8 Principles of the motion of fluids ∗

Here are treated the elements of the theory of the motion of fluids in general, 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 1 carefully 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 2 if the portion expanded into a larger volume, the continuity of the particles was violated, these were dispersed and no longer clinged 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, the 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, till 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. In the second part the author proceeds to the determination of the motion of a fluid produced by …