Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 10
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On the contrary, it seems to me that the theory, developed in all possiblerigor, gives, at least in several cases, a strictly vanishing resistance; a singular paradoxwhich I leave to future geometers for elucidation.Although d'Alembert ended his fluid dynamics on a paradox, he had achieved much onthe way. Through his dynamic principle and his equilibrium principle, he had obtainedhydrodynamic equations for the steady flow of an incompressible fluid that we mayretrospectively identify as the continuity equation, the condition of irrotational flow,and Bemoulli's Jaw.
Admittedly, he only wrote these equations for the cylindricallysymmetric and two-dimensional cases that were relevant to the fluid-resistance problem.The modem reader may wonder why he did not try to write general equations of fluid45D'Alembert [1752] pp. 60-2. Here d'A1embert discovered the Cauchy-Riemann condition for u and -v to bethe real and imaginary components, respectively, of an analytic function in the complex plane, as well as apowerful method to solve Laplace's equation Au = 0 in two dimensions. In [1761] p. 139, d'Alembert introducedthe complex potential rp + iop such that (u - iv)(dx + i dy) = d(rp + i,P).
The real part rp of this potential is thevelocity potential introduced by Euler in 1752; its imaginary part ofr is the so-called stream function, which is aconstant on any line of current, as d'Alembert noted.46D'Alembert gave a proof of this equivalence, which he did not regard as obvious.47D'Alembert had already discussed fluid resistance in part Ill of his treatise of 1744.
There he used amolecular model in which momentum was transferred by impact from the moving body to a layer of hardmolecules. He believed, however, that this molecular process would be negligible if the fluid molecules were tooclose to each other, for instance, when fluid was forced through the narrow space between the body and acontaining cylinder. In this case ([1744] pp.
205--6), he assnmed a parallel-slice flow and computed the fluidpressure on the body through Bernoulli's law. For a head-tail symmetric body, this pressure does not contribute tothe resistance if the flow has the same symmetry. After noting this difficulty, d'Alembert evoked the observedstagnancy of the fluid behind the body to retain only the Bernoulli pressure on the prow...D'Alembert [1768] p. 138. In his memoir of 1749, besides the Bernoulli pressure, d'Alembert evoked avelocity-proportional friction of the fluid on the body, and the tenacite of the fluid, according to which a certain(velocity-independent) force was required to separate the fluid molecules from each other at the prow of the body([1752] pp.
106--8). For a modern, more general derivation of the paradox, see Appendix A.THE DYNAMICAL EQUATIONS23motion in Cartesian coordinate form. The answer is plain: he was following an oldertradition of mathematical physics according to which general principles, rather thangeneral equations, were applied to specific problems.D'Alembert obtained his basic equations without recourse to the concept of pressure.Yet he had a concept ofinternal pressure, which he used to derive Bernoulli's law.
Then wemay wonder why he did not pursue the other approach sketched in his theory of winds,that is, the application of Newton's second law to a fluid element subjected to a pressuregradient. Plausibly, he favored a derivation that was based on his own principle ofdynamics and thus avoided obscure internal forces.D'Alembert knew well, however, that his equilibrium principle was simply the conditionof uniform integrability for the force density f. Had he cared to introduce the integral, sayP, he would have found the equilibrium equation f = \7P that makes P the internalpressure. Applying his dynamic principle, he would have reached the equation of motionf-Pdv= \JP,dt(1.31)which is simply Euler's equation.
But he did not proceed along these lines, and ratherwrote equations of motion that did not involve internal pressure.491.4Euler's equations1.4.1 The Latin memoirUnlike d'Alembert, the Swiss geometer and Berlin Academician Leonhard Euler did notbelieve that a new dynamic principle was necessary for continuous or connected systems,and he had no objection to internal forces. In 1740, he congratulated Johann Bernoulli forhaving 'determined most accurately the pressure in every state of the water.' In 1750, heclaimed that the true basis of continuum mechanics was Newton's second law applied tothe infinitesimal elements of bodies. Among the forces acting on the elements, he included'connection forces' acting on the boundary of the elements.
