Thermodynamic properties of a perfect gas

Most of the problems studied in Chapters 4 and 5 did not involve considerations of compressibility, the only exception being §4.4 where the static equilibrium of compressible fluids was considered. We now wish to study the dynamics of compressible fluids. We saw in §4.7 that the irrotational flow of an incompressible fluid around an object gives rise to the Laplace equation, which is an elliptic partial differential equation. The similar problem of high-speed flow of a compressible fluid around an object can give rise to a hyperbolic partial differential equation, provided the flow speed is larger than the sound speed. In other words, the mathematical character of the equations governing high-speed compressible flows can be quite different from that of the equations governing incompressible flows (although we begin from the same hydrodynamic equations!), and consequently the solutions can also be of profoundly different nature. We give here only a brief introduction to gas dynamics, which is the branch of hydrodynamics dealing with compressible flows. Even the subject of supersonic flows past solid objects just mentioned above, which leads to a two-dimensional problem, is not treated in this elementary introduction. See Landau and Lifshitz (1987, Chapter XII) or Liepmann and Roshko (1957, Chapter 8) for a discussion of this subject. We restrict ourselves to a discussion of one-dimensional gas dynamics problems only.

The mathematical analysis of compressible fluids becomes more manageable if we assume the fluid to behave as a perfect gas.