## E258 -- Principia motus fluidorum

(Principles of the motion of fluids)

Summary:
##### (based on Clifford A. Truesdell's introduction to Opera Omnia Series II, Volume 12)
This work is important in the history of rational mechanics. In it, Euler treats the theory of the motion of fluids in general: "given a mass of fluid, either free or confined in vessels, when an arbitrary motion shall have been impressed upon it, and meanwhile it is acted upon by arbitrary forces, the motion in which its several particles are to travel shall be determined, and at the same time the pressure with which the several parts act, as well mutually upon each other as also upon the sides of a vessel, shall be ascertained." Euler reduces the whole theory to pure analysis. This work is broken up into two parts.

In part 1, Euler shows that the motion of particles in fluids is much less restricted than that of particles in solids, although this is not to say that the particles of a fluid are not forced to satisfy certain laws. He assumes that a fluid cannot be forced into a smaller space and also that it cannot have its continuity interupted, along with the assumption that each part of the fluid is incompressible.

In part 2, Euler turns to the working of the forces that produce the actual motion of a fluid. He argues that any fluid that completely fills a closed vessel must stay in equilibrium, even when it is subject to arbitrary forces, since the pressure changes with time. This part is famous because it contains the derivations of the continuity equation u/x + v/y + w/z = 0 and the dynamical equations for ideal incompressible fluids, thus separating for the first time the kinematical from the dynamical aspects of the theory of continua. This part of the paper also contains many new ideas. Among them are the following:
• secular expansion;
• a geometrical interpretation for the sum of the r-rowed principal minors of any n × n matrix;
• differential equations of hydrostatics; and
• a really general treatment of the theory of flow in narrow tubes;
In addition, he proves that a necessary condition for the potential flow in a homogeneous incompressible fluid is the completeness of the differential Qdx + qdy + j dz. He also generalizes the earlier theory of friction in tubes and gives a rule for finding the most general homogeneous harmonic degree n polynomial. Euler proves that the only rigid potential motion is a state of uniform translation.
###### a.s.
Note: This is in fact the first part of a three-part treatise. For the rest of Euler's "De motu Flouidorum" see:
• E396 -- Sectio secunda de principiis motus fluidorum
• E409 -- Sectio tertia de motu fluidorum

According to C. G. J. Jacobi, a treatise with thie title: "De motu fluidorum in genere" was read to the Berlin Academy on August 31, 1752.

Publication:
• Originally published in Novi Commentarii academiae scientiarum Petropolitanae 6, 1761, pp. 271-311
• Opera Omnia: Series 2, Volume 12, pp. 133 - 168
• According to Jacobi, the manuscript of the printed treatise can be found in the archive of the Berlin Academy.
Documents Available:
• Original publication: E258
• A translation into English has been done by Enlin Pan.

• The Euler Archive attempts to monitor current scholarship for articles and books that may be of interest to Euler Scholars. Selected references we have found that discuss or cite E258 include:
• Craik ADD., "The origins of water wave theory." Annual Review of Fluid Mechanics, 36, pp. 1-28 (2004).
• Greenberg JL., "Fontaine, Alexis Fluxio-differential method and the origins of the calculus of several-variables." Annals of Science, 38 (3), pp. 251-290 (1981).