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.