Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 88, pp. 1–12. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu COUPLED POROSITY-FLUID CONCENTRATION FLUX-TEMPERATURE WAVES IN ISOTROPIC POROUS MEDIA ALESSIO FAMÀ, LILIANA RESTUCCIA Communicated by Giovanni Molica Bisci Abstract. In this article, we study a problem on propagation of coupled porosity, fluid concentration flux, and temperature waves. We use a model formulated in previous papers for porous media saturated by a fluid flow, in the framework of non-equilibrium thermodynamics. We derive three modes of propagation in the one-dimensional and perfect isotropic case, and then we test the validity of the model. The waves propagation velocities are represented in diagrams as functions of the wave number. The derived results have appli- cations in technological sectors such as seismology, medical sciences, geology and nanotechnology. 1. Introduction In a previous article [5] a problem of propagation of coupled porosity and fluid concentration waves was studied, using a theory developed in [6, 23, 24, 25, 26]. These papers use a theory describing porous media filled by a fluid flow formulated using the procedures of extended thermodynamics with internal variables; see [1, 3, 11, 12, 13, 15, 17, 18, 19, 22]. In this article, we focus our attention on a problem of coupled porosity, fluid concentration flux, and temperature waves, in perfect isotropic porous media. This problem has applications in technological sectors such as seismic waves, medical sciences, biology, geology, and nanotechnology. In nanostructures the volume ele- ment size L along a direction is comparable or smaller than the free mean path of the heat carriers l, i.e. l L � 1. Furthermore there are situations of propagation of high-frequency waves and the rate variation of properties of these porous media are faster than the time scale of the relaxation times of the fluxes to their equilibrium values. In Section 2 the temperature equation and the rate equations for the porosity field, its flux, the heat flux and the fluid concentration flux for the considered media are presented in the anisotropic case (see [25, 26]). In Section 3 we particularize the above equations in a special case and when the geometric, transport and thermal properties of the media are invariant for all rotations and inversions of the frame axes. In Section 4, assuming that the porous medium filled by a fluid flow occupies the whole space, we derive the propagation velocities of the coupled waves of the 2020 Mathematics Subject Classification. 74A15, 80A015. Key words and phrases. Porous media; non-equilibrium thermodynamics; plane waves. ©2021. This work is licensed under a CC BY 4.0 license. Submitted February 18, 2021. Published October 20, 2021. 1 2 A. FAMÀ, L. RESTUCCIA EJDE-2021/88 porosity, fluid concentration flux, and thermal fields in the one-dimesional case. For a given numerical set of the several coefficients present in the equations, the waves propagation speeds are represented in diagrams. Then the validity of the model is tested. The appendices present a detailed derivation of the rate and temperature equations in the case of perfect isotropic media. Monographs [4, 27] present a study of the porous media filled by a fluid flux. While the authors in [7] study a thermodynamic model for erosion and/or deposition in elastic porous media. 