## Abstract

A thermodynamic description of transient heat conduction at small length and timescales is proposed. It is based on extended irreversible thermodynamics and the main feature of this formalism is to elevate the heat flux vector to the status of independent variable at the same level as the classical variable, the temperature. The present model assumes the coexistence of two kinds of heat carriers: diffusive and ballistic phonons. The behaviour of the diffusive phonons is governed by a Cattaneo-type equation to take into account the high-frequency phenomena generally present at nanoscales. To include non-local effects that are dominant in nanostructures, it is assumed that the ballistic carriers are obeying a Guyer–Krumhansl relation. The model is applied to the problem of transient heat conduction through a thin nanofilm. The numerical results are compared with those provided by Fourier, Cattaneo and other recent models.

## 1. Introduction

It is well recognized that Fourier's law of heat conduction ** q**=−

*λ*∇

*T*with

**the heat flux vector, ∇**

*q**T*the temperature gradient and

*λ*the heat conductivity is only valid at low frequencies and large space scales. To cope with high-frequency processes, Fourier's law has been generalized by Cattaneo (1948) into the non-steady form 1.1wherein

*τ*designates the heat flux relaxation time. Cattaneo's relation reduces to Fourier';s law in the limit of vanishing values of

*τ*. However, Cattaneo';s equation is not able to describe highly non-local effects characterizing small-scale systems. To account for non-localities, a generalization has been proposed by Guyer & Krumhansl (1966

*a*,

*b*), who derived the following equation on the basis of the kinetic theory: 1.2where the quantity

*l*stands for the mean free path of the heat carriers, namely phonons; non-locality is expressed through the second-order space derivatives in ∇⋅

**and ∇∇⋅**

*q***. The objective of this work is to describe transient heat conduction at micro- and nanoscales based on extended irreversible thermodynamics (EIT) (Lebon**

*q**et al*. 2008; Jou

*et al*. 2010

*a*,

*b*). The main idea underlying this theory is to elevate the fast variables, such as the heat flux, to the status of independent variables at the same level as the slow variables such as energy, or temperature. In the next section, it is shown that the Cattaneo and Guyer–Krumhansl equations can be directly derived from the extended thermodynamics formalism. In the present paper, it is assumed that heat propagation is governed by two kinds of phonons: ballistic and diffusive ones. The idea is not new and was essentially initiated by Chen (2001, 2002), who proposed a so-called ‘ballistic–diffusion model’ mixing kinetic theory and macroscopic considerations. By contrast, our approach is purely macroscopic and rests on the assumption that the motion of the diffusive phonons is governed by Cattaneo's equation while the ballistic phonons, which are dominant when the dimensions of the system are equal to or smaller than the mean free path of the phonons, will obey the Guyer–Krumhansl relation.

The paper is set out as follows. In §2, the main ingredients of EIT are recalled; in particular, it is shown under which conditions the Cattaneo and Guyer–Krumhansl equations can be derived from this formalism. The ballistic–diffusion model is analysed in §3 and is applied to the problem of transient heat conduction in nanofilms, as shown in §4. The numerical results are analysed and compared with those provided by Fourier, Cattaneo and more recent descriptions by Joshi & Majumdar (1993), Chen (2002) and Alvarez & Jou (2010). General conclusions are drawn in §5.

## 2. Brief review of extended irreversible thermodynamics

The main idea underlying EIT is to consider the dissipative fluxes, such as the flux of heat in heat transport problems, as independent basic variables, on the same footing as the classical variables such as energy or temperature. Elevating dissipative fluxes to the status of independent variables amounts to introducing memory and non-local effects into the formalism. It is also assumed that there exists a non-equilibrium entropy *s* depending on the whole set of variables, including the fluxes. In the particular case of heat conduction, *s* is assumed to be a function of the internal energy *u* and the heat flux vector ** q**,
2.1with

