### A heat conduction problem involving phase change and its numerical solution by finite difference methods

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Extended thermodynamics is based on a set of equations of balance which are supplemented by local and instantaneous constitutive equations so that the field equations are quasi-linear differential equations of first order. If the constitutive functions are subject to the requirements of the entropy principle, one may write them in symmetric hyperbolic form by a suitable choice of fields. The kinetic theory of gases, or the moment theories based on the Boltzmann equation, provide an explicit example...

We characterize the general solutions to certain symmetric systems of linear partial differential equations with tensor functionals as unknowns. Then we determine the solutions that are physically meaningful in suitable senses related with the constitutive functionals of two simple thermodynamic bodies with fading memory that are globally equivalent, i.e. roughly speaking that behave in the same way along processes not involving cuts. The domains of the constitutive functionals are nowhere dense...

In this paper, by applying a simple mathematical model imitating the equation of state, behaviour of the phase transition curve near the critical point is investigated. The problem of finding the unique vapour-liquid equilibrium curve passing through the critical point is reduced to solving a nonlinear system of differential equations.

We review some of the most important phenomena due to the phonon-wall collisions in nonlocal heat transport in nanosystems, and show how they may be described through certain slip boundary conditions in phonon hydrodynamics. Heat conduction in nanowires of different cross sections and in thin layers is analyzed, and the dependence of the thermal conductivity on the geometry, as well as on the roughness is pointed out. We also analyze the effects of the roughness of the surface of the pores on the...

Nonlocal effects on heat transport beyond a simple Fourier description are analyzed in a thermodynamical model. In the particular case of hot nanosystems cooled through a graphene layer, it is shown that these effects may increase in a ten percent the amount of removed heat, as compared with classical predictions based on the Fourier law.