2. Model equations We consider a porous structure presenting a network of very thin tubes saturated by a fluid flow, whose mechanical, thermal and transport properties are analyzed using a model formulated in the framework of the extended thermodynamics (see [23, 24, 25, 26]). There the porosity field is described by an internal variable, the structural permeability tensor by rij as in Kubik [16], its gradient by rij,k and its flux by Vijk. (a comma in the lower indices indicates the spatial derivation.) Also, we assume that medium is elastic, and the inside the mechanical phenom- ena are described by the symmetric stress tensor τij and the small-strain tensor εij . The thermal processes are described by the temperature T , its gradient T,i, and the heat flux qi. The fluid flux through the porous channels is described by the fluid concentration c, its gradient c,i, and its flux jci . Thus, we choose the thermodynamic state vector C = {εij , c, T, rij , jci , qi, c,i, T,i, rij,k,Vijk}, where εij = 1 2 (ui,j +uj,i), with ui the displacement field. We refer to the configura- tion at time t, Kt, and use the standard Cartesian tensorial notation in rectangular coordinate systems. Furthermore, we assume that the porous skeleton filled by a fluid flow is a mixture of two components, so that we have ρ = ρ1 + ρ2, (2.1) with ρ the density of the medium as a whole, ρ1 the density of the fluid, and ρ2 the density of the elastic skeleton. We consider the continuity equation, where the source term has been neglected (see [3, 25]) ρċ+ jci,i = 0, (2.2) where a superimposed dot indicates the material derivative (i.e. d dt = ∂ ∂t + xγ ∂ ∂xγ , where Einstein convention for repeated indices is used). The concentration of the fluid is defined by c = ρ1/ρ and its flux jci by jci = ρ1(v1i − vi), with v1i the fluid velocity and vi the barycentric velocity of the mixture. These velocities satisfy ρvi = ρ1v1i + ρ2v2i, where v2i is the porous structure velocity. In the following the mass density ρ will be assumed constant. In [25, 26] the constitutive equations and rate equations were obtained (to close the systems of balance equations, see [25]) obeying the objectivity and frame indifference principles [9, 20, 21]. In particular the rate equations for rij , and the fluxes jci , qi and Vijk have the form ṙij + Vijk,k = β1 ijklεkl + β2 ijklrkl + β3 ijkj c k + β4 ijkqk + β5 ijklmrkl,m + β6 ijkc,k + β7 ijkT,k, (2.3) τ q q̇i = χ1 ijj c j − qi + χ3 ijklrjk,l + χ4 ijc,j − χ5 ijT,j , (2.4) EJDE-2021/88 COUPLED WAVES IN ISOTROPIC POROUS MEDIA 3 τ j c j̇ci = −jci + ξ2 ijqj + ξ3 ijklrjk,l − ξ4 ijc,j + ξ5 ijT,j , (2.5) V̇ijk = γ1 ijklj c l + γ2 ijklql + γ3 ijklmnVlmn + γ4 ijklc,l + γ5 ijklT,l + γ6 ijklmnrlm,n, (2.6) where the phenomenological tensors are assumed constant. In this article for the sake of simplicity we choose Vijk = −Dνrij,k, (2.7) with Dν a diffusive coefficient, and thus equation (2.3) keeps the form ṙij −Dνrij,kk = β1 ijklεkl + β2 ijklrkl + β3 ijkj c k + β4 ijkqk + β5 ijklmrkl,m + β6 ijkc,k + β7 ijkT,k. (2.8) In [26] the generalized telegraph temperature equation was deduced as τ qT̈ + Ṫ = −γij(τ q ε̈ij + ε̇ij) + ϕ(τ q c̈+ ċ) + ηij(τ q r̈ij + ṙij) +KijT,ji − ν1 ijj c j,i +Dνν 3 ijklrjk,li − ν4 ijc,ji, (2.9) where the phenomenological coefficients are assumed constant, Kij is the thermal diffusivity tensor, and (2.7) has taken into account. Equations (2.3)–(2.6), (2.8), (2.9) describe disturbances having finite velocity of propagation and their own relaxation times to reach their respective thermodynamic equilibrium values. In (2.4) the phenomenological tensors χ1 ij , χ 4 ij , and χ3 ij are the thermodiffusive kinetic tensor, thermodiffusive tensor, and phenomenological tensor. These tensors describe the influences of the fluid concentration flux, the concentration gradient, and the porosity field gradient on the heat flux, respectively. The phenomenological tensor χ5 ij is the thermal conductibility. Equation (2.4) is a generalization of the anisotropic transport equation Maxwell- Vernotte-Cattaneo for the heat flux τ q q̇i = −qi−χ5 ijT,j , where τ q is the relaxation time of the field qi, having finite propagation velocity. When the relaxation time τ q is null this equation reduces to the anisotropic Fourier law qi = −χ5 ijT,j describing thermal signals having infinite velocities of propagation (see [2, 8]). In equation (2.5) the phenomenological