*s*and

*u*denoting quantities measured per unit volume. The entropy

*s*obeys a time-evolution equation of the form 2.2wherein

*J*^{s}denotes the entropy flux vector,

*σ*

^{s}the rate of entropy production per unit volume, a positive-definite quantity according to the second law of thermodynamics. As the global velocity of the material is supposed to be equal to zero, partial and material time derivatives are identical. It is well known that Fourier's law can directly be derived from the classical theory of irreversible processes based on the local equilibrium hypothesis and developed, among others, by Onsager (1931), Prigogine (1961) and De Groot & Mazur (1962). To illustrate the range of applications of EIT, we now show that the Cattaneo and Guyer–Krumhansl equations enter naturally into the framework of this formalism. In differential form, relation (2.1) takes the form 2.3As usual, define the non-equilibrium temperature by

*T*

^{−1}=∂

*s*/∂

*u*and assume moreover that ∂

*s*/∂

**=−**

*q**α*(

*T*)

**with**

*q**α*(

*T*) a material coefficient allowed to depend on

*T*, the minus sign being introduced for convenience and nonlinear contributions in

**being omitted; substitution of these expressions in equation (2.3) yields 2.4The time derivative ∂**

*q**u*/∂

*t*is given by the first law of thermodynamics, which, for rigid heat conductors at rest and absence of internal heat sources, reads as 2.5Our next task is to formulate the time-evolution equation of

**as shown below; simple forms are provided by the Cattaneo and (or) Guyer–Krumhansl equations.**

*q*### (a) The Cattaneo equation

Making use of equation (2.5), relation (2.4) is written as
2.6By comparison with general expression (2.2) of the evolution equation for *s*, one obtains the following results for *J*^{s} and *σ*^{s}, respectively:
2.7The simplest way to ensure positiveness of *σ*^{s} is to assume a linear relation between the flux ** q** and the so-called thermodynamic force represented by the terms between parentheses,
2.8wherein

*μ*is a positive-definite coefficient. It is also shown within the general framework of EIT (Jou

*et al*. 2010

*a*) that

*α*≥0 in order to comply with the concavity property of entropy. By introducing the notation 2.9one recovers from expression (2.8) the familiar form of Cattaneo's equation—namely 2.10with

*τ*designating the relaxation time of the heat flux and

*λ*the positive heat conductivity. Both quantities are generally dependent on the temperature.

### (b) The Guyer–Krumhansl equation

It is observed that expression (2.7*a*) of the entropy flux *J*^{s} is the same as in the classical irreversible thermodynamics (i.e. the heat flux divided by the temperature). When non-localities become important, it is natural to expect that *J*^{s} will in addition depend on the gradients of ** q**. In that respect and without loss of generality, we find it justified to write

*J*^{s}in the following form involving terms in ∇

**and ∇⋅**

*q***: 2.11wherein**

*q**γ*is a coefficient to be identified later on; the factor 2 in the last term is not essential but has been introduced to recover the Guyer–Krumhansl kinetic equation. Starting from relation (2.2) and replacing ∂

*s*/∂

*t*and

*J*^{s}by expressions (2.4) and (2.11), respectively, it is found that 2.12or, reassembling the terms containing the factor

**, 2.13The simplest way to guarantee that the entropy production is positive-definite is to assume that there exists a linear relationship between the flux**

*q***and its conjugated force represented by the terms enclosed in the brackets and that**

*q**γ*is a positive factor; as a consequence, one is led to 2.14wherein

*μ*is a positive phenomenological coefficient. Introducing the identifications 2.15one finds again the Guyer–Krumhansl original law 2.16while the entropy production takes the form 2.17positivity of

*σ*

^{s}demands that the heat conductivity

*λ*be positive-definite. The derivation of the Guyer–Krumhansl equation given here is new and has the advantage of being rather simple; it also clearly exhibits that not only the Cattaneo but also the Guyer–Krumhansl relation can be derived by assuming that the entropy

