Euler L. Principles of the motion of fluids (794385), страница 5

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Since its integral is U =dSdt ,we shall havedp = αd x − 2udu − 2vdv − 2dUfrom where it appears by integration:2dSdtwith a given function S of the coordinates x, y and t themselves,whose differential, for dt = 0 is ud x + vdy.52. In order to understand better the nature of these formulas,let√ us consider the true velocity of the point l, which is V =(uu + vv). And the pressure will be: p = Const. + αx −2dSVV −R dt : in which the last term dS denotes the differential ofS = (ud x + vdy) itself, where the time t is allowed to vary.53.

If we now wish to also take friction into account, let usset it proportional to the pressure p. While the point l travelsthe element ds, the retarding force arising from the friction is= pf ; so that, setting dSdt = U, our differential equation will befor constant tpdp = αd x − ds − VdV − 2dU,fp = Const. + αx − uu − vv −from where we obtain by integration, taking e for the numberwhose hyperbolic20 logarithm is = 1,Z−ssfp=ee f (αd x − 2VdV − 2dV) orZs1 −sp = αx − VV − 2U − e fe f (αx − VV − 2U)ds.f54. The criterion of the motion which drives the fluid inreality consists in this that, fixing the time t, the differentialud x + vdy has to be complete: also continuity and constantuniform density demand that ddux + dvdy = 0, hence it followstoo that this differential udy − vd x will have to be complete.21From where both velocities u and v jointly must be functions ofthe coordinates x and y with the time t in such a way that bothdifferential formulas ud x + vdy and udy − vd x 22 be completedifferentials.55.

Let us set up the same investigation in general, givingthe point λ three velocities directed parallel to the axes AL,20 Natural.21 The published version has ud x + vdy, a mistake not present in Euler, 1752.22 Previous mistake repeated in the published version.du = Ld x + ldy + λdz + Ldtdv = Md x + mdy + µdz + Mdtdw = Nd x + ndy + νdz + Ndt.Although here the time t is also taken as variable, nonethelessfor the motion to be possible, by the preceding condition23 wehave L + m + ν = 0, or, which reexpresses the samedudvdw++= 0,dxdydza condition on which the present examination does not depend.56.

After the passage of time interval dt the point λ is carriedto π , and it travels a distance udt parallel to the axis AL, adistance vdt parallel to the axis AB and a distance wdt parallelto the axis AC. Thus the three velocities of the point which hasmoved from λ to π will be:parallel to AL = u + Lu dt + lv dt + λw dt + L dt;parallel to AB = v + Mu dt + mv dt + µw dt + M dt;parallel to AC = w + Nu dt + nv dt + νw dt + N dt,and the accelerations parallel to the same directions will bepar.

AL = 2(Lu + lv + λw + L);par. AB = 2(Mu + mv + µw + M);par. AC = 2(Nu + nv + νw + N).57. If we take the axis AC to be vertical, in such a way thatthe remaining two AL and AB are horizontal, the acceleratingforce due to gravity arises parallel to the axis AC with thevalue −1. Then indeed, denoting the pressure at λ by p, itsdifferential, at constant time isd p = R d x + r dy + ρdz,from which we obtain the three accelerating forcespar. AL = R;par.

AB = −r ;par. AC = −ρwhich are in fact easily collected in the same manner as wasdone in §§. 44 and 45, so that it is not necessary to repeat thesame computation. Hence we obtain the following equations24R = −2(Lu + lv + λw + L)r = −2(Mu + mv + µw + M)ρ = −1 − 2(Nu + nv + νw + N).58. Since the differential formula d p = Rd x + r dy + ρdzhas to be a complete differential, we havedRdr=;dydxdRdρ=;dzdxdrdρ=.dzdy23 From Part I.24 These are the three dimensional Euler equations.1849L.