tensors ξ2 ij , ξ 3 ij , and ξ5 ijkl describe the influences of the heat flux, porosity field gradient, and temperature gradient on the fluid concentration flux field, respectively. Furthermore, ξ4 ik is the diffusion tensor. Equation (2.5) generalizes the anisotropic transport equation Fick-Nonnenmacher for the fluid concentration flux τ j c j̇ci = −jci −ξ4 ijc,j , where τ j c is the relaxation time of the field jci , having finite propagation velocity. When the relaxation time τ j c is vanishing, this equation reduces to the anisotropic Fick law jci = −ξ4 ijc,j , where the fluid concentration flux has infinite propagation velocity. Equations (2.3) and (2.6) describe the evolution of the porosity field and its flux, and in their right-hand sides the sources terms represent contributions of several fields. Also, from the evolution equation (2.9) of the thermal field, it is seen that several fields influence the propagation of the field T . 3. Equations governing the evolution of porosity, fluid concentration flux, and temperature fields in a special case For the treatment of the problem of coupled porosity, fluid concentration flux and temperature waves, we take into account the system of differential equations (2.8), (2.5) and (2.9), and we assume the following: 4 A. FAMÀ, L. RESTUCCIA EJDE-2021/88 (i) the considered porous medium is at rest, (ii) in equation (2.8) the influence of the field εij can be neglected, (iii) in the rate equation (2.5) the influence of the fields qi and c,i can be disre- garded, (iv) in equation (2.9) the influence of the first time and second time derivatives of the small deformations field εij and the concentration field c, the second time derivative of the porous field rij , the fluid concentration flux gradient and the gradient of the concentration gradient can be neglected. Thus, we obtain ∂rij ∂t −Dνrij,kk = β2 ijklrkl + β3 ijkj c k + β4 ijkqk + β5 ijklmrkl,m + β6 ijkc,k + β7 ijkT,k, (3.1) τ j c ∂jci ∂t = −jci + ξ3 ijklrjk,l + ξ5 ijT,j , (3.2) τ q ∂2T ∂t2 + ∂T ∂t = ηij ∂rij ∂t +KijT,ji +Dνν 3 ijklrjk,li. (3.3) In the rate equation (3.1), because of the symmetry of rij = rji, the phenomenolog- ical coefficients βs (s = 2, . . . , 7) present some symmetries. For the fourth tensor β2 ijkl, present in equation (3.1), we have β2 ijkl = β2 jikl and β2 ijkl = β2 ijlk, (3.4) which are equivalent to β2 ijkl = β2 jikl = β2 ijlk = β2 jilk. Furthermore, βpijk = βpjik (p = 3, 4, 6, 7), β5 ijklm = β5 jiklm = β5 ijlkm = β5 jilkm. (3.5) Also, from the symmetry properties of rij and rjk,l, rjk,li (in the indexes {j, k} and {l, i} respectively) and of T,ji (in the indexes {j, i}), in the rate equations (3.2) and (3.3) we have for the following phenomenological symmetries: ξ3 ijkl = ξ3 ikjl, ηij = ηji, Kij = Kji, ν3 ijkl = ν3 ikjl, ν3 ijkl = ν3 ljki . (3.6) Relations (3.6)4 and (3.5)5 are equivalent to ν3 ijkl = ν3 ikjl = ν3 ljki = ν3 lkji. (3.7) The symmetry relations (3.4)-(3.7) reduce the number of the significant compo- nents of the above phenomenological tensors in equations (3.1)-(3.3). The number of these significant components can have a further reduction if we suppose the considered media are perfect isotropic. 3.1. Perfect isotropic media. In this subsection we consider perfect isotropic systems, having invariant symmetry properties with respect all rotations and the inversion of the frame axes. These properties simplify the form of the temperature and rate equations (3.1)–(3.3) in such a way that the number of the significant Cartesian components of the phenomenological tensors have a further reduction (see [6, 10, 14]). In fact, in this case the phenomenological tensors of order two ξ5 ij , Kij , and ηij take the form ξ5 ij = ξ5δij , Kij = Kδij , ηij = ηδij ; (3.8) EJDE-2021/88 COUPLED WAVES IN ISOTROPIC POROUS MEDIA 5 the tensors of order three and five vanish, β3 ijk = β4 ijk = β5 ijklm = β6 ijk = β7 ijk = 