Thermodynamics for a non-local viscous fluid. Part 1.

A generalization of the constitutive equation of an incompressible viscous second order fluid is considered. The presence of the histories of the strain rate tensor $mathbf{D}$ and of$ nabla mathbf{D}$ yields a non-simple material, for which the Laws of Thermodynamics assume a modified form, in terms of the internal power which is evaluated for the fluid. The thermodynamic constraints on the constitutive equation are derived. Then, some classical free energy functionals are adapted to this fluid and an explicit formula for the minimum free energy is also given.