*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: etr258.pdf 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: etr396p1.pdf,
and Chapters 4-6 : etr396p2.pdf . 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

Part III is now complete. The fluid is taken to be water. Chapter 1 :
etr409ch1.pdf , 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 : etr409ch3pdf
, 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 etr409ch4pdf , 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 etr409ch5pdf
. 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 etr424ch3.pdf
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 etr424ch4.pdf 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 etr424ch5.pdf 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 etr424ch6.pdf 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.