e. 3 Eq. 1-4 
and wall conduction are negligible. In any numerical calculation it is also 
necessary to have equation of state and specific heat data for the materials 
involved. Upon these simple and unquestionably valid foundations it is pos- 
Sible to build a theory which 1s capable of predicting a considerable frac- 
tion of the observed facts and in addition providing information not yet 
capable of direct observation. 
2. Basic Differential Equations 
In regions of space and time in which no discontinuities occur, the 
basic equations of conservation of mass, momentum and energy can be expressed 
as differential equations. 
a. Conservation of mass, Consider a very thin slice of the mterial 
in the tube. Let the thickness of the slice be §: its pressure P, its 
density @ and the absolute velocity U at any instant. The position of the 
Slice along the length of the tube is given by the coordinate x. If the 
matter in the back face of the slice moves forward with the velocity U while 
that in the front face has the velocity y+ ke € , then the thickness of 
the slice will change with time as follows: 
dS _ eeu (1) 
The slice is a definite portion of mtter so that its mass must remin 
constant. Therefore; 
d ;. = 
a (80) =o, (2) 
or, combining equations (1) and (2), 
Ne cat (3) 
at ae? 
an equation which is an expression of the law of conservation of matter. 
b. Conservation of momentum. Newton's Jaw of motion can be expressed 
as follows. The force on the back face of the slice is P (for unit cross- 
section) while the force on the front face is -P- iZP 8 . Consequently the 
law of motion becomes ee 
fend ara (4) 
where ‘ has been cancelled from both sides. 
c. Conservation of energy. The law of conservation of energy can be 
stated in the form; the increase in the internal energy and kinetic energy 
of the slice in unit time equals the net work done on the slice by the forces 
on its faces in that time. The work done on the back face is the force times 
