Simultaneous Extrema in the Entropy Production for Steady-State Fluid Flow in Parallel Pipes

Steady-state flow of an incompressible fluid in parallel pipes can simultaneously satisfy two contradictory extremum principles in the entropy production, depending on the flow conditions. For a constant total flow rate, the flow can satisfy (i) a pipe network minimum entropy production (MinEP) principle with respect to the flow rates, and (ii) the maximum entropy production (MaxEP) principle of Ziegler and Paltridge with respect to the choice of flow regime. The first principle - different to but allied to that of Prigogine - arises from the stability of the steady state compared to non-steady-state flows; it is proven for isothermal laminar and turbulent flows in parallel pipes with a constant power law exponent, but is otherwise invalid. The second principle appears to be more fundamental, driving the formation of turbulent flow in single and parallel pipes at higher Reynolds numbers. For constant head conditions, the flow can satisfy (i) a modified maximum entropy production (MaxEPMod) principle of \v{Z}upanovi\'c and co-workers with respect to the flow rates, and (ii) an inversion of the Ziegler-Paltridge MaxEP principle with respect to the flow regime. The interplay between these principles is demonstrated by examples.Comment: Revised version 2; 5 figure

THE TIGHT-BINDING APPROACH TO THE DIELECTRIC RESPONSE IN THE MULTIBAND SYSTEMS

Starting from the random phase approximation for the weakly coupled multiband tightly-bounded electron systems, we calculate the dielectric matrix in terms of intraband and interband transitions. The advantages of this representation with respect to the usual plane-wave decomposition are pointed out. The analysis becomes particularly transparent in the long wavelength limit, after performing the multipole expansion of bare Coulomb matrix elements. For illustration, the collective modes and the macroscopic dielectric function for a general cubic lattice are derived. It is shown that the dielectric instability in conducting narrow band systems proceeds by a common softening of one transverse and one longitudinal mode. Furthermore, the self-polarization corrections which appear in the macroscopic dielectric function for finite band systems, are identified as a combined effect of intra-atomic exchange interactions between electrons sitting in different orbitals and a finite inter-atomic tunneling.Comment: 20 pages, LaTeX, no figure