*s*depends, in addition to the classical variable

*u*, only on one single extra flux variable, the heat flux vector

**.**

*q*## 3. The ballistic–diffusion model

Micro- and nanomaterials are characterized by the property that the ratio of the mean free path *l* of the heat carriers and the mean dimension *L* of the system, the Knudsen number *Kn*=*l*/*L*, is comparable to or larger than unity. In the present work, we assume the coexistence of two kinds of heat carriers: diffusive phonons that undergo multiple collisions within the core of the system and ballistic phonons originating at the boundaries and experiencing mainly collisions with the walls. This model is called the ballistic–diffusion one and was initially introduced by Chen (2001). The main point underlying Chen's approach is to split the distribution function *f* into two parts: *f*=*f*_{b}+*f*_{d}, with the subscripts ‘b’ and ‘d’ referring to ballistic and diffusive phonons, respectively. Subsequently, the internal energy and the heat flux are decomposed into a ballistic and a diffusive component in such a way that
3.1The construction of the present model proceeds in three steps.

*Step 1. Definition of the space of state variables*. According to decomposition (3.1) of *u* and ** q**, the state variables are selected as follows:

— the couple

*u*_{d},*q*_{d}to account for the diffusive behaviour of the heat carriers;— the couple

*u*_{b},*q*_{b}to provide a description of the ballistic motion of the carriers.

For future use, we also introduce the diffusive and ballistic quasi-temperatures *T*_{d} and *T*_{b} defined, respectively, by *T*_{d}=*u*_{d}/*c*_{d} and *T*_{b}=*u*_{b}/*c*_{b}, where *c*_{d} and *c*_{b} denote the heat capacities per unit volume and are positive quantities to guarantee stability of the equilibrium state. Admitting that the heat capacities are equal so that *c*_{d}=*c*_{b}=*c*, and defining the total quasi-temperature by *T*=*u*/*c*, it is verified that *T*=*T*_{d}+*T*_{b}. Although the quantities *T*_{d}, *T*_{b} and *T* bear some analogy to the classical definition of the temperature, it should however be realized that, strictly speaking, these quantities do not represent temperatures in the usual sense but must be considered as a measure of the internal energies; this justifies the use of the terminology ‘quasi-temperature’.

*Step 2. Establishment of the evolution equations.* After having defined the state variables, one must specify their behaviour in the course of time and space. The evolutions of the internal energies *u*_{d} and *u*_{b} are governed by the classical energy balance laws
3.2while the total internal energy, *u*=*u*_{d}+*u*_{b}, satisfies the first law of thermodynamics (2.5), the quantities *r*_{d} and *r*_{b} designate source terms which may be either positive or negative. By virtue of the first law (2.5), one has to satisfy *r*_{d}+*r*_{b}=0 in the absence of energy sources, so that *r*_{d}=−*r*_{b}. Based on kinetic theory considerations (Chen 2001, 2002), it is shown that
3.3the minus sign indicating that ballistic carriers can be converted into diffusive ones but that the inverse is not possible; *τ*_{b} is the relaxation time of the ballistic energy flux *q*_{b}.

The evolution equation for the fluxes remains to be derived. Concerning the diffusive phonons, it is assumed that they satisfy the Cattaneo equation to cope with their high-frequency properties, i.e.
3.4wherein the relaxation time *τ*_{d} and the heat conductivity coefficient *λ*_{d} are positive quantities to meet the requirements of the stability of equilibrium and positivity of the entropy production, respectively (Lebon *et al*. 2008; Jou *et al*. 2010*a*). However, expression (3.5) is not able to describe the ballistic regime, which is mainly influenced by non-local effects as most of the ballistic carriers cross the system without experiencing collisions except with the boundaries. As shown before, this situation is satisfactorily described through the Guyer–Krumhansl equation
3.5where *l*_{b} is the mean free path of the ballistic phonons; the terms involving the space derivatives of the heat flux vector account for the non-local effects and are important when the spatial scale of variation of the heat flux is comparable to the mean free path of the heat carriers. From the kinetic point of view, Guyer and Krumhansl have shown that *τ*_{b} can be identified with the collision time *τ*_{R} of the resistive phonons' collisions (non-conserving momentum collisions), and that *l*^{2}_{b}=(1/5)*v*^{2}*τ*_{R}*τ*_{N} with *v* the mean velocity of phonons and *τ*_{N} the collision time of normal (momentum conserving) phonons' collisions. Let us also mention that the relaxation times, the mean free paths and the heat conductivities are not independent but, according to the phonon kinetic theory, they are related by
3.6wherein *v*_{d}=*l*_{d}/*τ*_{d} and *v*_{b}=*l*_{b}/*τ*_{b} designate the mean velocity of the diffusive and ballistic phonons, respectively. Expressions (3.2), (3.4) and (3.5) provide the basic set of the eight scalar evolution equations for the eight unknowns *u*_{d}, *u*_{b}, *q*_{d} and *q*_{b}.

