Thermodynamics of viscoelastic fluids: the temperature equation

From the thermodynamics with internal variables we will derive the temperature equation for viscoelastic fluids. We consider the type of storage of mechanical energy, the dissipation of mechanical energy, the compressibility of the fluid, the nonequilibrium heat capacity and thermal expansion, and deformation induced anisotropy of the heat conduction. The well-known stress differential models that fit into the thermodynamic theory will be treated as an example. Adapting a power-law scaling of the shear moduli on temperature and density, as is usual in rubber elasticity, we will derive an approximation of the temperature equation in measurable quantities. This equation will be compared with experimental results.