In the case of fluids, theseinternal forces were to be identified with the pressure. The acceleration of the fluidelements therefore depended on the combined effect of the pressure gradient and externalforces (gravity), as noted by d'Alembert in his memoir on winds. In hydraulic writings of1750/51, Euler thus obtained the differential versiondt = g -dPdvd.zof Johann Bernoulli's equation (1 .8) for parallel-slice efflux.5°(1 .32)49In this light, d'Alembert's later neglect of Euler's �pproach should not be regarded as a mere expression ofrancor.50Euler to J.
Bernoulli, 1 8 Oct. 1740, in Euler [1998] pp. 386-9; Euler [1750] p. 90 (the main purpose of thispaper was the derivation of the equations of motion of a solid). On the hydraulic writings, cf. Truesdell [1954] pp.XLI-XLV. These included Euler's evaluation of the pressure in the pipes that were being built to feed the fountainsof Sanssouci (Euler [1752]), nicely discussed in Eckert [2002]. There Euler used the generalization (1.9) ofBernoulli's law to non-permanent flow, which he derived from eqn (1.32). As Eckert explains, the failure of thefountains project and an ambiguous letter from the King of Prussia to Voltaire have led to the myth of Euler'sincapacity in concrete matters.WORLDS OF FLOW24In a Latin memoir of 1752, probably stimulated by the two memoirs of d'Alembert hehad reviewed for the Berlin Academy, Euler obtained the general equations of fluidmotion for an incompressible fluid in terms of the internal pressure P and the Cartesiancoordinates of the velocity v.
For this purpose, he simply applied Newton's second law to acubic element of fluid subjected to the gravity g and to the pressure P acting on the cube'sfaces. By a now familiar reasoning, this procedure yields (for unit density)&v+ (v 'i7)v = gat·-'i7P.(1 .33)Euler also obtained the continuity equation'i7 · V = 0,(1 .34)and eliminated P from the equation of motion to obtain[*J+ (v 'i7) ('i7 x v) - (('i7 x v) 'i7]v = 0.··(1 .35)Interestingly, Euler repeated d'Alembert's mistake of regarding 'i7 x v = 0 as a necessarycondition for the validity of the former relation, whereas it is only a sufficient condition.This error allowed him to introduce what later fluid theorists called the velocity potential,that is, the function <;o(r) such that v = 'i7 <p.
Equation (1.33) may then be rewritten as(1 .36)Spatial integration of this equation yields a generalization of Bemoulli's law:P=g r·-� if 88t�" + C,2-(1.37)where C is a constant (time dependence can be absorbed into the velocity potential).Lastly, Euler applied this equation to the flow through a narrow tube of variable section toretrieve the results of the Bemoullis. 51Although Euler's Latin memoir contained the basic hydrodynamic equations for anincompressible fluid, the form of exposition was still in flux. Euler often used specificletters (coefficients of differential forms) for partial differentials rather than Fontaine'snotation, and measured velocities and acceleration in gravity-dependent units.
He proceeded gradually, from the simpler two-dimensional case to the fuller three-dimensionalcase. His derivation of the continuity equation was more intricate than we would nowexpect. In addition, he erred in believing in the general existence of a velocity potential.These characteristics make Euler's Latin memoir a transition between d'Alembert's fluiddynamics and the fully-modem foundation of this science found in the French memoirs of1 755. 5251Euler [1752] pp.
1 54-7.52Cf. Truesdell [1954] pp. LXII-LXXV.25THE DYNAMICAL EQUATIONS1.4.2The French memoirsThe first of these memoirs is devoted to the equilibrium of fluids, both incompressible andcompressible. Euler presumably realized that his new hydrodynamics contained a newhydrostatics based on the following principle: the action of the contiguous fluid on a given,internal element of fluid results from an isotropic, normal pressurePexerted on itssurface.
The equilibrium of an infinitesimal element subjected to this pressure and to theforce densityf of external origin then requiresf - "VP = 0.(1 .38)As Euler showed, all known results of hydrostatics follow from this simple mathematicallaw.53In his next memoir, Euler obtained the general hydrodynamic equations for compressible fluids:(1 .39)for the continuity condition, and 'Euler's equation'&vot + (v · "V)v =1P(f - 'VP),(1 .40)to which a relation between pressure, density, and heat must be added for completeness.Euler now realized that'V x v did not necessarily vanish, for example in the case of vortexflows.
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