Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 11
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In a sequel to this memoir, he showed that Bernoulli's law nonetheless remainedvalid along the stream lines of any steady flow of an incompressible fluid. Indeed, owing tothe identity( 1 .41)the integration of the convective acceleration term along a line of flow eliminatesand contributes the!if term of Bernoulli's law.54'V x vEuler deplored the difficulty of solving his equations. He could not really handle anyproblem that was not accessible to earlier methods, although he devoted much space to thegeneral conditions of integrability. His true achievement was a strikingly modern andcrystal-clear expression of the foundations of hydrodynamics.
Present derivations of thefundamental equations follow Euler's original procedures very closely. Unlike the earlierhydrodynamic writings of d'Alembert and of the Bernoullis, Euler's memoirs are immediately intelligible to the modern reader. They mark the emergence of a new style of mathematical physics in which fundamental equations take the place of fundamental principles.Yet we should not underestimate Euler' s debts to his predecessors. Euler himself paidtribute to the Bernoullis and to d'Alembert, despite his obscure role in d'Alembert's failure53Euler [1755a] p. 5.54Euler [1755b] pp.
63, 65; [1755c] 1 17. Cf. Truesdell (1954] pp. LXXV-C.WORLDS OF FLOW26to win the Berlin prize on winds.55 These authors anticipated essential features of Euler'sapproach. Johann Bernoulli had a concept of internal pressure, and some sort of convective derivative (the gurges). D'Alembert had particular cases of the partial differentialequations of continuity and motion, as well as the general idea of deriving the equationsof motion by balancing acceleration, external forces, and pressure gradient.
Euler's rolewas to prune unnecessary and unclear elements in the abundant writings of his predecessors, and to combine the elements he judged most fundamental in the clearest and mostgeneral manner.1.5 Lagrange's analysis1 .5.1Methods ofresolutionIn his celebrated memoir of 1781 on fluid motion, Joseph Louis Lagrange judged that thefoundations of this subject had been sufficiently established by d'Alembert and hisfollowers.
But he deplored the lack of efficient, rigorous methods for solving practicalquestions of fluid motion. Having already done much work on the integration of partialdifferential equations, he knew that the general integral of this kind of equation dependedon an arbitrary function that could only be determined through the boundary conditions.A first condition for the determination of specific flows was a clear and complete statement of the boundary conditions. 56Already known were the condition that the velocity of the fluid on the walls of itscontainer should be parallel to the walls, and the condition that the pressure on the freesurface should be equal to the external pressure. Lagrange added the condition that a fluidparticle initially on the free surface of the fluid should retain this property 'so that the fluiddoes not divide itself but always forms a continuous mass.' lff (r, t) = 0 is the equation ofthe fluid surface, this condition implies8f8t- + (v \i')f = 0(1 .42)·on the surface.57In order to ease the resolution of Euler's equation, Lagrange systematically introducedthe velocity potential cp, which reduces the number of unknown functions from three toone.
It was therefore important to him to determine the condition under which thispotential existed. The following, important theorem answered this question: wheneverthe motion of an incompressible fluid is prompted by forces that derive from,,a potential(gravity or externalpressure), a velocity potential exists.(if)In order to prove this, Lagrange multiplied Euler's equation (1 .40) by dr to obtain8v&- ·dr + (\7xv) (v x dr) = - f dr - - - d·1p·dPp2:55Cf. Grimberg [1998] pp. 8-10.56Cf. Truesdell [1955] pp.
XC-CV.57Lagrange [1781] p. 704. On later criticism of tbis condition, cf. Truesdell [1955] p. XCI.(1 .43)THE DYNAMICAL EQUATIONS27If the pressure is a function of density only (which is, of course, the case for an incompressible fluid) and iff/p derives from a potential, then the right-hand side of this equationis an exact differential. After noting this, Lagrange applied his favorite method, powerseries development, to the functions v(t) and '11 x v(t). As the left-hand side of eqn (1 .43)must be an exact differential, the vanishing of the coefficients of '11 x v(t) up to order nimplies that the (n + l)th coefficient of v dr is an exact differential or, equivalently, thatthe (n + l)th coefficient of '11 x v(t) vanishes. As, by hypothesis, v dr is an exact differential for t = 0, the first term of the development of 'V x v(t) must vanish. By induction, allother terms must then vanish.
Therefore, '11 x v(t) vanishes at any time and there exists avelocity potential at any positive time.58Although Lagrange seems to have believed that the conditions of his theorem were metfor most flows in nature, he gave one example in which they were not, namely tidal motion(since the Coriolis forces do not derive from a potential). Lagrange also (incorrectly)argued that the velocity potential existed for small motions in which the second-order term(v 'V)v could be neglected. As either this condition or that of the previous theorem seemedto hold in many cases of motion, Lagrange believed he could restrict his analysis topotential flows without much loss of generality. He gave the propagation of sound in acompressible fluid as an example of the applicability of the second condition. He gave themotion of an incompressible fluid under the sole effect of gravity as an example of theapplicability of the first condition.
59In the latter case, the equations for the velocity potential were still too complicated toallow integration in finite terms. Lagrange assumed one of the dimensions of the fluid tobe very small, so that one of the coordinates of the fluid particles could be taken to bemuch smaller than the other coordinates. Then the velocity potential could be expressed asa power series with respect to this coordinate.
