Some generalizations in steady one-dimensional gas dynamics

Abstract

In a stationary weak sound wave the four gas dynamical quantities, entropy, stagnation enthaply, mass flow per unit area and impulse per unit area are constant. The six processes in which pairs of these four variables are kept constant are studied for the case of any single phase fluid or mixture of fluids in equilibrium, and it is shown that the remaining variables are stationary at sonic points. Such points are shown to occur once only in each process and are identified as maxima or minima, on the assumption that the fluid is a normal one which expands on heating at constant pressure and for which (∂²p/∂v²)_s is positive.

Newton's theory of sound assumed that the fluid temperature was invariant. Across a stationary Newtonian sound wave, the four quantities, temperature, mass flow and impulse per unit area, and a dynamical variant of the Gibbs function are constant. The six processes in which pairs of these four variables are kept constant are studied, and it is shown that the remaining variables are stationary at points where the fluid speed equals Newton's sound velocity. These points are shown to occur once only in each process and are identified as maxima or minima with the further proviso that (∂²p/∂v²)_T is positive.

The two sets of processes have one member in common, that usually referred to as the Rayleigh line. The Fanno line and the usual isentropic ‘nozzle’ process also belong to the first set.

Finally, the variation of stagnation temperature and pressure in some of the processes and in stationary shocks is investigated.