For his whole life Euler was interested in fluids and fluid mechanics, especially their applications to shipbuilding and navigation. He first wrote on fluid mechanics in his Paris Prize essay of 1727, [E4], Meditationes super problemate nautico, de implantatione malorum …, (Thoughts about a navigational problem on the placement of masts), an essay that earned the young Euler an acessit, roughly an honorable mention, from the Paris Academy. Euler's last book, Théeorie complete de la construction et de la manoeuvre des vaisseaux, (Complete theory of the construction and maneuvering of ships) [E426], published in 1773, also dealt with practical applications of fluid mechanics. We could summarize Euler's contribution to the subject by saying that he extended the principles described by Archimedes in On floating bodies from statics to dynamics, using calculus and partial differential equations. Indeed, he made some of the first practical applications of partial differential equations.

Euler's work is very well known among people who study fluid mechanics. Several of the fundamental equations that describe non-turbulent fluid flow are known simply as “the Euler equations,” and the problem of extending those equations to turbulent flow, the Navier-Stokes equations, is one of the great unsolved problems of our age.

This month, we are going to look at [E258], Principia motus fluidorum, (Principles of the motion of fluids), in which Euler derives the partial differential equations that describe two of the basic properties of fluids:

1. the differential equations for the continuity of incompressible fluids, and

2. the dynamical equations for ideal incompressible fluids.

We will examine the first of these derivations in detail.

Euler begins by warning us how much more complicated fluids are than solids. If we know the motions of just three points of a rigid solid, then we can determine the motion of the entire body. For fluids, though, different parts of the fluid can have very different motions. Even knowing the flows of many points still leaves infinitely many possible flows.