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Steady-state flow of an incompressible fluid in parallel pipes cansimultaneously satisfy two contradictory extremum principles in the entropyproduction, 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 entropyproduction (MaxEP) principle of Ziegler and Paltridge with respect to thechoice of flow regime. The first principle - different to but allied to that ofPrigogine - arises from the stability of the steady state compared tonon-steady-state flows; it is proven for isothermal laminar and turbulent flowsin parallel pipes with a constant power law exponent, but is otherwise invalid.The second principle appears to be more fundamental, driving the formation ofturbulent flow in single and parallel pipes at higher Reynolds numbers. Forconstant head conditions, the flow can satisfy (i) a modified maximum entropyproduction (MaxEPMod) principle of \v{Z}upanovi\'c and co-workers with respectto the flow rates, and (ii) an inversion of the Ziegler-Paltridge MaxEPprinciple with respect to the flow regime. The interplay between theseprinciples is demonstrated by examples.

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