Euler L. Principles of the motion of fluids (794385), страница 7
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56 that three accelerating forces are required,which are here repeated. If an element of fluid is conceivedhere, whose volume, or mass is d xdydz, the moving forcesrequired for the motion areand by triple integration these formulas ought to be extendedthroughout the mass of the fluid; thus forces equivalent to all44and their directions may be obtained. Truly this discussion isfor a later investigation, which I shall not deepen here.81. Furthermore, the quantity T = uu+vv+ww+2U, whichis analyzed in this calculation, furnishes a simpler formula forexpressing the pressure through the height p; we have indeedp = C − z − T when the particles of the fluid are pressedupon solely by the gravity.
But if an arbitrary particle λ isacted upon by three accelerating forces which are Q, q and Φ,acting parallel to the directions of the axes AF, AB and AC,respectively, after a calculation similar to the previous one hasbeen carried out, the pressure will be given byZp = C + (Qd x + qdy + Φdz) − T.par. AL = 2d xdydz(Lu + lv + λw + L) =dudududu2d xdydz u+v+w+;dxdydzdtpar. AB = 2d xdydz(Mu + mv + µw + M) =dvdvdvdv2d xdydz u+v+w+;dxdydzdtpar. AC = 2d xdydz(Nu + nv + νw + N) =dwdw dwdw+v+w+,2d xdydz udxdydzdtThus it is plain that the differential Q + qdy + Φdz mustbe complete, as otherwise a state of equilibrium, or at leasta possible one, could not exist. That this condition must beimposed on the acting forces Q, q and Φ was shown very clearlyby the most famous Mr.
Clairaut.4582. Here are, therefore, the principles of the entire doctrineof the motion of fluids, which, even if they at first sightmay seem insufficiently fruitful, nevertheless embrace almosteverything treated both in hydrostatics and in hydraulics, so thatthese principles must be regarded as having very broad extent.For this to appear more clearly, it is worthwhile to show howthe precepts learned in hydrostatics and hydraulics follow.83. Let us therefore consider first a fluid in a state of rest, sothat we have u = 0, v = 0 and w = 0; in view of T = 2U, thepressure in an arbitrary point λ of the fluid isZp = C + (Qd x + qdy + Φdz) − 2U.so that by triple integration the components of the total forceswhich must act on the whole mass of fluid may be obtained.79.
But since the second condition requires that ud x +vdy +wdz be a complete differential, whose integral is S, let us put asbefore, with time allowed to vary, dS = ud x+vdy+wdz+Udt.dv dudw dudUSince dudy = d x ; dz = d x ; dt = d x those three moving forcesemerge41 :udu + vdv + wdw + dUpar.
AL = 2d xdydzdx39 In the printed version, but not in Euler, 1752, there are several signmistakes.40 Here, internal forces are meant.41 There is a misprint in the printed version, w instead of +.par. AB = kd xdydzpar. AC = κd xdydzHere, U is a function of the time t itself which we take asconstant. Indeed, we investigate the pressure at a given time;42 There is a misprint: u instead of κ.43 Here is again a misprint: k instead of κ.44 The pressure forces.45 Clairaut, 1743.1853L. Euler / Physica D 237 (2008) 1840–1854the quantity U can be included in the constant C, so that weobtainZp = C + (Qd x + qdy + Φdz)where Q, q an Φ are the forces acting on the particle of waterλ, parallel to the axes AL, AB and AC.84.
The pressure p can only depend on the position ofthe point λ thatR is on the coordinates x, y and z; it is thusnecessary that (Qd x + qdy + Φdz) be a prescribed functionof them, which therefore admits integration. Thus it is firstlyclear that in the manner indicated the fluid cannot be sustainedin equilibrium, unless the forces acting on each element of thefluid are such that the differential formula Qd x + qdy + Φdz iscomplete. Thus, if its integral is denoted P, the pressure at λ willbe p = C + P. Therefore, if the only force present is gravity,impelling parallel to the direction CA, we shall have p = C− z;hence, if the pressure is fixed at one point λ, the constant C canbe obtained.
From which the pressure at a given time will bedefined completely at all points of the fluid.85. However, with time passing, the pressure at a given placecan change; and this plainly occurs, if variability is assumedfor the forces impelling on the water, whose calculation cannotbe made from those forces which are assumed to act on eachelement of the fluid,46 but in such a way that they keep eachother in equilibrium and produce no motion. But if, moreover,these forces are not subject to any change, the letter C willindeed denote a constant quantity, not depending on time t;and at a given location λ we will always find the same pressurep = C + P.86. It is possible to determine the extremal shape of a fluidin a permanent state, when it is not subjected to any force.47Certainly, at the extreme surface of the fluid at which the fluidis left to itself and not contained within the walls of the vasein which it is enclosed, the pressure must be zero.
