A simple model of porous media with elastic deformations and erosion or deposition
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
A simple model of porous media with elastic deformations and erosion or deposition A. Fam` a, L. Restuccia and D. Jou
Abstract. This paper deals with a model for solids with porous channels filled by an incompressible isotropic fluid. The Darcy– Brinkman–Stokes law is derived, that represents a rate equation for the local mass flux of the fluid, presenting relaxation times in which this flux evolves towards its local-equilibrium value, viscous effects and a permeability tensor referring to a response of the system to an external agent, i.e. the fluid flow produced by a pressure gradient. The erosion/deposition phenomena in an elastic porous matrix are also studied and particular thermal porous metamaterials, that have interesting functionality, like in fluid flow cloaking, are illustrated as application of the obtained results. This derived model is completely in agreement with a theory formulated in the framework of the rational irreversible thermodynamics, where two internal variables are introduced (a symmetric structural porosity tensor and a symmetric second order tensor influencing viscous phenomena, that is interpreted as the symmetric part of the velocity gradient), when the results are considered in a first approximation and some suitable assumptions are done. The constitutive theory is worked out by using Liu’s and Wang’s theorems. The obtained theory has applications in several technological sectors, like physics of soil, pharmaceutics, physiology, etc., and contributes to the study of new porous metamaterials. Mathematics Subject Classification. 74A15, 74D10, 74F10. Keywords. Porous solids, Non-equilibrium thermodynamics with internal variables, Porous matrix with erosion/deposition, Constitutive relations for porous media, Application Liu’s theorem.
1. Introduction In previous papers, using a thermodynamical model [1,2], formulated in the frame of extended thermodynamics [3–12] with internal variables, two of us derived some constitutive equations for elastic porous media [13,14], and the coupled waves of the fluid-concentration and porosity fields [15], described using the structural porosity tensor rij `a la Kubik [16] as internal variable. In this work, besides considering the effects of elastic coupling of the fluid with the porous solid matrix, we also consider erosive effects of the fluid flow on the solid matrix or, inversely, deposition effects, which lead to ageing effects of the porous medium. To this aim, we focus our attention on a particular case sufficiently interesting and simple to explicitly illustrate some practical applications of suitable constitutive equations, that are derived in the present paper by the elaboration of an appropriate thermodynamic model. The description of porous media with effects of erosion or deposition are of great interest in the physics of soil, in pharmaceutics (controlled release of medicals from solid matrices) and in physiology. The organization of the paper is the following. In Sect. 2, an incompressible isotropic fluid
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