  The Glansdorff and Prigogine General Evolution Criterion (GEC) is an
inequality that holds for macroscopic physical systems obeying local
equilibrium and that are constrained under timeindependent boundary conditions.
The latter, however, may prove overly restrictive for many applications
involving fluid flow in physics, chemistry and biology. We therefore analyze in
detail a physically more-encompassing evolution criterion for time-dependent
convective viscous flow with time-dependent boundary conditions. The result is
an inequality involving the sum of a bulk volume and a surface contribution,
and reduces to the GEC if and only if the surface term is zero. We use the
closed-form analytical solution of the starting flow problem in straight
cylindrical pipes to confirm this extended general evolution criterion. We next
validate the starting flow problem and evolution criterion numerically.
Numerical methods are used to validate the extended evolution criterion in
non-fully developed flows inside complex geometries with curvature and torsion,
such as encountered in sections of helical pipes. Knowledge of only the
algebraic sign of the surface contribution is sufficient for predicting how the
volume thermodynamic forces evolve in time and how the system approaches its
non-equilibrium stationary state consistent with the boundary conditions.
