  The theory for the non-isothermal rheology of polymer fluids proposed in [14]
used several approximations including the so-called linear gradient
approximations for the temperature field and Brownian forces. While it had the
significant advantage of dealing with linear equations, the approximations
involved may have led to several non-physical predictions. This work is a
continuation of [14] in that it obtains the corresponding non-linear
configurational probability density equation in dimensionless form without the
linear gradient approximations for the temperature field and Brownian forces.
It does so for incompressible diluted polymer solutions with polymer molecules
being modeled as FENE (F initely E xtensible N onlinear E lastic) chains. Next
we prove the existence of temperature dependent, positive variational solutions
for the probability density equation of the FENE model.
