Real processing V The balance of entropy production

Abstract

In part A of the paper we start from the fact that balances can be brought to a common form. The same holds for the entropy production. We introduce the latter into the former and obtain a general balance of entropy production which allows to derive special balances for any process. In particular, we treat the momentum balance, including the Navier–Stokes equation. The treatment of the convection term leads us to the convection condition which has to be fulfilled by the function space of the velocity. By the way, we show how the principle of minimal entropy production can be derived from balances for stationary processes and generalised for non-stationary processes. The balance of entropy production allows us to treat turbulence without models. We give a derivation and generalisation of the Orr–Sommerfeld equation and show that for treating turbulence the Orr–Sommerfeld equation in any form can be replaced by the entropy production balance with considerable simplifications. In part B we give a summary of the publications on real processing l–V. They report a development of irreversible thermodynamics in the years from 1996 to 2001, which seems to be an essential contribution to mathematical physical irreversible thermodynamics bridging the statement of Helmholtz and Rayleigh 1868 over the Principle of Minimal Entropy Production of Prigogine 1947 and his and Glansdorffs Evolution Principle 1964 to our time 2001. The most decisive points seem to be the generalisation of the Helmholtz–Rayleigh condition by the Clebsch-Ansatz, the derivation of school-book laws like Hagen–Poiseuille, the treatment of convective processes, the determination of turbulent eigenstates and perhaps the entropy statements. The intention for the future is the application of synergetics, in particular the Master- and the Fokker-Planck equation, to transport processes, in particular, to the determination of the complete turbulent states composed by the eigenstates obtained from the balance of entropy production! In the appendix we show that the special continuity condition can be applied under certain conditions even if only the general one would hold.