Darrigol O., Frisch U. «From Newton's mechanics to Euler's equations» (1123933), страница 7
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In these works of 1750–51, Eulerobtained the equation of motion for parallel-slice pipe flow bydirectly relating the acceleration of the fluid elements to thecombined effect of the pressure gradient and gravity. He thusobtained the differential versionAn English translation of the Latin memoir will be includedin these Proceedings.This relative failure did not discourage Euler. Equipped withhis new principle of mechanics and probably stimulated by thetwo memoirs of d’Alembert, which he had reviewed, he setout to formulate the equations of fluid mechanics in their fullgenerality.
A memoir entitled “De motu fluidorum in genere”was read in Berlin on 31 August 1752 and published under thetitle “Principia motus fluidorum” in St. Petersburg in 1761 aspart of the 1756–1757 proceedings. Here, Euler obtained thegeneral equations of fluid motion for an incompressible fluid interms of the internal pressure P and the Cartesian coordinatesof the velocity v.52In the first part of the paper, he derived the incompressibilitycondition. For this, he studied the deformation during a timedt of a small triangular element of water (in two dimensions)and of a small triangular pyramid (in three dimensions).
Themethod here is a slight generalization of what d’Alembert did inhis memoir of 1749 on the resistance of fluids. Euler obtained,in his own notation:4. Euler’s equationsdvdP=g−dtdz(31)of Johann Bernoulli’s equation (7) for parallel-slice efflux.From this, he derived the generalization (8) of Bernoulli’slaw to non-permanent flow, which he applied to evaluate thepressure surge in the pipes that would feed the fountains ofSanssouci.49Although d’Alembert had occasionally used this kind ofreasoning in his theory of winds, it was new in a hydrauliccontext.
As we saw, the Bernoullis did not rely on internalpressure in their own derivations of the equations of fluidmotion. In contrast, Euler came to regard internal pressure asa key concept for a Newtonian approach to the dynamics ofcontinuous media.In a memoir of 1750 entitled “Découverte d’un nouveauprincipe de mécanique,” he claimed that the true basis ofcontinuum mechanics was Newton’s second law applied to theinfinitesimal elements of bodies. Among the forces acting onthe elements he included “connection forces” acting on theboundary of the elements. In the case of fluids, these internalforces were to be identified to the pressure.50Euler’s first attempt to apply this approach beyond theapproximation of parallel-slices was a memoir on the motions47 In this light, d’Alembert’s later neglect of Euler’s approach should not beregarded as a mere expression of rancor.48 Reported by Libri, 1846: 51.49 Euler, 1752.
On the hydraulic writings, cf. Truesdell, 1954:XLI–XLV; Ackeret, 1957. On Euler’s work for the fountains of Sanssouci, cf.Eckert, 2002, 2008. As Eckert explains, the failure of the fountains project andan ambiguous letter of the King of Prussia to Voltaire have led to the myth ofEuler’s incapacity in concrete matters.50 Euler, 1750: 90 (the main purpose of this paper was the derivation of theequations of motion of a solid).dvdwdu++= 0.dxdydz(32)In the second part of the memoir, he applied Newton’ssecond law to a cubic element of fluid subjected to the gravityg and to the pressure P acting on the cube’s faces. By a nowfamiliar bit of reasoning, this procedure yields (for unit density)in modern notation:∂v+ (v · ∇)v = g − ∇ P.∂t(33)Euler then eliminated the pressure gradient (basically by takingthe curl) to obtain what we now call the vorticity equation:∂+ (v · ∇) (∇ × v) − [(∇ × v) · ∇]v = 0,(34)∂t51 Euler, 1760, Truesdell, 1954: LVIII–LXII.52 Euler, 1756–1757.
Cf. Truesdell, 1954: LXII–LXXV. D’Alembert’s role(also the Bernoullis’s and Clairaut’s) is acknowledged by Euler somewhatreluctantly in a sentence at the beginning of the third memoir cited in epigraphto the present paper.O. Darrigol, U. Frisch / Physica D 237 (2008) 1855–1869in modern notation. He then stated that “It is manifest thatthese equations are satisfied by the following three values[∇ × v = 0], in which is contained the condition providedby the consideration of the forces [i.e. the potential characterof the r.h.s.
