  In this paper, we use a generic and general variational method to obtain
solutions to the flow of generalized Newtonian fluids through circular pipes
and plane slits. The new method is not based on the use of the Euler-Lagrange
variational principle and hence it is totally independent of our previous
approach which is based on this principle. Instead, the method applies a very
generic and general optimization approach which can be justified by the
Dirichlet principle although this is not the only possible theoretical
justification. The results that were obtained from the new method using nine
types of fluid are in total agreement, within certain restrictions, with the
results obtained from the traditional methods of fluid mechanics as well as the
results obtained from the previous variational approach. In addition to being a
useful method in its own for resolving the flow field in circular pipes and
plane slits, the new variational method lends more support to the old
variational method as well as for the use of variational principles in general
to resolve the flow of generalized Newtonian fluids and obtain all the
quantities of the flow field which include shear stress, local viscosity, rate
of strain, speed profile and volumetric flow rate. The theoretical basis of the
new variational method, which rests on the use of the Dirichlet principle, also
provides theoretical support to the former variational method.
