Euler L. Principles of the motion of fluids (794385), страница 2

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When thisoccurs in separate portions, any expansion into a larger volume,or compression into a smaller volume is prohibited. And, ifwe turn attention to this only property, we can have only suchmotion that the fluid is not permitted to expand or compress.Furthermore, what is said here about arbitrary portions of thefluid, has to be understood for each of its elements; so that thevolume of its elements must constantly preserve its value.8.

Thus, assuming that this condition holds, let an arbitrarymotion be considered to occur at each point of the fluid;moreover, given any element of the fluid, consider the brieftranslations of each of its boundaries. In this manner thevolume, in which the element is contained after a very shorttime, becomes known. From there on, this volume is posed tobe equal to the one occupied previously, and this equation willprescribe the calculation of the motion, in so far as it will bepossible.

Since all elements occupy the same volumes duringall periods of time, no compression of the fluid, nor expansioncan occur; and the motion is arranged in such a way that thisbecomes possible.9. Since we consider not only the velocity5 of the motionoccurring at every point of the fluid but also its direction, bothaspects are most conveniently handled, if the motion of eachpoint is decomposed along fixed directions. Moreover, this decomposition is usually carried out with respect to two or threedirections6 : the former is appropriate for decomposition, if themotion of all points is completed in the same plane; but if theirmotion is not contained in the same plane, it is appropriate todecompose the motion following three fixed axes. Because thelatter case is more difficult to treat, it is more convenient to begin the investigation of possible motions with the former case;once this has been done, the latter case will be easily completed.10.

First I will assign to the fluid two dimensions in such away that all of its particles are now not only found with certaintyin the same plane, but also their motion is performed in it.Let this plane be represented in the plane of the table (Fig. 1),let an arbitrary point l of the fluid be considered, its positionbeing denoted by orthogonal coordinates AL = x and Ll = y.The motion is decomposed following these directions, giving a4 See the English translation of “General laws of the motion of fluids” inthese Proceedings.5 Meaning here the absolute value of the velocity.6 Depending on the dimension: Euler treats both the two- and the threedimensional cases.1842L. Euler / Physica D 237 (2008) 1840–1854Fig.

1.velocity lm = u parallel to the axis AL and ln = v parallel tothe√other axis AB: so that the true future velocity of this pointis (uu + vv), and its direction with respect to the axis AL isinclined by an angle with the tangent uv .11. Since the state of motion, presented in a way which suitseach point of the fluid, is supposed to evolve, the velocitiesu and v will depend on the position l of the point and willtherefore be functions of the coordinates x and y. Thus, we putupon a differentiationdu = Ld x + ldyanddv = Md x + mdy,which differential formulas, since they are complete,7 satisfydldMdmfurthermore dLdy = d x and dy = d x .

Here it is noted thatin such expression dLdy , the differential of L itself or dL, isunderstood to be obtained from the variability with respect to y;in similar manner in the expression dl/d x, for dl the differentialof l itself has to be taken, which arises if we take x to vary.12.

Thus, it is in order to be cautious and not to take indl dMdmsuch fractional expressions dLdy , d x , dy , and d x the numeratorsdL, dl, dM, and dm as denoting the complete differentialsof the functions L, l, M and m; but they designate suchdifferentials constantly that are obtained from variation ofonly one coordinate, obviously the one, whose differentialis represented in the denominator; thus, such expressionswill always represent finite and well defined quantities.Furthermore, in the same way are understood L = ddux , l = dudy ,M = ddvx and m = dvdy ; which notation of ratios has been usedfor the first time by the most enlightened Fontaine,8 and I willalso apply it here, since it gives a non negligible advantage ofcalculation.13.

Since du = Ld x + ldy and dv = Md x + mdy, here itis appropriate to assign a pair of velocities to the point which is7 Exact differentials.8 A paper “Sur le calcul intégral” containing the notation d f for the partialdxderivative of f with respect to x was presented by Alexis Fontaine desBertins to the Paris Academy of Sciences in 1738, but it was published onlya quarter of a century later (Fontaine, 1764). Nevertheless, Fontaine’s paperwas widely known among mathematicians from the beginning of the 1740s,and, particularly, was discussed in the correspondence between Euler, DanielBernoulli and Clairaut; cf.

Euler, 1980: 65–246.at an infinitely small distance from the point l; if the distanceof such a point from the point l parallel to the axis AL is d x,and parallel to the axis AB is dy, then the velocity of this pointparallel to the axis AL will be u + Ld x + ldy; furthermore,the velocity parallel to the other axis AB is v + Md x + mdy.Thus, during the infinitely short time dt this point will becarried parallel to the direction of the axis AL the distancedt (u + Ld x + ldy) and parallel to the direction of the otheraxis AB the distance dt (v + Md x + mdy).14.

