Euler L. Principles of the motion of fluids

Physica D 237 (2008) 1840–1854www.elsevier.com/locate/physdPrinciples of the motion of fluidsILeonhard EulerAvailable online 5 May 2008AbstractThe elements of the theory of the motion of fluids in general are treated here, the whole matter being reduced to this: given a mass of fluid,either free or confined in vessels, upon which an arbitrary motion is impressed, and which in turn is acted upon by arbitrary forces, to determine themotion carrying forward each particle, and at the same time to ascertain the pressure exerted by each part, acting on it as well as on the sides of thevessel. At first in this memoir, before undertaking the investigation of these effects of the forces, the Most Famous Author1 carefully evaluates allthe possible motions which can actually take place in the fluid.

Indeed, even if the individual particles of the fluid are free from each other, motionsin which the particles interpenetrate are nevertheless excluded, since we are dealing with fluids that do not permit any compression into a narrowervolume. Thus it is clear that an arbitrary small portion of fluid cannot receive a motion other than the one which constantly conserves the samevolume; even though meanwhile the shape is changed in any way.

It would hold indeed, as long as no elementary portion would be compressed atany time into a smaller volume; furthermore2 if the portion expanded into a larger volume, the continuity of the particles was violated, these weredispersed and no longer clung together, such a motion would no longer pertain to the science of the motion of fluids; but individual droplets wouldseparately perform their motion. Therefore, this case being excluded, motion of the fluids must be restricted by this rule that each small portionmust retain for ever the same volume; and this principle restricts the general expressions of motion for elements of the fluid.

Plainly, considering anarbitrary small portion of the fluid, its individual points have to be carried by such a motion that, when at a moment of time they arrive at the next location, until then they occupy a volume equal to the previous one; thus if, as usual, the motion of a point is decomposed parallel to fixed orthogonaldirections, it is necessary that a certain established relation hold between these three velocities, which the author has determined in the first part.In the second part the author proceeds to the determination of the motion of a fluid produced by arbitrary forces, in which matter the wholeinvestigation reduces to this that the pressure with which the parts of the fluid at each point act upon one another shall be ascertained; whichpressure is denoted most conveniently, as customary for water, by a certain height; this is to be understood thus, that each element of the fluidsustains a pressure the same as if were pressed by a heavy column of the same fluid, whose height is equal to that amount.

Thus, in such wayin each point of the fluid the height referring to the state of the pressure will be given; since it is not equal to the one in the neighbourhood, itwill perturb the motion of the elements. But this pressure depends as well on the forces acting on each element of the fluid, as on those, acting inthe whole mass; thus, by the given forces, the pressure in each point and thereupon the acceleration of each element – or its retardation – can beassigned for the motion, all which determinations are expressed by the author through differential formulas. But, in fact, the full development ofthese formulas mostly involves the greatest difficulties.

But nevertheless this whole theory has been reduced to pure analysis, and what remains tobe completed in it depends solely upon subsequent progress in Analysis. Thus it is far from true that purely analytic researches are of no use inapplied mathematics; rather, important additions in pure analysis are now required.c 2008 Published by Elsevier B.V.I.

First partI This is an English adaptation by Walter Pauls of Euler’s memoir ‘Principiamotus fluidorum’ (Euler, 1756–1757). Updated versions of the translation maybecome available at http://www.oca.eu/etc7/EE250/texts/euler1761eng.pdf.For a detailed presentation of Euler’s fluid dynamics papers, cf. Truesdell,1954, which has also been helpful for this translation. Euler’s work is discussedin the perspective of eighteenth century fluid dynamics research by Darrigoland Frisch (2008). The help of O.

Darrigol, U. Frisch, G. Grimberg and G.Mikhailov is also acknowledged. Explanatory footnotes and references havebeen supplied where necessary; Euler’s memoir had neither footnotes nor a listof references.1 Summaries, which at that time were not placed at the beginning of thecorresponding paper, were published under the responsibility of the Academy;c 2008 Published by Elsevier B.V.0167-2789/$ - see front matter doi:10.1016/j.physd.2008.04.0191. Since liquid substances differ from solid ones by the factthat their particles are mutually independent of each other, theycan also receive most diverse motions; the motion performed byan arbitrary particle of the fluid is not determined by the motionthe presence of the words “Most Famous Author”, rather common at the time,cannot be taken as evidence that Euler usually referred to himself in this way.2 In the original, we find “verum quoniam”; the literal translation “sinceindeed” does not seem logically consistent.L.

Euler / Physica D 237 (2008) 1840–1854of the remaining particles to the point that it cannot move inany other way. The matter is very different in solid bodies,which, if they were inflexible, would not undergo any changein their shape; in whatsoever way they be moved, each of theirparticles would constantly keep the same location and distancewith respect to other particles; it thus follows that, the motionof two or, if necessary, three of all the particles being known,the motion of any other particle can be defined; furthermorethe motion of two or three particles of such a body cannot bechosen at will, but must be constrained in such a way that theseparticles preserve constantly their positions with respect to eachother.32.

But if, moreover, solid bodies are flexible, the motionof each particle is less constrained: because of bending, thedistance as well as the relative position of each particle canbe subject to change. However, the manner itself of bendingconstitutes a certain law which various particles of such a bodyhave to obey in their motion: certainly what has to be takencare of is that the parts that experience in their neighbourhoodsuch a strong bending with respect to each other are neither tornapart from the inside nor penetrate into each other. Indeed, aswe shall see, impenetrability is demanded for all bodies.3.

In fluid bodies, whose particles are united amongthemselves by no bond, the motion of each particle is muchless restricted: the motion of the remaining particles is notdetermined from the motion of any number of particles. Evenknowing the motion of one hundred particles, the future motionpermitted to the remaining particles still can vary in infinitelymany ways. From which it is seen that the motion of thesefluid particles plainly does not depend on the motion of theremaining ones, unless it be enclosed by these so that it isconstrained to follow them.4. However, it cannot happen that the motion of all particlesof the fluid suffers no restrictions at all. Furthermore, onecannot at will invent a motion that is conceived to occur foreach particle. Since, indeed, the particles are impenetrable,it is immediately clear that a motion cannot be maintainedin which some particles go through other particles and,accordingly, penetrate each other: also, because of this reasonsuch motion certainly cannot be conceived to occur in the fluid.Therefore, infinitely many motions must be excluded; after theirdetermination the remaining ones are grouped together.

It isseen worthwhile to define them more accurately regarding theproperty which distinguishes them from the previous ones.5. But before the motion by which the fluid is agitated at anyplace can be defined, it is necessary to see how every motion,which can definitely be maintained in this fluid, be recognized:these motions, here, I will call possible, which I will distinguishfrom impossible motions which certainly cannot take place.

Wemust then find what characteristic is appropriate to possiblemotions, separating them from impossible ones. When this isdone, we shall have to determine which one of all possiblemotions in a certain case ought actually to occur. Plainly wemust then turn to the forces which act upon the water, so that3 Here Euler refers to the motion of rigid solid bodies treated previously inEuler, 1750.1841the motion appropriate to them may be determined from theprinciples of mechanics.6. Thus, I decided to inquire into the character of the possiblemotions, such that no violation of impenetrability can occurin the fluid. I shall assume the fluid to be such as never topermit itself to be forced into a lesser space, nor should itscontinuity be interrupted.

Once the theory of fluids has beenadjusted to fluids of this nature, it will not be difficult to extendit also to those fluids whose density is variable and which donot necessarily require continuity.47. If, thus, we consider an arbitrary portion in such a fluid,the motion, by which each of its particles is carried has to be setup so that at each time they occupy an equal volume.