Euler L. Principles of the motion of fluids (794385), страница 6
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Suppose we had attributed another direction to thegravity or even adopted arbitrary variable forces acting on theparticles of the fluid. Differences would arise in the values ofthe pressure, but the law which the three velocities of the fluidhave to obey would not suffer any changes. Thus, whatever theacting forces, the three velocities u, v and w have to satisfy theconditions that the differential formula ud x + vdy + wdz bedwcomplete and that ddux + dvdy + dz = 0 should hold. Therefore,the three velocities u, v and w can be fixed in infinitely manyways while satisfying the two conditions; and then it is possibleto prescribe the pressure at every point of the fluid.3066.
However, much more difficult would be the followingquestion: given the acting forces and the pressure at all places,to determine the motion of the fluid at all points. Indeed, wewould then have some equations31 of the form p = C − z −uu − vv − ww − 2U, from which the relation of the functionsu, v and w would have to be defined in such a way that notonly the equations themselves would be satisfied, but also thepreviously contributed rules32 would have to be obeyed; thiswork would certainly require the greatest force of calculation.It is fitting therefore to inquire in general into the nature offunctions proper to satisfy both criteria.67. Most conveniently therefore let us begin with thecharacterization of the integral quantity S, whose differentialis ud x + vdy + wdz, when time is held constant. Let thus29 This is basically the Bernoulli pressure law for potential flow.30 Many statements in this paragraph are rendered invalid by the generallyincorrect assumption of potential flow.31 The plural is here used probably because this relation has to be satisfied atall points.32 Incompressibility and potentiality.S be a function of x, y and z, the time t being contained inconstant quantities.
When S is differentiated, the coefficientsof the differentials d x, dy and dz are the velocities u, v andw which at the present time suit the point of fluid λ, whosecoordinates are x, y and z. The question thus arises here to finddwthe functions S of x, y and z such that ddux + dvdy + dz = 0; now,since we have u =ddSdx2ddSdy 2ddSdz 2dSdx , v0.33=dSdyand w =dSdzit follows that++=68. Since it is not plain how this can be handled in general, Ishall consider certain rather general cases.
LetS = (Ax + By + Cz)n .We havedS= nA(Ax + By + Cz)n−1 anddxddS= n(n − 1)AA(Ax + By + Cz)n−2dx2and the expressions forhave to satisfyddSdy 2andddSdz 2will be similar. Thus wen(n − 1)(Ax + By + Cz)n−2 (AA + BB + CC) = 0which is plainly satisfied when either n = 0 or n = 1. Thus wehave the solutions S = Const. and S = Ax + By + Cz, wherethe constants A, B and C are arbitrary.69. But if n is neither 0, nor 1, we necessarily have: AA +BB + CC = 0: and then S is given byS = (Ax + By + Cz)nfor any value of the exponent n; even the time t itself willpossibly enter in n.
Furthermore we can add up arbitrarily manysuch S and obtain yet another solution.34 The functionS = α + βx + γ y + δz + (Ax + By + Cz)n +000ζ (A0 x + B0 y + C0 z)n + η(A00 x + B00 y + C00 z)n +000θ (A000 x + B000 y + C000 z)n etc.will satisfy the condition only if we have:AA + BB + CC = 0;A0 A0 + B0 B0 + C0 C0 = 0;A00 A00 + B00 B00 + C00 C00 = 0 etc.70. Here suitable values are given for S in which thecoordinates x, y, z have either one, or two, or three, or fourdimensions35I. S = AII.
S = Ax + By + CzIII. S = Ax x + Byy + Czz + 2Dx y + 2Ex z + 2Fyz with A +B+C=033 This is what will later be called Laplace’s equation.34 In modern terms, Euler is here using the linear character of the Laplaceequation.35 In modern terms we would say “which are polynomials in x, y, z of degreesup to four”.1851L. Euler / Physica D 237 (2008) 1840–1854IV. S = Ax 3 +By 3 +Cz 3 +3Dx x y +3Fx x z +Hyyz +6Kx yz +3Ex yy +3Gx zz +3Iyzz with A+E+G = 0; B+D+I =0; C + F + H = 073. Moreover, the second formula S = Ax + By + Cz, afterdifferentiation, gives these three velocities to the point λ:V.Thus simultaneously, all points of the fluid are carried by anidentical motion in the same direction.
From which the wholefluid moves in the same manner as a solid body, carried onlyby a forward motion. But at different times the velocities aswell as the direction of this motion are able to be varied in anarbitrary way, depending on what the extrinsic driving forcesrequire. Therefore, the pressure at the point λ at the time t onwhich A, B, C depend, is36 p = C − z − AA − BB − CC −dBdC2x dAdt − 2y dt − 2z dt .74. The third formula S = Ax x + Byy + Czz + 2Dx y +2Ex z + 2Fyz, where A + B + C = 0, gives the followingthree velocities37 of the point λ: u = 2Ax + 2Dy + 2Ez;v = 2By + 2Dx + 2Fz; w = 2Cz + 2Ex + 2Fy, or w =2Ex + 2Fy − 2(A + B)z.
Here, at a given instant, differentpoints of the fluid are carried by different motions; moreover,in the time development an arbitrary motion of a given pointis permitted, because A, B, D, E, F can be arbitrary functionsof the time t. Finally, a much greater variety can take place, ifmore elaborate values are given to the function S.75. In the second case the motion of the fluid wascorresponding to the forward motion of a solid body in which,plainly, at any instant the different parts are carried by a motionequal and parallel to itself.