0; (3.9) the tensors of order four have the form Lijkl = L1δijδkl + L2δikδjl + L3δilδjk, (3.10) where Ls (s = 1, 2, 3) are the 3 significant components of Lijkl, so that β2 ijkl, ξ 3 ijkl and ν3 ijkl have three significant components. Because of these tensors satisfy also the symmetry properties (3.4)3, (3.6)1, and (3.7), in the appendix we show that the tensors β2 ijkl, ξ 3 ijkl, and ν3 ijkl have only two significant components. Furthermore, in the following we will consider only the scalar (or spherical) part rδij of rij , with r defined by r = 1 3 rkk, (3.11) having split rij in its deviatoric part r̃ij = rij − rδij , and its spherical part rδij . By this assumption we have rij = rδij , rij,kk = r,kkδij , rkl,m = r,mδkl, rjk,l = r,lδjk, rjk,li = r,liδjk. (3.12) From equations (3.1)-(3.3), taking into consideration the results (3.12) and re- lations (3.8), (3.9), (4.2), (4.4), and (4.5) for the phenomenological tensors (see detailed calculations in Appendices 4–7), we derive the following simplified sys- tem of equations governing the evolution of porosity, fluid concentration flux and temperature fields ∂r ∂t −Dνr,kk = −αrr, (3.13) τ j c ∂jci ∂t = −jci + αcr,i + βcT,i, (3.14) τ q ∂2T ∂t2 + ∂T ∂t = KT,kk + αT r,kk − 3ηαrr, (3.15) where αr = (τ r) −1 > 0 (3.16) is the inverse of the relaxation time of the field r, given by relation (5.3) of Appendix 5, αc, βc, and αT are coupling coefficients reflecting some cross-kinetic effects of the porosity gradient field and the temperature gradient on the the fluid concentration flux, and the effect of the field r,kk on the temperature field, respectively (see relations (6.3) and (7.4) of Appendices 6 and 7, respectively). A detailed derivation of equations (3.13)-(3.15) has been obtained in Appendix 5–7. 3.2. Propagation velocities of the coupled waves. The aim of this Subsection is to find the dispersion relation, the propagation velocities of the coupled porosity, fluid concentration flux, and temperature waves as functions of the wave number. We assume that the porous medium occupies the whole space, and confine our study to one-dimensional waves, that propagate along the x direction, so that jc = (jc, 0, 0). Thus, we assume that the solutions of the set of equations (3.13)-(3.15) keep the form r(x, t) = r̂eik(x−vt), jc(x, t) = ĵceik(x−vt), T (x, t) = T̂ eik(x−vt), (3.17) 6 A. FAMÀ, L. RESTUCCIA EJDE-2021/88 where v is the wave velocity, k is the wave number and r̂, ĵc and T̂ are the amplitudes of the waves r(x, t), jc(x, t) and T (x, t) (3.17); v is defined by v = ω/k [m s−1], with ω the angular frequency, ω = 2πf [s−1], being f the wave frequency, k the wave number, given by k = 2π/λ [m−1], and λ the wavelength. Thus, inserting the relations (3.17) and their suitable derivatives into (3.13)- (3.15), we obtain the system of equations( Dνk 2 + αr − ikv ) r̂ = 0, (3.18) αcikr̂ + ( τ j c ikv − 1 ) ĵc + βcikT̂ = 0, (3.19) (−αT k2 − 3ηαr)r̂ + ( τ qk2v2 + ikv −Kk2 ) T̂ = 0, (3.20) which has non-trivial solutions only if its determinant vanishes, i.e. D = ∣∣∣∣∣∣ Dνk 2 + αr − ikv 0 0 αcik τ j c ikv − 1 βcik −αT k2 − 3ηαr 0 τ qk2v2 + ikv −Kk2 ∣∣∣∣∣∣ = 0. (3.21) Developing D we obtain the following dispersion relation for the waves propagation velocities v concerning possible propagation modes: τ j c τ qk3v4 + ik2 [ τ j c + τ q + τ j c τ q ( Dνk 2 + αr ) ] v3 − k [ ( τ j c + τ q ) ( Dνk 2 + αr ) + τ j c Kk2 + 1 ] v2 − i [( Kτ j c k2 + 1 ) ( Dνk 2 + αr ) +Kk2 ] v +Kk ( Dνk 2 + αr ) = 0. (3.22) From the real part of this dispersion relation, we obtain τ j c τ qk2v4 − [( τ j c + τ q ) ( Dνk 2 + αr ) + τ j c Kk2 + 1 ] v2 +K ( Dνk 2 + αr ) = 0, (3.23) from which we derive two possible modes v(1) = √ A + √ A 2 −B, v(2) = √ A − √ A 2 −B, (3.24) where A = τ j cKk2 + 1 + ( τ j c + τ q ) ( Dνk 2 + αr ) 2τ jcτ qk2 , with A > 0, (3.25) B = K ( Dνk 2 + αr ) τ jcτ qk2 , with B > 0. (3.26) The velocity v(1) is real when A 