Euler L. Principles of the motion of fluids (794385), страница 3
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Torepresent its location we use three fixed axes AL, AB and ACorthogonal to each other (Fig. 2). Let the triple coordinatesparallel to these axes be AL = x, Ll = y and lλ = z; whichare obtained if firstly a perpendicular λl is dropped from thepoint λ to the plane determined by the two axes AL and AB;and then a perpendicular lL is drawn from the point l to theaxis AL. In this manner the location of the point λ is expressedthrough three such coordinates in the most general way and canbe adapted to all points of the fluid.23. Whatever the later motion of the point λ, it can beresolved following the three directions λµ, λν, λo, parallel tothe axes AL, AB and AC.
For the motion of the point λ we setthe velocity parallel to the direction λµ = u,the velocity parallel to the direction λν = v,the velocity parallel to the direction λo = w.Since these velocities can vary in an arbitrary manner fordifferent locations of the point λ, they will have to be consideredas functions of the three coordinates x, y and z. Afterdifferentiating them, let us put to proceeddu = Ld x + ldy + λdzdv = Md x + mdy + µdzdw = Nd x + ndy + νdz.9 This is the two-dimensional incompressibility condition, which in a slightlydifferent form has already been established by D’Alembert, 1752; cf. alsoDarrigol and Frisch, 2008:§III.1844L. Euler / Physica D 237 (2008) 1840–1854Henceforth the quantities L, l, λ, M, m, µ, N, n, ν will befunctions of the coordinates x, y and z.24.
Because these formulas are complete differentials, weobtain as abovedLdl dLdλ dldλ=;=;=dyd x dzd x dzdydMdm dMdµ dmdµ=;=;=dyd x dzd x dzdydNdn dNdν dndν=;=;=,dyd x dzd x dzdywhere it is assumed that the only varying coordinate is thatwhose differential appears in the denominator.1025. Thus, this point λ will be moved in the time dt by thisthreefold motion, which is considered to take place at the pointX; hence it movesAP − ALAQ−AMAR − ALAS − AL====u dt(u + L d x) dt(u + l dy) dt(u + λ dz) dtP p − LlQq − MmRr − LnSs − Ll====v dt(v + M d x) dt(v + m dy) dt(v + µ dz) dtpπ − lλqΦ − mµrρ − nνsσ − lo====w dt(w + N d x) dt(w + n dy) dt(w + ν dz) dt.Thus the three coordinates for these four points π, Φ, ρ andσ will beP p = y + vdt;parallel to the axis AL the distance = udtparallel to the axis AB the distance = vdtparallel to the axis AC the distance = wdt.AP = x + udt;The true velocity of the point λ, denoted by V , which clearlyarises from the composition of this triple motion, is √given inview of orthogonality of the three directions by V = (uu +vv + ww) and the elementary distance, which is travelled intime dt through its motion, will be V dt.26.
Let us consider an arbitrary solid element of the fluid tosee whereto it is carried during the time dt; since it amounts tothe same, let us assign a quite arbitrary shape to that element,but of a kind such that the entire mass of the fluid can be dividedinto such elements; to investigate by calculation, let the shapebe a right triangular pyramid, bounded by four vertices λ, µ, νand o, so that for each one there are three coordinatesqΦ = z + (w + Nd x)dtw.r.t.
ALw.r.t. ABw.r.t. ACof point λxyzof point µx + dxyzof point νxy + dyzpπ = z + wdtRQ = x + d x + (u + Ld x)dt;of point oxyz + dz.Since the base of this pyramid is λµν = lmn = 12 d xdy andthe hight λo = dz, its volume will be = 61 d xdydz.27. Let us investigate, whereto these vertices λ, µ, ν and oare carried during the time dt: for which purpose their threevelocities parallel to the directions of the three axes must beconsidered. The differential values of the velocities u, v and ware given byVelocityw.r.t. ALw.r.t.
