Non-Equilibrium Thermodynamics of porous media filled by a fluid flow and of rigid bodies with an internal tensorial field influencing the thermal phenomena.

This thesis consists of two Parts, every one dedicated, in the framework of non-equilibrium thermodynamics to the study of the behavior of fluid-saturated porous media and of rigid bodies with an internal tensorial field influencing the thermal phenomena, respectively. Furthermore, three Appendices are present, that clarify some aspects of the arguments presented in the second, third, fourth, and seventh Chapters. Regarding the first Part, the influence of porous channels filled by fluid on the other fields occurring inside the media is illustrated by the introduction of a structural permeability tensor à la Kubik, giving a macroscopic characterization of a porous structure and coming from the use of volume and area averaging procedures. Understanding the influence of porous tubes on mechanical and transport properties in miniaturized systems is an interesting topic because by experimental and theoretical studies it was found that the porous density has a minor effect on the thermal conductivity for porous defects densities smaller than a characteristic value dependent on the material and temperature but for higher values than this value, the thermal conductivity decreases, and this situation influences the nanodevices performances. In particular, in Chapter 1, using a model for porous media filled by a fluid flow, in the anisotropic and linear case the constitutive relations, the temperature, and energy equations and the rate equations for the porosity field, its flux, the fluid-concentration flux, and the heat flux are derived to close the system of equations describing the media under consideration. In Chapter 2, the case of porous media isotropic with respect to rotations and inversions of frame axes is treated and symmetry properties of phenomenological tensors of a higher order than two (until six) are derived. In Appendix A special forms of isotropic tensors up to six orders are deduced. In Chapter 3 a simple model for solids with porous channels, filled by an incompressible isotropic fluid and presenting erosion/deposition phenomena is given. The Darcy-Brinkman-Stokes law is obtained, which represents a rate equation for the local mass flux of the fluid, with a relaxation time in which this flux evolves towards its local-equilibrium value. In Appendix B the objective representations of scalar, vectorial, and tensorial functions are presented, clarifying some equations deduced in this Chapter. In Chapters 4, 5, and 6 applications of the theories developed in the first and second Chapters are done. In particular, in Chapter 4 a study of a problem of propagation of coupled porosity and fluid-concentration waves in isotropic porous media is worked out, deducing the wave propagation velocities as functions of the wavenumber. Also in this Chapter, some expressions of isotropic tensors with special symmetries are deduced in Appendix A (see the article \cite{FR-1}). In Chapter 5 following Boillat's methodology for quasi-linear and hyperbolic systems of the first order, we obtain Bernoulli's equation governing the propagation of weak discontinuities in isotropic porous media filled by a fluid. In Chapter 6 a general method to construct approximate smooth solutions for nonlinear hyperbolic partial differential equations is illustrated and applied in the case where interactions between the fluid-concentration field, the porosity field, and their fluxes in porous isotropic media are considered. Regarding the second Part of this thesis, the deepening knowledge of mechanical, thermal, and transport properties in rigid bodies with an internal variable influencing thermal phenomena is very interesting in several technological sectors, such as in material sciences and nanotechnology. In Chapter 7 general constitutive equations of heat transport with second sound and ballistic propagation in isotropic rigid heat conductors are given using non-equilibrium thermodynamics with internal variables. Appendix C is addressed to a two-dimensional symmetric explicit representation of the conductivity matrix, that appears in the expression of entropy production deduced in this Chapter.