Having noted these things, let us consider a triangularelement lmn of water, and let us seek the location into whichit is carried by the motion during the time dt. Let lm be theside parallel to the axis AL and let ln be the side parallel tothe axis AB: let us also put lm = d x and ln = dy; or let thecoordinates of the point m be x + d x and y; the coordinates ofthe point n be x and y + dy. It is plain, since we do not definethe relation between the differentials d x and dy, which can betaken negative as well as positive, that in thought the wholemass of fluid may be divided into elements of this sort, so thatwhat we determine for one in general will extend equally to all.15.

To find out how far the element lmn is carried duringthe time dt due to the local motion, we search for the pointsp, q and r , to which its vertices, or the points l, m and n aretransferred during the time dt. SinceVelocity w.r.t. AL=Velocity w.r.t. AB=of point luvof point mu + Ld xv + Md xof point nu + ldyv + mdyin the time dt the point l reaches the point p, chosen such that:AP − AL = udtandP p − Ll = vdt.Furthermore, the point m reaches the point q, such thatAQ − AM = (u + Ld x)dt andQq − Mm = (v + Md x)dt.Also, the point n is carried to r , chosen such thatAR − AL = (u + ldy)dtandRr − Ln = (v + mdy)dt.16.

Since the points l, m and n are carried to the points p,q and r , the triangle lmn made of the joined straight lines pq,pr and qr , is thought to be arriving at the location defined bythe triangle pqr . Because the triangle lmn is infinitely small,its sides cannot receive any curvature from the motion, andtherefore, after having performed the translation of the elementof water lmn in the time dt, it will conserve the straight andtriangular form. Since this element lmn must not be eitherextended to a larger volume, nor compressed into a smaller one,the motion should be arranged so that the volume of the trianglepqr is rendered to be equal to the area of the triangle lmn.17.

The area of the triangle lmn, being rectangular at l, is1d2 xdy, value to which the area of the triangle pqr should beput equal. To find this area, the pair of coordinates of the pointsp, q and r must be examined, which are:AP = x + udt;AQ = x + d x + (u + Ld x)dt;AR = x + (u + ldy)dt; P p = y + vdtQq = y + (v + Md x)dt, Rr = y + dy + (v + mdy)dt.1843L. Euler / Physica D 237 (2008) 1840–1854Then, indeed, the area of the triangle pqr is found from the areaof the succeeding trapezoids, so thatpqr = P pr R + RrqQ − P pqQ.Since these trapezoids have a pair of sides parallel to andperpendicular to the base AQ, their areas are easily found.18.

Plainly, these areas are given by the expressions1PR(P p + Rr )21RrqQ = RQ(Rr + Qq)21P pqQ = PQ(P p + Qq).2P pr R =Fig. 2.By putting these together we find:∆ pqr =111PQ.Rr − RQ.P p − PR.Qq.222Let us set for brevityAQ = AP + Q;AR = AP + R;Qq = P p + q;andRr = P p + r,so that PQ = Q, PR = R, and RQ = Q − R, and we have∆ pqr = 12 Q(P p + r ) − 12 (Q − R)P p − 21 R(P p + q) or∆ pqr = 12 Q.r − 21 R.q.19.

Truly, from the values of the coordinates representedbefore it follows thatQ = d x + Ld xdt;R = ldydt;q = Md xdtr = dy + mdydt,upon the substitution of these values, the area of the triangle isobtained11d xdy(1 + Ldt)(1 + mdt) − Ml d xdydt 2 ,221pqr = d xdy(1 + Ldt + mdt + Lmdt 2 − Mldt 2 ).2pqr =orThis should be equal to the area of the triangle lmn, that is= 12 d xdy; hence we obtain the following equationLdt + mdt + Lmdt 2 − Mldt 2 = 0orL + m + Lmdt − Mldt = 0.20.

Since the terms Lmdt and Mldt are negligible for finiteL and m, we will have the equation L + m = 0. Hence, for themotion to be possible, the velocities u and v of any point l haveto be arranged such that after calculating their differentialsdu = Ld x + ldy,anddv = Md x + mdy,one has L+m = 0. Or, since L= ddux and m = dvdy , the velocitiesu and v, which are considered to occur at the point l parallel tothe axes AL and AB, must be functions of the coordinates xand y such that ddux + dvdy = 0, and thus, the criterion of possible9motions consists in this that ddux + dvdy = 0; and unless thiscondition holds, the motion of the fluid cannot take place.21.

We shall proceed identically when the motion of the fluidis not confined to the same plane. Let us assume, to investigatethis question in the broadest sense, that all particles of thefluid are agitated among themselves by an arbitrary motion,with the only law to be respected that neither condensation norexpansion of the parts occurs anywhere: in the same way, weseek which condition should apply to the velocities that areconsidered to occur at every point, so that motion be possible:or, which amounts to the same, all motions that are opposed tothese conditions should be eliminated from the possible ones,this being the criterion of possible motions.22. Let us consider an arbitrary point of the fluid λ.

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