ABw.r.t. ACof point λuvwof point µu + Ld xv + Md xw + Nd xof point νu + ldyv + mdyw + ndyof point ou + λdzv + µdzw + odz28. If we let the points λ, µ, ν and o be transferred tothe points π, Φ, ρ and σ in the time dt, and establish thethree coordinates of these points parallel to the axes, the smalldisplacement parallel to these axes will be10 The partial differential notation was so new that Euler had to remind thereader of its definition.AR = x + (u + ldy)dt;Qq = y + (v + Md x)dt;Rr = y + dy + (v + mdy)dt;rρ = z + (w + ndy)dtAS = x + (u + λdz)dt;Ss = y + (v + µdz)dt;sσ = z + dz + (w + νdz)dt.29.
Since after time dt has elapsed the vertices λ, µ, ν ando of the pyramid are transferred to the points π, Φ, ρ and σ ,πΦρσ defines a similar triangular pyramid. Due to the natureof the fluid the volume of the pyramid πΦρσ should be equal tothe volume of the pyramid λµνo put forward, that is 16 d xdydz.Thus, the whole matter is reduced to determining the volume ofthe pyramid πΦρσ . Clearly, it remains a pyramid, if the solidpqr πΦρσ is removed from the solid pqr π Φρσ ; the lattersolid is a prism orthogonally incident to the triangular basispqr , and cut by the upper oblique section πρΦ.30.
The other solid pqr πΦρσ can be divided by similarlyinto three prisms truncated in this manner, namelyI. pqr sπ Φσ ;II. pr sπρσ ;III.qr sΦρσ.This has to be accomplished in such a way that1d xdydz = pqsπ Φσ + pr sπρσ + qr sΦρσ − pqr πΦρ.6Since such a prism is orthogonally incident to its lower base,and furthermore has three unequal heights, its volume is foundby multiplying the base by one third of the sum of these heights.31.
Thus, the volumes of these truncated prisms will be1pqs( pπ + qΦ + sσ )31pr sπρσ = pr s( pπ + rρ + sσ )31qr sΦρσ = qr s(qΦ + rρ + sσ )31pqr πΦρ = pqr ( pπ + qΦ + rρ).3pqsπΦσ =L. Euler / Physica D 237 (2008) 1840–1854Since pqr = pqs + pr s + qr s, the sum of the first three prismswill definitely be small, or11111d xdydz = − pπ.qr s − qΦ. pr s − rρ. pqs + sσ. pqr,63333ord xdydz = 2 pqr.sσ − 2 pqs.rρ − 2 pr s.qΦ − 2qr s. pπ.32.
Thus, it remains to define the bases of these prisms: butbefore we do this, let us putAQ = AP + Q; Qq = P p + q; qΦ = pπ + Φ;AR = AP + R; Rr = P p + r ; rρ = pπ + ρ;AS = AP + S; Ss = P p + s; sσ = pπ + σ,in order to shorten the following calculations. After thesubstitution of these values, the terms containing pπ willannihilate each other, and we shall haved xdydz = 2 pqr.σ − 2 pqs.ρ − 2 pr s.Φso that the value of the bases to be investigated is smaller.33.
Furthermore the triangle pqr is obtained by removingthe trapezoid P pqQ from the figure P prqQ, the latter being thesum of the trapezoids P pr R and RrqQ; from which it followsthat∆ pqr =111PR(P p + Rr ) + RQ(Rr + Qq) − PQ(P p + Qq);222or, because of PR = R; RQ = Q − R; and PQ = Q we shallhave∆ pqr =1111R(P p − Qq) + Q(Rr − P p) = Qr − Rq.2222In the same manner we have∆ pqs =11PS(P p + Ss) + SQ(Ss + Qq)221− PQ(P p + Qq),2or∆ pqs =11S(P p + Ss) + (Q − S)(Ss + Qq)221− Q(P p + Qq),2from where it follows that:∆ pqs =1111S(P p − Qq) + Q(Ss − P p) = Qs − Sq.2222And finally∆ pr s =111PR(P p + Rr ) + RS(Rr + Ss) − PS(P p + Ss),222or111∆ pr s = R(P p + Rr ) + (S − R)(Rr + Ss) − S(P p + Ss)222from where it follows that∆ pr s =1111R(P p − Ss) + S(Rr − P p) = Sr − Rs.2222184534.