Thus weshall obtain the following equation: P = const; the shape ofthe external surface of the fluid is then expressed through arelation between the three coordinates x, y and z. And if for theexternal circumference held P = E, since C = −E, in anotherarbitrary internal location λ the pressure would be p = P − E.In this manner, if the particles of the fluid are driven by gravityonly, and because p = C − z, the following will hold at for theexternal surface z = C; from which the external free surface isperceived to be horizontal.87. Next, everything which has so far been brought outconcerning the motion of a fluid through tubes is easily derivedfrom these principles. The tubes are usually regarded as verynarrow, or else are assumed to be such that through any sectionnormal to the tube the fluid flows across with equal motion:from there originates the rule, that the speed of the fluid at anyplace in the tube is reciprocally proportional to its amplitude.Let therefore λ be an arbitrary point of such a tube, of whichthe shape is expressed by two equations relating the three46 That is the internal pressure forces.47 Here, Euler will comment on the shape of the free (extreme) surface of afluid contained in an open vessel.coordinates x, y and z, so that thereupon for any abscissa xthe two remaining coordinates y and z can be defined.88.
Let henceforth the cross section of this tube at λ be rr ; inanother fixed location of the tube, where the cross-section is f f ,let the velocity at the present time be ; now after time dt haselapsed, let the velocity become + d , so that is a functionof time t, and similarly with ddt . Hence the true velocity of thefluid at λ will be at the present time V = frrf . Since now y andz are obtained from the shape of the tube, we have dy = ηd xand dz = θd x; thus the three velocities of the point λ in thefluid, parallel to directions AL, AB and AC, areη1ffff;; v=√√rr(1 + ηη + θ θ )rr(1 + ηη + θ θ )θffw=,√rr(1 + ηη + θ θ )u=4and hence, uu + vv + ww = VV = f r 4 : and rr is functionof x itself, thus of the dependent variables y and z.89.
Since ud x + vdy + wdz must be a complete differential,the integral of which is denoted = S, we have:√f f d x(1 + ηη + θθ )ff=d x (1 + ηη + θ θ).√(1 + ηη + θ θ )rrrr√Moreover, d x (1 + ηη + θ θ ) expresses the element of thedS =tube itself; if we denote it by ds, we shall obtain dS = f frr ds :although is a function of the time,48 here we fix the time and,furthermore, the quantities s and rr do not dependR on time butf f dsonly on the shape of the tube; thus we have S =rr .90. Turning now to the pressure p which is found at the pointof the tube λ, the quantity U has to be considered; it arisesfrom the differentiation of the quantity S, if the time only isconsidered as variable, so that we have U = dSdt .
Thus, sinceR f dsthe integral formula f rrdoes not involve time t, on the oneR f f dsdUhand we shall have dSdt = U = dtrr , and on the otherhand it will follow from §. 80 that:Zf42df f dsT=+.dtrrr4Therefore, after introducing arbitrary actions of forces Q, q andΦ, the pressure at λ will beZZf42df f ds−p = C + (Q d x + q dy + Φ dz) −.dtrrr4This is that same formula which is commonly written for themotion of a fluid through tubes; but now much more widelyvalid, since arbitrary forces acting on the fluid are assumedhere, while this formula is commonly restricted to gravityalone. Meanwhile it is in order to remember that the threeforces Q, q and Φ must be such that the differential formulaQ d x + q dy + Φ dz be complete, that is, admit integration.48 As was stated in §.
88.1854L. Euler / Physica D 237 (2008) 1840–1854ReferencesD’Alembert Jean le Rond, 1752 Essai d’une Nouvelle Théorie de la Résistancedes Fluides, Paris.Clairaut Alexis, 1743 Théorie de la Figure de la Terre, Tirée des Principes del’Hydrostatique, Paris.Darrigol Olivier, Frisch Uriel, 2008 ‘From Newton’s mechanics to Euler’s equations’, these Proceedings. Also at www.oca.eu/etc7/EE250/texts/darrigolfrisch.pdf.Euler Leonhard, 1750 ‘Découverte d’un nouveau principe de mécanique’.Académie Royale des Sciences et des Belles-Lettres de Berlin, Mémoires, 6[printed in 1752], 185–217. Also in Opera omnia, ser. 2, 5, 81–108, E177.Euler Leonhard, 1752 ‘De motu fluidorum in genere’, early version ofEuler, 1756–1757 [handwritten copy], deposited at the Berlin-BrandenburgAkademie der Wissenschaften, Akademie-Archiv call number: I-M 120.Euler Leonhard, 1755 ‘Principes généraux du mouvement des fluides’,Académie Royale des Sciences et des Belles-Lettres de Berlin, Mémoires 11[printed in 1757], 274–315.
Also in Opera omnia, ser. 2, 12, 54–91,[Eneström index] E226. English translation in these Proceedings.Euler Leonhard, 1756–1757 ‘Principia motus fluidorum’ [written in 1752].Novi commentarii academiae scientiarum Petropolitanae, 6 [printed in1761], 271–311. Also in Opera omnia, ser. 2, 12, 133–168, E258.Euler Leonhard, 1980 Commercium epistolicum, ser. 4A, 5, eds. A. Juskevicand R. Taton. Basel.Fontaine Alexis, 1764 Le calcul intégral.—Mémoires donnés à l’Académieroyale des sciences, non imprimés dans leur temps, Paris.Mikhailov Gleb K., 1999 ‘The origins of hydraulics and hydrodynamics in thework of the Petersburg Academicians of the 18th century’. Fluid dynamics,34, 787–800.Truesdell Clifford, 1954 ‘Rational fluid mechanics, 1657–1765’.
In Euler,Opera omnia, ser. 2, 12 (Lausanne), IX–CXXV..
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