*Step 3. Elimination of the fluxes* *q*_{d} *and* *q*_{b}. This operation is easily achieved and is shown in appendix A. Assuming that all the transport coefficients are constant, one is led to the two second-order linear coupled differential equations,
3.7and
3.8Expressions (3.7) and (3.8) are the key relations of our model. Setting *τ*_{d}=*τ*_{b}=*τ*, *λ*_{d}=*λ*_{b}=*λ*, and *c*_{d}=*c*_{b}=*c*, making use of the energy balance (3.2*b*) for the ballistic phonons, one directly recovers Chen's basic result from expression (3.7)—namely
3.9This relation differs from the telegraph equation by the presence of the term ∇⋅*q*_{b}. In Chen's formalism, the heat flux vector *q*_{b} has been obtained by using the kinetic definition of the heat flux and by solving the Boltzmann equation. Here, we do not refer to a kinetic approach but solve the problem exclusively at the macroscopic level. It should also be emphasized that our model is more general than that of Chen, who introduced, without any justification, the simplifying assumptions that *τ*_{d}=*τ*_{b}. Moreover, Chen remains silent about the signs of *τ*_{i}, *λ*_{i} and *c*_{i} (*i*=*d*,*b*).

## 4. Application: transient temperature distribution in thin films

The foregoing model will be applied to the study of transient heat conduction in a one-dimensional thin film of thickness *L*, which may be of the same order of magnitude as, or even smaller than, the mean free path *l* of the phonons. Heat capacity and heat conductivity are assumed to be constant and to take the same values for the diffusive and ballistic phonons; internal energy sources are absent (*r*=0). Initially, the system is at uniform energy *u*_{0} or, using an equivalent terminology, at the ‘quasi-temperature’ *T*_{0} related to *u*_{0} by *u*_{0}=*cT*_{0}. The lower surface *z*=0 is suddenly brought at *t*=0 to the ‘quasi-temperature’ *T*_{1}=*T*_{0}+Δ*T*, while the upper surface *z*=*L* is kept at the ‘quasi-temperature’ *T*_{0}. In addition, we introduce the Knudsen numbers *Kn*_{i}=*l*_{i}/*L* (*i*=*d*,*b*), which, by virtue of expression (3.6), can be given the more general form
4.1Having in mind numerical solutions, it is convenient to use dimensionless quantities
4.2with *θ*_{d},*θ*_{b} and *θ*(=*θ*_{d}+*θ*_{b}) designating the non-dimensional energy (or quasi-temperature) associated with the ballistic, diffusive and total energy, respectively. The corresponding evolution equations (3.7) and (3.8) now take the form
4.3and
4.4

### (a) Initial conditions

At *t*=0, the sample is at uniform temperature *T*_{0}, which implies that the total energy is given by *u*(*z*,0)=*u*_{d}(*z*,0)+*u*_{b}(*z*,0)=*cT*_{0}. But it is reasonable to suppose that, at short times, the ballistic phonons are dominant so that the initial energy will be essentially of ballistic nature, leading to *u*_{b}(*z*,0)=*cT*_{0} or, in dimensionless notation,
4.5Throughout the sample, at time *t*=0, the heat flux ** q** is also zero; as a consequence of the energy balance (2.5), it is checked that initially ∂