After the substitution of these values we will obtaind xdydz = (Qr − Rq)σ + (Sq − Qs)ρ + (Rs − Sr )Φ;thus the volume of the pyramid π Φρσ will be111(Qr − Rq)σ + (Sq − Qs)ρ + (Rs − Sr )Φ.666From the values of the coordinates presented above in §. 28followsQ = d x + Ld xdt q = Md xdt Φ = Nd xdtR = ldydt r = dy + mdydt ρ = ndydtS = λdzdt s = µdzdt σ = dz + νdzdt.35. Since here we haveQr − Rq = d xdy(1 + Ldt + mdt + Lmdt 2 − Mldt 2 )Sq − Qs = d xdz(−µdt − Lµdt 2 + Mλdt 2 )Rs − Sr = dydz(−λdt − mλdt 2 + lµdt 2 )the volume of the pyramid πΦρσ is found to be expressed as1 +L dt +Lm dt 2 +Lmν dt 3 +m dt −Ml dt 2 −Mlν dt 3 1+ν dt +Lν dt 2 −Lnµ dt 3d xdydz,+mν dt 2 +Mnλ dt 3 6−nµ dt 2 −Nmλ dt 3 23−Nλ dt+Nlµ dtwhich (volume), since it must be equal to that of the pyramidλµνo = 16 d xdydz, will satisfy, after performing a division bydt the following equation11 .0 = L + m + ν + dt (Lm + Lν + mν − Ml − Nλ − nµ)+ dt 2 (Lmν + Mnλ + Nlµ − Lnµ − Mlν − Nlµ).36.
Discarding infinitely small terms, we get this equation:12L + m + ν = 0, through which is determined the relationbetween u, v and w, so that the motion of the fluid be possible.dwSince L = ddux , m = dvdy and ν = dz , at an arbitrary point ofthe fluid λ, whose position is defined by the three coordinatesx, y and z, and the velocities u, v and w are assigned in thesame manner to be directed along these same coordinates, thecriterion of possible motions is such thatdvdwdu++= 0.dxdydzThis condition expresses that through the motion no part of thefluid is carried into a greater or or lesser space, but perpetuallythe continuity of the fluid as well as the identical density isconserved.37. This property is to be interpreted further so that at thesame instant it is extended to all points of the fluid: of course,the three velocities of all the points must be such functions ofdwthe three coordinates x, y and z that we have ddux + dvdy + dz = 0:11 This is the calculation to which Euler refers in his later French memoirEuler, 1755.12 This is the three-dimensional incompressibility condition.1846L.
Euler / Physica D 237 (2008) 1840–1854in this way the nature of those functions defines the motionof every point of the fluid at a given instant. At another timethe motion of the same points may be howsoever different,provided that at an arbitrary point of time the property holds forthe whole fluid. Up to now I have considered the time simply asa constant quantity.38. If however, we also wish to consider time as variable sothat the motion of the point λ whose position is given by thethree coordinates AL = x, Ll = y and lλ = z has to be definedafter the elapsed time t, it is certain that the three velocitiesu, v and w depend not only on the coordinates x, y and z butadditionally on the time t, that is they will be functions of thesefour quantities x, y, z and t; furthermore, their differentials aregoing to have the following formdu = Ld x + ldy + λdz + Ldt;dv = Md x + mdy + µdz + Mdt;dw = Nd x + ndy + νdz + Ndt;Meanwhile we shall always have L + m + ν = 0, having inview that at every arbitrary instant the time t is considered tobe constant, or dt = 0.
Howsoever the functions u, v and wvary with time t, it is necessary that at every moment of timethe following holds:dudvdw++= 0.dxdydzSince the condition expresses that any arbitrary portion of thefluid is carried in a time dt into a volume equal to itself, thesame will have to happen, due to the same condition, in the nexttime interval, and therefore in all the following time intervals.II.
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