  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.