*θ*(

*z**,

*t**)∂

*t**=0; this result remains, in particular, satisfied under the assumptions 4.6

### (b) Boundary conditions

The formulation of the boundary conditions is a more delicate problem. Their importance has to be emphasized because, in nanomaterials, their influence is felt throughout the whole system. To satisfy the conditions *θ*(0,*t**)=1 and *θ*(1,*t**)=0, the simplest suggestion would be to suppose that, at *z**=0, *θ*_{b}(0,*t**)=1 together with *θ*_{d}(0,*t**)=0, while, at *z**=1, the temperature of both the ballistic and diffusive constituents would be zero. However, such expressions are too simple and do not, in particular, cope with temperature jumps owing to thermal boundary resistance as discussed in several papers (Swartz & Pohl 1989; Joshi & Majumdar 1993; Chen 2002; Naqvi & Waldenstrom 2005). This is the reason why we have considered the following boundary conditions for the ballistic carriers:
4.7The quantity *a*, which represents the temperature jump of the ballistic phonons at the face *z**=0 at *t**=0, is taken to be equal to 1/2. This value may be understood statistically: as the temperature boundary condition at *z**=0 actually represents an internal energy boundary condition, it can be said that the ballistic phonons which are generated at the heated face are formed by half of the carriers at the initial internal energy *θ*_{b}=0, and the other half at the value *θ*_{b}=1 corresponding to the energy at the face where the temperature is suddenly increased. This result is confirmed by Chen (2002), who was able to determine the explicit expression of *θ*_{b} (*z**,*t**) by solving the Boltzmann equation, from which results that indeed *θ*_{b}(0,*t**)=1/2, at the heated boundary *z**=0. *A posteriori*, it is shown later on that this value leads to results which match satisfactorily well with other different approaches. Concerning the diffusive carriers, along with Chen (2002) we also assume that both of the interfaces are black phonon emitters and absorbers, implying that the boundaries are made only of incident diffusive carriers. Combining the Cattaneo equation and the Marshak boundary condition (Modest 1993) for black body thermal radiation, one obtains (Chen 2002)
4.8with the positive and negative signs at the right-hand side corresponding to the lower *z**=0 and upper *z**=1 faces, respectively; the factor (*Kn*_{d}/*Kn*_{b})^{2} is not present in Chen's developments because of his hypothesis of equality of relaxation times.

### (c) Discussion of the results

In the first stage, we have assumed that *Kn*_{b}=*Kn*_{d}=*Kn* because it is necessary to check the validity of our model by comparing with previous different approaches. In particular, we have compared our results with those of Joshi & Majumdar (1993), who solved the Boltzmann equation of phonons' radiative transfer (EPRT) model, the Chen (2001, 2002) ballistic–diffusive model and the model proposed by Alvarez & Jou (2010), who used a modified Fourier law with a heat conductivity depending on the Knudsen number. A modified version of Alvarez and Jou's work was recently proposed by Xu & Hu (2011), who based their analysis on a coarse graining of the Boltzmann equation. In addition, for the sake of completeness, we have solved the hyperbolic Cattaneo and the parabolic Fourier equations for the identical geometry and boundary conditions.