In other cases the motion of thefluid could be suspected to correspond to solid-body motion,either rotational or anomalous. It suffices to put forward sucha hypothesis – beyond the second case – to find that it cannottake place. Indeed, in order to happen, not only would it benecessary that the pyramid πΦρσ would be equal,38 but alsosimilar to the pyramid λµνo, or that the following holds√πΦ = λµ = d x = (QQ + qq + ΦΦ)√πρ = λν = dy = (RR + rr + ρρ)√π σ = λo = dz = (SS + ss + σ σ )√Φρ = µν = (d x 2 + dy 2 ) =√((Q − R)2 + (q − r )2 + (Φ − ρ)2 )√Φσ = µo = (d x 2 + dz 2 ) =√((Q − S)2 + (q − s)2 + (Φ − σ )2 )√ρσ = νo = (dy 2 + dz 2 ) =√((R − S)2 + (r − s)2 + (ρ − σ )2 ),433+ Ax + 6Dx x yy + 4Gx y + 4Hx y + 12Nx x yzS = + By 4 + 6Ex x zz + 4Ix 3 z + 4Kx z 3 + 12Ox yyz+ Cz 4 + 6Fyyzz + 4Ly 3 z + 4Myz 3 + 12Px yzzwithA+D+E=0G+H+P=0B+D+F=0I+K+O=0C+E+F=0L + M + N = 0.71.
Hence it is clear how these formulas are to be obtainedfor any order. First, simply give to the various terms thenumerical coefficients which belong to them from the law ofpermutation, or, equivalently, which arise when the trinomialx + y + z is raised to that same power. Let indefinite lettersA, B, C, etc., be adjoined to the numerical coefficients. Then,ignoring the coefficients, observe whenever there occur threeterms of the type LZx 2 + MZy 2 + NZz 2 having a commonfactor Z formed from the variables. Whenever this occurs, setthe sum of the literal coefficients L + M + N equal to zero.
Forexample, for the fifth power we haveS = Ax 5 + 5Dx 4 y + 5Dx 4 z + 10Gx 3 yy + Gx 3 zz +20Kx 3 yz + 30Nx yyzz +Bx 5 + 5Ex 4 y + 5Ex 4 z + 10Hx 3 yy + Hx 3 zz +20Lx 3 yz + 30Ox yyzz+ Cx 5 + 5Fx 4 y + 5Fx 4 z + 10Ix 3 yy + Ix 3 zz +20Mx 3 yz + 30Px yyzzand the following determinations of the coefficient letters areobtainedA + G + G = 0;D + H + O = 0;D + I + P = 0;B + H + H = 0;E + G + N = 0;G + F + P = 0;K + L + M = 0;C + I + F = 0;F + G + N = 0;F + H + O = 0.In the same way for the sixth order such determinations willgive 15, for the seventh 21, for the eighth 28 and so on.72.
In the very first formula S = A the coordinates x, y and zare clearly not intertwined. Thus the three velocities u, v and ware equal to zero, and hence this describes a quiet state of fluid.Also the pressure at an arbitrary point for different times willbe able to vary in an arbitrary manner. Indeed A is an arbitraryfunction of time and, for a given time t, the pressure at the pointλ is p = C − 2dAdt − z. Through this formula is revealed thestate of the fluid, when it is subjected at an arbitrary instant toarbitrary forces, which nevertheless balance each other, so thatno motion in the fluid can arise from them: where it happens, ifthe fluid is enclosed in a vase from which it can nowhere escape,it is also compressed by suitable forces inside.u = A;v=Bandw = C.where we applied the values taken from §.
32.76. Then the three latter equations, combined with theformer, are reduced to these:QR + qr + Φρ = 0; QS + qs + Φσ = 0 andRS + r s + ρσ = 0.36 The printed version, but not Euler, 1752, has a missing BB in the formula.37 In both the printed version and in Euler, 1752, the first velocity componentis mistakenly denoted by α.38 In volume.1852L. Euler / Physica D 237 (2008) 1840–1854Moreover, if the values assigned in §. 34 are substituted for theletters Q, R, S, q, r , s, Φ, ρ, σ and the higher-order terms forthe rests are neglected, the three former will give1 = 1 + 2Ldt; l + M = 0;1 = 1 + 2mdt; λ + N = 0;1 = 1 + 2νdt; µ + n = 0,so that we have L = 0 m = 0 and ν = 0, M = −l, N = −λ andn = −µ.77. Thus, the three velocities of this point λ would have tobe compared to the condition that the following hold39udu + vdv + wdw + dUpar. AB = 2d xdydzdyudu + vdv + wdw + dUpar.
AL = 2d xdydz.dz80. Let us set now uu + vv + ww + 2U = T. The function Tdepends on the coordinates x. y, z; take it at a given instant oftime t:42dT = Kd x + kdy + κdz.The three moving forces of the element d xdydz are43du = ldy + λdz;dv = −ld x + µdz;dw = −λd x − µdy.par. AL = Kd xdydzBut the second condition demands a motion of the fluid suchthat l = M, λ = N and n = µ; hence all the coefficients l,λ and µ vanish; also the velocities u, v and w will take thesame value everywhere in the fluid. Therefore it is plain thatthe motion of the fluid cannot correspond to solid-body motionother than pure translational.78. To ascertain the effect of the forces which act from theoutside upon the fluid, it is first necessary to determine thoseforces40 which are required for effecting the motion which wehave assumed to exist in the fluid. These are equivalent to theforces which in fact work upon the fluid; furthermore we haveseen above in §.
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