of (33)]”. He thus concluded that the velocity waspotential, repeating here d’Alembert’s mistake of confusinga necessary condition with a sufficient condition. This errorallowed him to introduce what later fluid theorists called thevelocity potential, that is, the function ϕ(r) such that v = ∇ϕ.Eq. (33) may then be rewritten as:∂1 (∇ϕ) + ∇ v 2 = g − ∇ P.(35)∂t2Spatial integration of this equation yields a generalization ofBernoulli’s law:1∂ϕP = g · r − v2 −+ C,(36)2∂twherein C is a constant (time-dependence can be absorbed inthe velocity potential). Lastly, Euler applied this equation to theflow through a narrow tube of variable section to retrieve theresults of the Bernoullis.Although Euler’s Latin memoir contained the basichydrodynamic equations for an incompressible fluid, the formof exposition was still in flux.
Euler frequently used specificletters (coefficients of differential forms) for partial differentialsrather than Fontaine’s notation, and he measured velocitiesand accelerations in gravity-dependent units. He proceededgradually, from the simpler two-dimensional case to the fullerthree-dimensional case. His derivation of the incompressibilityequation was more intricate than we would now expect.
Andhe erred in believing in the general existence of a velocitypotential. These characteristics make Euler’s Latin memoira transition between d’Alembert’s fluid dynamics and thefully modern foundation of this science found in the Frenchmemoirs.534.3. The French memoirsAn English translation of the second French memoir will beincluded in these Proceedings.The first of these memoirs “Principes généraux de l’étatd’équilibre des fluides” is devoted to the equilibrium of fluids,both incompressible and compressible. Euler realized that hisnew hydrodynamics contained a new hydrostatics based onthe following principle: the action of the contiguous fluid ona given, internal element of fluid results from an isotropic,normal pressure P exerted on its surface.
The equilibrium ofan infinitesimal element subjected to this pressure and to theforce density f of external origin then requires:f − ∇ P = 0.The second French memoir, “Principes généraux dumouvement des fluides,” is the most important one. Here, Eulerdid not limit himself to the incompressible case and obtainedthe “Euler’s equations” for compressible flow:∂t ρ + ∇ · (ρv) = 0,1∂t v + (v · ∇)v = (f − ∇ P),ρ(38)(39)to which a relation between pressure, density, and heat must beadded for completeness.55The second French memoir is not only the coronation ofmany decades of struggle with the laws of fluid motion bythe Bernoullis, d’Alembert and Euler himself, it also containsmuch new material.
Among other things, Euler now realizedthat ∇ × v needed not vanish, as he had assumed in his Latinmemoir, and gave an explicit example of incompressible vortexflow in which it did not.56 In a third follow-up memoir entitled“Continuation des recherches sur la théorie du mouvement desfluides,” he showed that even if it did not vanish, Bernoulli’s lawremained valid along any stream line of a steady incompressibleflow (as he had anticipated on his memoir of 1750–1751 onriver flow).
In modern terms: owing to the identity1 2(v · ∇)v = ∇v − v × (∇ × v),(40)2the integration of the convective acceleration term along a lineof flow eliminates ∇ × v and contributes the v 2 /2 term ofBernoulli’s law.57In his second memoir, Euler formulated the general problemof fluid motion as the determination of the velocity at any timefor given values of the impressed forces, for a given relationbetween pressure and density, and for given initial values offluid density and the fluid velocity.
He outlined a generalstrategy for solving this problem, based on the requirementthat the form (f − ρ v̇) · dr should be an exact differential (inorder to be equal to the pressure differential). Then he confinedhimself to a few simple, soluble cases – for instance uniformflow (in the second memoir), or flow through a narrow tube(in the third memoir). In more general cases, he recognized theextreme difficulty of integrating his equations under the givenboundary conditions:58We see well enough . . .
how far we still are from a complete knowledge ofthe motion of fluids, and that what I have explained here contains but a feeblebeginning. However, all that the Theory of fluids holds, is contained in the twoequations above [Eq. (1)], so that it is not the principles of Mechanics whichwe lack in the pursuit of these researches, but solely Analysis, which is not yetsufficiently cultivated for this purpose. Thus we see clearly what discoveriesremain for us to make in this Science before we can arrive at a more perfectTheory of the motion of fluids.(37)As Euler showed, all known results of hydrostatics follow fromthis simple mathematical law.5453 Cf. Truesdell, 1954: LXII–LXXV.54 Euler, 1755a.186755 Euler, 1755b: 284/63, 286/65.
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