Oldroyd fluids with hyperbolic heat conduction

Development of constitutive equations of Oldroyd fluids with hyperbolic heat conduction, directly from the free energy and dissipation functions, is the focus of this work. Two models from hyperbolic thermoelasticity of solids – theory with one thermal relaxation time and theory with two thermal relaxation times – provide a stepping-stone for this task. The setting for these models is offered, respectively, by the primitive thermodynamics of Edelen and the thermodynamic orthogonality of Ziegler, both cases involving an internal parameter: the inelastic strain. The identified free energy and dissipation functions show that, in the first case, the hyperbolic (telegraph-like) heat conduction is governed by the Maxwell-Cattaneo model involving one thermal relaxation time, whereas, in the second case, by the Fourier law and two thermal relaxation times in the constitutive laws for stress and entropy. In both cases, the thermal fields vanish from the mechanical equations as the Oldroyd fluid becomes incompressible, whereas the mechanical fields are present in the telegraph-type equations for temperature.