Principles of the motion of fluids

The 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 the motion 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 the vessel. At first in this memoir, before undertaking the investigation of these effects of the forces, the Most Famous Author 11Summaries, which at that time were not placed at the beginning of the corresponding paper, were published under the responsibility of the Academy; the 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. ully evaluates all the possible motions which can actually take place in the fluid. Indeed, even if the individual particles of the fluid are free from each other, motions in which the particles interpenetrate are nevertheless excluded, since we are dealing with fluids that do not permit any compression into a narrower volume. Thus it is clear that an arbitrary small portion of fluid cannot receive a motion other than the one which constantly conserves the same volume; even though meanwhile the shape is changed in any way. It would hold indeed, as long as no elementary portion would be compressed at any time into a smaller volume; furthermore 22In the original, we find “verum quoniam”; the literal translation “since indeed” does not seem logically consistent. e portion expanded into a larger volume, the continuity of the particles was violated, these were dispersed and no longer clung together, such a motion would no longer pertain to the science of the motion of fluids; but individual droplets would separately perform their motion. Therefore, this case being excluded, motion of the fluids must be restricted by this rule that each small portion must retain for ever the same volume; and this principle restricts the general expressions of motion for elements of the fluid. Plainly, considering an arbitrary 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 orthogonal directions, it is necessary that a certain established relation hold between these three velocities, which the author has determined in the first part. second part the author proceeds to the determination of the motion of a fluid produced by arbitrary forces, in which matter the whole investigation reduces to this that the pressure with which the parts of the fluid at each point act upon one another shall be ascertained; which pressure is denoted most conveniently, as customary for water, by a certain height; this is to be understood thus, that each element of the fluid sustains 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 way in 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, it will 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 in the whole mass; thus, by the given forces, the pressure in each point and thereupon the acceleration of each element–or its retardation–can be assigned for the motion, all which determinations are expressed by the author through differential formulas. But, in fact, the full development of these formulas mostly involves the greatest difficulties. But nevertheless this whole theory has been reduced to pure analysis, and what remains to be completed in it depends solely upon subsequent progress in Analysis. Thus it is far from true that purely analytic researches are of no use in applied mathematics; rather, important additions in pure analysis are now required.