The non-dimensional temperature profiles for different *Kn* values (*Kn*=0.1; 1 and 10) versus the distance at different times are represented in figures 1–3. To emphasize the specific roles of the two constituents, we have made explicit the contributions of the total, ballistic and diffusive components. The region close to the hot side is mainly dominated by the ballistic component contribution, which decreases with space, while the diffusive component is increasing up to a maximum, after which one observes a descent towards zero; the descent is the steepest as *Kn* becomes smaller. As expected, the influence of the ballistic constituent becomes more important as *Kn* is increased, while the role of the diffusive constituent is dominant for small and intermittent *Kn*'s and is growing with time. This observation reflects the conversion of the ballistic internal energy into the diffusive one as time passes. It is also shown that, for *Kn*=10, the steady state is reached rather soon (after *t**=1) and is decreasing linearly with space (figure 3). The results are in qualitative accord with the aforementioned formalisms with, however, small discrepancies at small times (*t**<0.1), especially for *Kn*=10. To avoid overloaded graphs, we have deliberately not plotted the results of the EPRT, Chen and Alvarez–Jou models as they are very close to ours. It is concluded that our description matches the results derived from various points of view, ranging from macroscopic, microscopic and mixed micro–macro approaches. Note also that, for increasing values of *Kn* (especially *Kn*=10), a temperature jump is observed at the cold face. This indicates that the ballistic part exhibits a strong wall resistance not only at the hot but also at the cold face (especially at large *Kn*'s; figure 3). The small bump just before the temperature jump is caused by numerical errors owing to the abrupt temperature change.

It is clearly seen that both the Cattaneo and Fourier descriptions lead to unrealistic results. Neither of these models predicts the temperature jumps at the boundary. Moreover, they yield overestimated values for the temperature profiles as they do not integrate the specific properties of heat transport at nanoscales; this is particularly true at large *Kn*'s. This is not surprising as the Cattaneo and Fourier laws give rise to an overestimated heat conductivity (Zhang 2007; Alvarez & Jou 2008). As observed in figures 2*b* and 3*a*, the Cattaneo equation exhibits a temperature discontinuity, propagating like an attenuating wave, the attenuation being due to the diffusion; at large time values, both the Cattaneo and Fourier limits show the same linear behaviour with respect to the spatial coordinate (figures 1*c*, 2*c* and 3*c*).

To better understand the specific contributions of the ballistic and diffusive constituents when the corresponding relaxation times are unequal, we have in the second step considered different values of *Kn*_{d} and *Kn*_{b}. To be explicit, we have fixed *Kn*_{d}=0.1 with *Kn*_{b} taking the values 1 and 10. The results, which are plotted in figures 4 and 5, exhibit the same general tendency as in the case of equal *Kn* values with the ballistic contribution being dominant at the *z**=0 heated face; the ballistic carriers tend also to play a more important role at the cold face *z**=1 as time and *Kn*_{b} are becoming larger. We also notice that the peak in the diffusive distribution (figure 1*c*) is disappearing. It is not surprising to observe that the diffusive contribution becomes minute at large *Kn*_{b}/*Kn*_{d} ratios (see figure 5, for which *Kn*_{b}/*Kn*_{d}=100). We also note that, at these values, the distribution of the temperature is practically linear and quickly reaches its stationary value after *t**=1; indeed, calculated curves at *t**=10 indicate no change.

## 5. Final comments and conclusion

A thermodynamic description of transient heat transport at nanoscales based on EIT is proposed. The problem is important in the context of nano-electronics and heat transport in new materials. The model is original and purely macroscopic. The central assumption of the present work is that, contrary to previous approaches, the set of variables, namely the internal energy and the energy flux, is split into contributions of diffusive and ballistic nature. Heat transport is viewed as a two-fluid diffusion–reaction process with ballistic particles converting into diffusive ones. The latter are obeying a Cattaneo equation, while the behaviour of the ballistic phonons is governed by a Guyer–Krumhansl relation. This choice is motivated by the property that non-local effects are dominating in ballistic collisions.

The most important results of the present work are embodied in the differential equations (4.2) and (4.3), which describe the behaviour of the diffusive and ballistic internal energies. These relations have been derived after elimination of the ballistic and diffusive heat fluxes from the basic set of time-evolution equations constituted by the balance of energies, the Cattaneo and Guyer–Krumhnsl equations. The choice of the initial and boundary conditions was inspired by earlier work by several authors (Joshi & Majumdar 1993; Chen 2001; Alvarez & Jou 2010).

