Euler's Ideal Fluids:

Euler spent some time occasionally investigating the theory of ideal fluids, at a time when the physical properties of liquids such as viscosity and surface tension were not fully understood; such ideal fluids were ideally suited to a calculus based investigation. At present we have the file that we will call Part I: which includes some of the basic equations such as that for continuity for ideal fluids, without the messy contributions due to viscosity and surface tension, etc.,  and which are now presented by means of vector analysis. The following paper taken as Part II : E396 is presented here in two parts for convenience Chapters 1-3:  , and Chapters 4-6 : Here Euler is concerned more with establishing basic equations rather than solving individual problems, which were of course fundamental to the further development of the subject. Horace Lamb, for example, in his initial Treatise on the Motion of Fluids published in 1879, begins his work with the foundations laid by Euler in these papers; this book incidently was written here in Adelaide.

Part III is now complete. The fluid is taken to be water.  Chapter 1 :  , regarding fluid flow in one dimension or in tubes, which uses two fundamental equations described previously. Chapter 2  :  etr409ch2.pdf ,  considers the flow along capillary tubes of various geometries. Chapter 3  :  considers the detailed solutions of problems regarding the flow of water in various kinds of tubes, U tubes, etc., and the use of pumps as simple machines in raising water. Chapter 4   performs a detailed analysis of the use of individual pumps raising water with the aid of pistons, initially alone but finally with 2 or 4 pumps working together; the use of Euler's analysis might be able to be applied to the working of the human heart considered as 2 pumps working out of phase with each other, as considered here.  Chapter 5 is rather surprising  Here Euler is concerned with the properties of water flowing inside a closed glass circular tube set in the vertical plane; heat is supplied at one end of the horizontal diameter and cooling at the other end, and the laminar flow of water flowing as a convection current is analysed, making use of the density changes of water on heating or cooling water; later the locations of heating and cooling are changed to various angles, including the vertical; in the course of these experimental observations Euler and his experiments observed the anomalous expansion of water a few degrees above freezing point, and also the change of the convection currents from laminar to turbulent flow under certain conditions.

Part IV. Here the fluid is taken to be air; immediately numerous difficulties arise, as Euler attempts to use the same two basic equations developed for liquid flow. At present we offer Chapter I :  etr424ch1.pdf ; at this stage Euler was unable to provide a differential equation describing the motion of a small packet or particle of air, and made use of scales or orders of magnitude of the observable effects, such as pressure and density changes associated with the translations involved.  Chapter 2 is now presented in which a sound pulse or pulses are sent down a hollow tube  etr424ch2.pdf  ; still more difficulties arise as Euler considers the fundamental properties of sound waves, such as may arise as pulses ; some of the properties of the pulses , both rarefied and compressed, seem to be in order; other assumptions appear difficult to verify experimentally, and are derived from his desire to use the basic equations already established, in some form. Chapter 3   is an application of the theory to the notes produced by flutes; recall that Euler played the flute; it is clear now how Euler's theory is deficient, and although some properties of air pulses in tubes are shown correctly, the fact that the pressure and density changes induced are linear is a major deficiency; essentially the wave equation should be solved, resulting in s.h.m., as we well know. Chapter 4   is now complete; this chapter considers the continuous motion through tubes of varying cross-section, such as conoidal shapes and higher order curves; the analysis becomes increasingly difficult for higher order curves. Chapter 5   is now complete; this chapter considers the continuous motion through tubes of increasing  cross-section, in the form of hyperbolic conoids; the analysis becomes increasingly difficult for these higher order curves. In general it is found that such curves are not suitable for musical instruments, unless the air flow is minimal, according to Euler's analysis. Chapter 6   is now complete ; a comparison is made between conoidal and cylindrical tubes, either open or closed at each end; this is the last chapter in this book.