On the thermodynamic stability of steady-state adiabatic systems

This paper begins by reviewing Bethe's (1942) work on the subject. He considered the propagation of a normal shock wave in a medium with an arbitrary equation of state. Difficulties arise if one attempts to extend his theory to systems containing plane oblique shocks or the reflection or refraction of such shocks. The object of the present paper is to resolve these difficulties. General conditions for the local thermodynamic equilibrium and thermodynamie stability, of a non-equilibrium system in steady-state, adiabatic, flow are summarized by the principle of maximum entropy production, which gives \[ \Delta s\geqslant 0;\quad {\rm d}(\Delta s)= 0;\quad {\rm d}^2(\Delta s) < 0, \] for ht, constant, where s is the specific entropy and ht is the specific total enthalpy; it is deduced from the second law. Conversely the consequences of Δs < 0, d(Δs) ≠ 0, d2(Δs) = 0, are discussed and may lead to either an impossibility or to some form of instability such as unsteadiness, or a change in the structure of the system (a catastrophe).