One of our objectives was to convince the reader of the flexibility and wide range of applicability of EIT. It is shown that a rather simple model is able to cope with much of the results derived from more sophisticated approaches. It should however be kept in mind that the present work rests on several simplifying assumptions: for instance, from a fundamental point of view, questions may be raised about the definition of temperature at nanoscales. To circumvent this problem, in our analysis, temperature was understood to be a measure of internal energy; the quantities *θ*_{d} and *θ*_{b} of the diffusive and ballistic components must therefore be understood to be quasi-temperatures, defined as a measure of the corresponding energies *u*_{d} and *u*_{b} to which they are related by the simple expressions *θ*_{d}=*u*_{d}/*c*, *θ*_{b}=*u*_{b}/*c* with *c* designated as the heat capacity. Moreover, our approach is restricted to the linear domain as all nonlinear contributions are omitted. In addition, coupling between diffusive and ballistic heat fluxes has been neglected. It should be realized that the formalism discussed above represents only the first step towards a more elaborate description of heat transport at micro- and nanoscales. In particular, it is expected that higher order fluxes (the flux of the heat flux, the flux of the flux of the heat flux,…) (Jou *et al*. 2010*a*,*b*) should be introduced from the start to cope with the particulate behaviour of heat carriers at short wavelengths, but the difficulty is then the physical interpretation of these new variables coupled with the complexity of the mathematical formalism. The selection of the most appropriate set of state variables remains an open problem. Finally, as shown in the previous section, the establishment of appropriate boundary conditions remains a delicate task. In that respect, recent work (Jou *et al.* 2010*b*, 2011) describes interesting and original perspectives.

In spite of the above limitations, application of the model to the problem of transient heat conduction in materials with thickness of the order of magnitude of the mean free path of heat carriers has led to satisfactory results. Indeed, after comparison with earlier results derived from several works based on completely different approaches, we have obtained results exhibiting qualitative agreement.

## Acknowledgements

The present study is supported by a collaborative project between Wallonie–Bruxelles and Quebec under grant 06-809 (period 2009–2011). Discussions on an earlier version with Professors A. Valenti (University of Catania) and A. Palumbo (University of Messina) were highly appreciated. Useful comments by Professors P.C. Dauby and Th. Desaive (Liege University) are also acknowledged.

## Appendix A. Derivation of equations (3.7) and (3.8)

Application of operator ∇ on the Cattaneo equation (3.4) and use of *T*_{d}=*u*_{d}/*c*_{d} yields
A1wherein ∂_{t} denotes the time derivative. Moreover, the balance of total energy (2.5) can be written in the form
A2After differentiating equation (A2) with respect to time and substituting in equation (A1), one is led to
A3wherein ∇⋅*q*_{d} has been eliminated by means of equation (A2). We now eliminate by taking the time derivative of equation (3.2*b*) with *r*_{b}=−*u*_{b}/*τ*_{b}, and we multiply this equation by *τ*_{d}; the result is
A4

Substituting this result in equation (A3) and replacing, in the right-hand side of equation (A3), the two terms ∇⋅*q*_{b}+∂_{t}*u*_{b} by −*u*_{b}/*τ*_{b}, by virtue of equation (3.2*b*), we finally return to equation (3.7) after multiplying by (−1),
A5

To derive expression (3.8), we start from the time derivative of the energy balance equation (3.2*b*), which, multiplied by *τ*_{b}, takes the form
A6

To eliminate the term ∇⋅(∂_{t}*q*_{b}), we will use the Guyer–Krumhansl equation (3.6) to which we apply the operator ∇⋅, from which it follows that
A7

After substitution of equation (A7) in equation (A6) and using equation (3.2*b*) to eliminate ∇⋅*q*_{b}, one obtains, after some elementary arithmetic, relation (3.8)—namely
A8

- Received February 1, 2011.
- Accepted May 26, 2011.

- This journal is © 2011 The Royal Society