  We continue our investigation to the use of the variational method to derive
flow relations for generalized Newtonian fluids in confined geometries. While
in the previous investigations we used the straight circular tube geometry with
eight fluid rheological models to demonstrate and establish the variational
method, the focus here is on the plane long thin slit geometry using those
eight rheological models, namely: Newtonian, power law, Ree-Eyring, Carreau,
Cross, Casson, Bingham and Herschel-Bulkley. We demonstrate how the variational
principle based on minimizing the total stress in the flow conduit can be used
to derive analytical expressions, which are previously derived by other
methods, or used in conjunction with numerical procedures to obtain numerical
solutions which are virtually identical to the solutions obtained previously
from well established methods of fluid dynamics. In this regard, we use the
method of Weissenberg-Rabinowitsch-Mooney-Schofield (WRMS), with our adaptation
from the circular pipe geometry to the long thin slit geometry, to derive
analytical formulae for the eight types of fluid where these derived formulae
are used for comparison and validation of the variational formulae and
numerical solutions. Although some examples may be of little value, the
optimization principle which the variational method is based upon has a
significant theoretical value as it reveals the tendency of the flow system to
assume a configuration that minimizes the total stress. Our proposal also
offers a new methodology to tackle common problems in fluid dynamics and
rheology.