Euler / Physica D 237 (2008) 1840–1854After a differentiation and a division by −2 the following threeequations are obtained25 udL vdlwdλ dL++++ Ll + lm + λn =dydydydyI udM + vdm + wdµ + dM + ML + mM + µNdxdxdxdxudLvdlwdλdL++++ Lλ + lµ + λν =dzdzdzdzII udN + vdn + wdν + dN + NL + nM + νNdxdxdx dxudMvdmwdµdM++++ Mλ + mµ + µν =dzdzdzdzIII udNvdnwdνdN++++ Nl + nm + νn.dydydydy59. Moreover, because of the nature of the completedifferentials, we havedLdydµdxdLdzdνdxdMdzdνdy======dl;dxdM;dzdλ;dxdN;dzdµ;dxdn;dzdmdxdLdydldzdLdzdNdydMdz======dM;dydl;dtdλ;dydλ;dtdn;dxdµ;dtdλdydMdxdndxdNdxdmdzdNdy======dl;dzdMdtdN;dydN;dtdµ;dydn,dtafter substituting of which values those three equations will betransformed into these26dl − dMdl − dMdl − dM+u+v+dtdxdydl − dM+ (l − M)(L + m) + λn − µN = 0,wdzdλ − dNdλ − dNdλ − dN+u+v+dtdxdydλ − dNw+ (λ − N)(L + ν) + lµ − nM = 0,dzdµ − dndµ − dndµ − dn+u+v+dtdxdydµ − dn+ (µ − n)(m + ν) + Mλ − Nl = 0.wdz60.

Now it is manifest that these three equations are satisfiedby the following three valuesl = M;λ = N;µ=nnotation chosen we have27dudv=;dydxdudw=;dzdxdvdw=dzdythese conditions moreover are the same as those which arerequired in order that the formula ud x + vdy + wdz be acomplete differential. From which this criterion consists in thatthe three velocities u, v and w have to be functions of x, y andz together with t in such a manner that for fixed constant timethe formula ud x + vdy + wdz admits an integration.61. Taking the time t constant or dt = 0, we havedu = Ld x + Mdy + Ndzdv = Md x + mdy + ndzdw = Nd x + ndy + νdzmoreover, for R, r and ρ the values areR = −2(Lu + Mv + Nw + L)r = −2(Mu + mv + nw + M)ρ = −1 − 2(Nu + nv + νw + N).Regarding the pressure p, we obtain the following equationd p = −dz−2u(Ld x + Mdy + Ndz) = −dz − 2udu − 2vdv − 2wdw− 2Ld x − 2M − 2Ndz−2v(Md x + mdy + ndz)−2w(Nd x + ndy + νdz)−2Ld x − 2Mdy − 2Ndz.62. Since in truth L =integrationdudt ;M=p = C − z − uu − vv − ww − 2Z dvdt ;N=dwdt ,we obtain bydudvdwdx +dy +dz .dtdtdtBy the previously ascertained condition ud x + vdy + wdzis integrable.

Let us denote its integral by S, which can alsoinvolve the time t; taking also the time t variable, we havedS = ud x + vdy + wdz + Udt,dU dvdU dwdUand we have dudt = d x ; dt = dy ; dt = dz . From where, withtime generally taken constant, it can be assumed in the aboveintegral thatdUdUdUdx +dy +dz = dU,dxdydzand we obtain28p = C − z − uu − vv − ww − 2U, ordSp = C − z − uu − vv − ww − 2 .dtin which is contained the criterion furnished by theconsideration of the forces. Here therefore follows that in the63. Thus, uu + vv + ww is manifestly expressing the squareof the true velocity of the point λ, so that, if the true velocity of25 The printed version contains mistakes not present in Euler, 1752: in theformula labelled II, instead of L there is L; in the formula labelled III there is av instead of u.26 These are the equations for the vorticity.27 Here Euler repeats the mistake of assuming that the only solution is zerovorticity flow; in Euler, 1755 this will be corrected.28 The published version has a ds in the denominator, instead of the correctdt, found in Euler, 1752.1850L.

Euler / Physica D 237 (2008) 1840–1854this point is denoted V , the following equation is obtained forthe pressure29p = C − z − VV −2dS.dtTo use this, firstly one must seek the integral S of theformula ud x + vdy + wdz which should be complete. Thisis differentiated again, taking only the time t as variable. Afterdivision by dt, one obtains the value of the formula dSdt , whichenters into the expression for the state of the pressure p.64. But before we may add here the previous criterion,regarding possible motion, the three velocities u, v and w mustbe such functions of the three coordinates x, y and z, and oftime t that, firstly, ud x + vdy + wdz be a complete differentialdwand, secondly, that the condition ddux + dvdy + dz = 0 holds.The whole motion of fluids endowed with invariable density issubjected to these two conditions.Furthermore, if we take also the time t to be variable, andthe differential formula ud x + vdy + wdz + Udt is a completedifferential, the state of the pressure at any point λ, expressedas an altitude p, will be given byp = C − z − uu − vv − ww − 2U,if only the fluid enjoys the natural gravity and the plane BAL ishorizontal.65.

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