2 −B ≥ 0 and A + √ A 2 −B ≥ 0. (3.27) Condition (3.27)1 is satisfied when[ τ j c Kk2 + 1 + ( τ j c + τ q )( Dνk 2 + αr )]2 − 4τ j c τ qKk2 ( Dνk 2 + αr ) ≥ 0, (3.28) whereas (3.27)2 is always satisfied, if (3.28) holds, because it is a sum of two positive terms. The velocity v(2) is real when: (i) expression (3.28) holds, and from (3.24)2 we have A − √ A 2 −B ≥ 0, (3.29) EJDE-2021/88 COUPLED WAVES IN ISOTROPIC POROUS MEDIA 7 from which we derive B ≥ 0, that is always true. From the imaginary part of the dispersion relation (3.22), we derive k2 [ τ j c + τ q + τ j c τ q ( Dνk 2 + αr ) ] v3 − [ ( Kτ j c k2 + 1 ) ( Dνk 2 + αr ) +Kk2 ] v = 0, (3.30) from which we obtain the values v(3) = 0, v(4) = √ Kk2 + (Kτ jck2 + 1) (Dνk2 + αr) k2 [τ jc + τ q + τ jcτ q (Dνk2 + αr)] . (3.31) Notice that the velocity v(4) is real for all k 6= 0 because in (3.31) the radicand is always positive. Thus, we have obtained three possible modes of propagation: v(1), v(2) and v(4). In Figures 1–3 the propagation speeds v(1), v(2), and v(4) as functions of k, solving the real part (3.23) or the imaginary part (3.30) of the dispersion relation (3.22), are represented in the case where, as an example, we have considered a given numerical set of the several coefficients present in the equations of the examined problem: Dν = 10−2 m2 s−1, K = 10−4 m2 s−1, τ j c = 10−2 s, τ q = 10−2 s, and αr = 102 s−1. In this assumption condition (3.28) is satisfied for all k and furthermore the velocities v(1) and v(2) are real. We recall that the velocity v(4) is real for all k 6= 0. Figure 1. Representation of the wave propagation speed v(1) (in blue color) as function of k, for a given numerical set of the several coefficients present in the examined problem. The horizontal line in fuchsia color is its horizontal asymptote. Conclusions In this article we worked out for a perfect isotropic porous media filled by a fluid flow, a system of rate equations for the porosity, a fluid concentration flux, and temperature fields to study the propagation of coupled waves of these fields. We used a model formulated in previous papers, in the framework of rational extended irreversible thermodynamics with internal variables. A structural permeability ten- sor rij , its gradient rij,k, and its flux Vijk were introduced in the thermodynamic state vector and the mass density of the mixture consisting of the porous skeleton 8 A. FAMÀ, L. RESTUCCIA EJDE-2021/88 Figure 2. Representation of the wave propagation speed v(2) (in blue color) as function of k, for a given numerical set of the several coefficients present in the studied problem.The horizontal line in fuchsia color is its horizontal asymptote. Figure 3. Representation of the wave propagation speed v(4) (in blue color) as function of k, for a given numerical set of the several coefficients present in the examined problem. The horizontal line in fuchsia color is its horizontal asymptote. and the fluid flowing inside of it was assumed constant. The body was supposed occupying the whole space. The dispersion relation was derived, three possible propagation modes were obtained, and the corresponding wave propagation veloc- ities as functions of the wave number k, were represented in diagrams, for a given set of the several phenomenological coefficients present in the studied problem. 