI8z, zI8t);and the velocity variation perbeing a coordinate in the direction ofparallel motion).
His style was more algebraic than his son's, with recourse to dimensionalquantities including the acceleration g of gravity. But his reliance on partial differentialswas only implicit.Section1.3 is devoted to the contributions of Jean le Rond d'Alembert. In 1743/44, theFrench philosopher and geometer rederived the results of the Bernoullis by means of a newprinciple of dynamics, according to which a moving system must be in equilibrium withrespect to fictitious forces obtained by subtracting from the real (external) forces acting onthe parts of the system the product of their mass and their acceleration.
In the particularcase of parallel flow, or for the oscillations of a compound pendulum, this method isequivalent to those of the Bernoullis. D' Alembert, however, was innovative in explicatingthe partial differentials in the expression of the fluid acceleration. Most importantly, hismethod enabled him to consider two-dimensional flows (with infinitely many degrees offreedom), whereas the Bernoullis were confined to flows with only one degree of freedom.3D'Alembert achieved this tremendous generalization in his memoir on winds ofand in his memoir on fluid resistance of17471749.
There he obtained particular cases ofEuler'sequations, for the two-dimensional or axially-symmetric flow of an incompressible fluid.More precisely, his equations were those we would now obtain by eliminating the pressurefromEuler's equations (vorticity equation). The reason for this peculiarity is that d' Alembert's principle essentially short-cuts the introduction of internal contact forces such aspressures. D' Alembert nevertheless had a concept of internal pressure, which he used in hisexpression of Bernoulli's law. In his memoir on winds he even indicated an alternativeroute to the equations of fluid motion, by balancing the pressure gradient, the gravity, andthe inertial force at any point of the fluid.Soon after studying d'Aiembert's memoirs, Leonhard Euler showed how to derivecompletely general equations of fluid motion through a similar method, by applyingNewton's second law to each fluid element and taking into account the pressure fromthe surrounding fluid.
This achievement is described in Section1.4. The clarity andmodernity ofEuler's approach has lent itself to the myth of a sudden emergence ofEuler'shydrodynamics in Euler's magic hands. In reality, Euler struggled for many years todevelop a satisfactory theory of fluid motion. He only reached his aim after integratingdecisive contributions by Johann Bernoulli and by d' Alembert. His famous memoir of1755 did not reflect sudden, isolated inspiration.43New insights into d'A1embert's fluid dynamics and mathematical methods are found in Grimberg [1998](thesis directed by Michel Paty)."Truesdell [1954] has a detailed, competent analysis ofEuler's memoirs on fluid mechanics.4WORLDS OF FLOWNor should Euler's memoir be regarded as the last word on the foundations of perfectfluid mechanics.
As explained in Section1.5, the last section in this chapter, Lagrangeoffered an alternative foundation to Euler's equations, based on the general principles ofhis analytical mechanics. He also specified the boundary conditions, without which Euler'sequations would largely remain an empty formal scheme; he obtained a fundamentaltheorem about the existence of a velocity potential; and he gave a general method ofapproximation for solving the equations of narrow flows. Due to these advances, he couldprove the approximate validity of the old hypothesis of parallel-slice motion and solve theproblem of small waves on shallow water. Together with Euler's fundamental memoirs,these brilliant results were the starting-point for most of later hydrodynamics.1.1 Daniel Bernoulli's Hydrodynamica1738 the Swiss physician and geometer Daniel Bemoulli published his Hydrodynamica,sive de viribus et motibus jluidorum commentarii (hydrodynamics, a dissertation on theforces and motions of fluids).
He coined the word hydrodynamica to announce a new,Inunified approach of hydrostatics and hydraulics. Although he did not create the modemscience of fluid motion, his treatise marks a crucial transition: with novel and uniformmethods, it solved problems that belonged to a long-established tradition.5Since Greek and Roman hydraulics, an important problem of fluid motion was the flowof water from a vessel through an opening or a short pipe.
The Renaissance and theseventeenth century saw the first experimental studies of this problem, as well as the firstattempts to subject it to the laws of mechanics. Other topics of practical interest were theworking of hydraulic machines and waterwheels, and ship resistance. Topics of philosophical interest were the elasticity of gases and Cartesian vortices. TheHydrodynamicacovered all these subjects, except fluid resistance, which Bemoulli probably judged to bebeyond the grasp of contemporary mathematics. His newest results concerned efflux. Healso introduced the concept of work(vis absoluta)inaugurated the kinetic theory of elastic fluids. 61.1.1done by hydraulic machines, and heThe principle of live forcesThe basic principle on which Daniel Bemoulli based his hydrodynamics was what hecalled 'the equality of potential ascent and actual descent'.
He thus alluded to ChristiaanHuygens' study of the center of oscillation of a compound pendulum in the celebratedHorologium Oscillatorium of 1673. In modem terms, we would say that Huygens obtainedthe length of the simple pendulum that is equivalent to a given compound pendulum byequating the kinetic energy of the system of oscillating masses at a given instant to its signreversed potential energy. In Bemoulli's terms, thepotential ascent means'the verticalaltitude which the center of gravity of the system would reach if the several particles,converting their velocities upward, are considered to rise as far as possible.' Thedescent denotesactual'the vertical altitude through which the center of gravity has descendedafter the several particles have been brought to rest.' The potential ascent corresponds to5Cf. Dugas [1950] pp.
274--<l, Truesdell [1954] pp. XXIII-XXXI, Mikhai1ov [2002].60n early hydraulics, cf. Rouse and !nee [1957] Chaps 2-9, Garbrecht [1987]. On D. Bernoulli's Hydrodynamica, cf. Calero [1996] pp. 422-59.THE DYNAMICAL EQUATIONS5our kinetic energy divided by the total weight, and the actual descent to the sign-reversedpotential energy divided by the total weight.7For a modem reader, Bemoulli's text is much harder to penetrate than this simpleidentification would suggest. The main difficulty comes from the lack of a theory of thecombination of dimensional quantities, and the now archaic appeal to Euclidean proportions and equivalent lengths.
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