Lagrange thus obtained the parallel-slicesolution of the effiux problem in a first approximation, and also corrections depending onhigher powers of the width of the vessel. Most originally, he showed that small surfacedisturbances on shallow water obeyed the equations of a vibrating string with a propagation velocity ../ifi, where h is the depth of the water.60Lagrange's equations and boundary conditions for the velocity potential of an incompressible fluid were the invariable basis of much of nineteenth-century hydrodynamics, for instance; the theories of waves by Poisson, Cauchy, Stokes, Boussinesq,Korteweg, and de Vries.
The resolution of these equations is intimately bound to thedevelopment of potential theory and Fourier analysis. To cite only two examples, Cauchyreinvented Fourier analysis in his memoir on waves, and Stokes obtained importanttheorems for the potential, which his friend Kelvin transposed to electric and magneticcontexts.
6 1···58Lagrange [1781] pp. 714-17; Lagrange to d'Alembert, 15 Apr. 1781, in Lagrange [1867-1892] vol. 13, pp.362-6. This proof only holds if the function v(t) is analytical. Cauchy [1827a] has the first rigorous, general proof.59/bid. pp. 713-18, 721-3, 728.60/bid. pp. 728-48. Cf. Chapter 2, pp. 35-37.61 Cf. Wise [1981], Darrigol [2000] pp. 128-9, Darrigol [2003].WORLDS OF FLOW281 .5.2Continuum dynamicsLagrange returned to hydrodynamics when he wrote his Mechanique analitique of 1 788. Asis well known, he there obtained the general equations for the dynamics of connectedsystems by combining the principle of virtual velocities and d'Alembert's principle.
Fromd'Alembert's viewpoint, fluids could not be treated on the same footing, because theirinternal composition was too complex to allow a deduction of the condition of equilibrium; this condition had to be obtained empirically. In his analytical mechanics, Lagrangeprided himself on eliminating this asymmetry between solid and fluid dynamics. He firstshowed that the condition of equilibrium of an incompressible fluid derived from theprinciple of virtual velocities applied to an ideal continuum.According to this principle, the moment (virtual work) ofthe force density f acting within a .fluid mass and of the pressure P exerted on the free surface of the fluid must vanish for anydisplacement 8r of the fluid particles that satisfies the condition of incompressibility\l · 8r = 0.
Through Lagrange's method of multipliers, this condition is equivalent toI (f·8r + A \l · 8r)d-r -I8r · PdS= 0,(1 .44)where A(r) is the Lagrange multiplier, and the displacement 8r is now arbitrary, except onsolid walls where it must be parallel to the walls. Integrating by parts the A term, this givesI (f -Hence\lA.) · 8r dT +J(A. - P)8r · dS = 0.( 1.45)f - \lA.
must vanish within the fluid, and the parameter A must be equal totheexternal pressure on the free surface. This is equivalent to Euler's equilibrium condition,the parameter A playing the role of the internal pressure. 62For a compressible fluid, there is no constraint on the displacement 8r (save forparallelism on solid walls), but the moment of the internal forces of elasticity must beadded to the moment of the external forces.
As the 'elasticity' P tends to increase thevolume dT of the particles of fluid, Lagrange wrote this new moment asIP8(d'T) =IP(\l · 8r)dT(1 .46)Consequently, the condition of equilibrium has the same form as in the case of incompressibility, and the 'elasticity' P plays the role of Euler's internal pressure. D�embert'sprinciple, combined with this condition, yields Euler's equations of fluid motion.
In all,Lagrange's purely analytical approach to the equilibrium and motion of fluids led to thesame set of fundamental equations as Euler's more intuitive approach. With a mathematical subtlety that prevented large diffusion, he subsumed the conditions of equilibrium of acontinuum nnder a general principle of statics. 6362Lagrange [1788] pp. 139-45, 438-41 (case of motion). A similar procedure was previously given in Lagrange[1761] pp. 435-59. Cf. Truesdell [1954] p. CXXIV.63Lagrange [1788] pp. 1 55-7, 492-93.
A similar procedure is found in Lagrange [1761] pp. 459-68 (although atthat time Lagrange used a generalization of a variational principle by Euler instead of d'Alembert's principle).29THE DYNAMICAL EQUATIONS1 .5.3The Lagrangian pictureCombining d'Alembert's principle and the principle of virtual velocities, Lagrangeobtained Euler's equation in the formf - P'Y-\1P =0.(1 .47)In his memoir of 178 1 , Lagrange followed d'Alembert's and Euler's original method ofcharacterizing the fluid motion through the velocity v as a function of the points of space rand the time t.
This leads to the 'Eulerian' form (1 . 1) of Euler's equation. In theique analitique,Mechanhowever, Lagrange judged that 'a distinct idea of the nature of thisequation' required another representation, in which the positionregarded as a function of time and of their positionMultiplying eqn (1 .47) by the differentialr of the fluid particles isR at the origin of time.64dr yields(f - P'Y) · dr - dP = 0,or, in terms of the coordinates(1.48)Rr, Rz, R3 , and t,(1 .49)This is the so-called 'Lagrangian' form of Euler's equation. For an incompressible fluid,the continuity condition further requires that the transformation Rvolumes, that is, 65det( 8r; )BRj=1.-t rlocally conserves(1.50)As Lagrange noted, this form of the equations of fluid motion is more complex than theEulerian form.
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