4. Appendix A: Perfect isotropic tensors with special symmetry properties Here we consider perfect isotropic tensors of fourth order, having special symme- try properties, and thus a reduced number of significant components. In particular, we demonstrate that the tensors β2 ijkl, ξ 3 ijkl and ν3 ijkl can be expressed only by two significant independent components. EJDE-2021/88 COUPLED WAVES IN ISOTROPIC POROUS MEDIA 9 Case (a) Let us consider the fourth order perfect isotropic tensor β2 ijkl, present in equation (3.1) and having the symmetries β2 ijkl = β2 jikl = β2 ijlk = β2 jilk. Using relation (3.10) we obtain β2 jikl = β2 aδjiδkl + β2 b δjkδil + β2 c δjlδik (4.1) and an analogous expression for β2 ijkl. Matching the two relations obtained by the help of (3.10), the tensor β2 ijkl can be written as β2 ijkl = β2 1δijδkl + β2 2(δikδjl + δilδjk), with β2 1 = β2 a, β 2 2 = (β2 b + β2 c )/2. (4.2) Case (b) Let us consider the fourth order perfect isotropic tensor ξ3 ijkl, present in equation (3.2) and having the symmetry ξ3 ijkl = ξ3 ikjl. From relation (3.10) we have ξ3 ikjl = ξ3 aδikδjl + ξ3 b δijδkl + ξ3 c δilδkj (4.3) and an analogous result for ξ3 ikjl. Matching the two results we have ξ3 ijkl = ξ3 1δilδjk + ξ3 2(δijδkl + δikδjl). (4.4) Case (c) The perfect isotropic fourth tensor ν3 ijkl, present in equation (3.3), has the symmetries ν3 ijkl = ν3 ikjl = ν3 ljki = ν3 lkji. Thus, by an analogous method used in the cases (a) and (b) the tensor ν3 ijkl can be written as ν3 ijkl = ν3 1δilδjk + ν3 2(δijδkl + δikδjl). (4.5) 5. Appendix B: Derivation of the rate equation for porosity field To obtain equation (3.13), we use (3.9), (3.12)1, (3.12)2, and the special form (4.2) assumed by the fourth order tensor β2 ijkl, so that (3.1) takes the form ∂r ∂t δij −Dνr,kkδij = [β2 1δijδkl + β2 2(δikδjl + δilδjk)]rδkl, (5.1) where β2 1 , β2 2 are the 2 significant independent components of the fourth tensor β2 ijkl. Then, from (5.1) we obtain ∂r ∂t δij −Dνr,kkδij = ( 3β2 1 + 2β2 2 ) rδij , (5.2) i.e. equation (3.13), when i = j and we define( 3β2 1 + 2β2 2 ) = −αr = − (τ r) −1 , (5.3) where τ r the relaxation time of the porosity field. 6. Appendix C: Derivation of the rate equation for the fluid concentration flux To derive (3.14), we use (3.2) and the special forms (3.8)1 and (4.4) for the tensors ξ5 ij and ξ3 ijkl, respectively, so that we obtain τ j c ∂jci ∂t = −jci + [ξ3 1δilδjk + ξ3 2(δijδkl + δikδjl)]r,lδjk + ξ5T,i, (6.1) 10 A. FAMÀ, L. RESTUCCIA EJDE-2021/88 where ξ3 1 , ξ3 2 are the 2 significant independent components of the fourth tensor ξ3 ijkl and ξ5 is the only significant component of the second order tensor ξ5 ij . Thus, equation (6.1) keeps the form τ j c ∂jci ∂t = −jci + (3ξ3 1 + 2ξ3 2)r,i + ξ5T,i, (6.2) i.e. equation (3.14), when we define βc = ξ5, αc = 3ξ3 1 + 2ξ3 2 . (6.3) 7. Appendix D: Derivation of temperature equation To deduce (3.15), we use (3.3) , (3.11), (3.12)1, and (3.12)5, and the special forms (3.8)2, (3.8)3 , and (4.5) of the tensors Kij , ηij and ν3 ijkl, so that we obtain τ q ∂2T ∂t2 + ∂T ∂t = 3η ∂r ∂t +KT,ii +Dν [ ν3 1δilδjk + ν3 2(δijδkl + δikδjl) ] r,liδjk, (7.1) where ν3 1 , ν3 2 are the 2 significant independent components of the fourth tensor ν3 ijkl and K, η are the only significant components of the second order tensors Kij and ηij . Then equation (7.1) reads τ q ∂2T ∂t2 + ∂T ∂t = 3η ∂r ∂t +KT,ii +Dν ( 3ν3 1 + 2ν3 2 ) r,ii. (7.2) Using (3.13), equation (7.2) assumes the form τ q ∂2T ∂t2 + ∂T ∂t = KT,ii +Dν ( 3ν3 1 + 2ν3 2 + 3η ) r,ii − 3ηαrr, (7.3) i.e. equation (3.15), when we define αT = Dν ( 3ν3 1 + 2ν3 2 + 3η ) . (7.4) Acknowledgements The authors thank Prof. David Jou, from Universitat Autonòma of Barcelona, Catalonia, Spain, for his very enlightening discussions and comments, and Prof. Julio G. 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Stagno d’Alcontres, Salita Sperone 31, 98166 Messina, Italy Email address: lrestuccia@unime.it 1. Introduction 2. Model equations 3. Equations governing the evolution of porosity, fluid concentration flux, and temperature fields in a special case 3.1. Perfect isotropic media 3.2. Propagation velocities of the coupled waves Conclusions 4. Appendix A: Perfect isotropic tensors with special symmetry properties 5. Appendix B: Derivation of the rate equation for porosity field 6. Appendix C: Derivation of the rate equation for the fluid concentration flux 7. Appendix D: Derivation of temperature equation Acknowledgements References