Gradient generalization to the extended thermodynamic approach and diffusive-hyperbolic heat conduction

We study a generalization of irreversible thermodynamics with nonlocal closing relation. Thus parabolic and hyperbolic models can be described within one single theory. In the 1-d case, Guyer-Krumhansl equation and classical Fourier heat conduction may be obtained, depending on the constitutive assumptions. The thermodynamical restrictions in form of the Clausius-Duhem inequality are studied taking into account an extra flux of entropy corresponding to nonlocal irreversible effects. Numerical solutions to the resulting initial-boundary value problem are calculated and compared with available experimental results.