## Abstract

There is increasing interest in convection in local thermal non-equilibrium (LTNE) porous media. This is where the solid skeleton and the fluid may have different temperatures. There is also increasing interest in thermal wave motion, especially at the microscale and nanoscale, and particularly in solids. Much of this work has been based on the famous model proposed by Carlo Cattaneo in 1948. In this paper, we develop a model for thermal convection in a fluid-saturated Darcy porous medium allowing the solid and fluid parts to be at different temperatures. However, we base our thermodynamics for the fluid on Fourier's law of heat conduction, whereas we allow the solid skeleton to transfer heat by means of the Cattaneo heat flux theory. This leads to a novel system of partial differential equations involving Darcy's law, a parabolic fluid temperature equation and effectively a hyperbolic solid skeleton temperature equation. This system leads to novel physics, and oscillatory convection is found, whereas for the standard LTNE Darcy model, this does not exist. We are also able to derive a rigorous nonlinear global stability theory, unlike work in thermal convection in other second sound systems in porous media.

## 1. Introduction

Convective flow problems in a porous medium where the fluid temperature, *T*_{f}, may be different from the solid skeleton temperature, *T*_{s}, are being increasingly studied. This situation, where the two temperatures may be different, is usually referred to as local thermal non-equilibrium (LTNE). One of the driving reasons for the increased attention of LTNE flows in porous media is the numerous amount of applications of this area in real life. For example, there are applications in tube refrigerators in space [1]; in nanofluid flows [2–4, ch. 8]; in fuel cells [5]; in resin flow, important in processing composite materials [6]; in nuclear reactor maintenance [7]; in heat exchangers [8]; in flows in microchannels [9]; in flow in porous metallic foams [10,11]; in textile transport [12]; and in convection in stellar atmospheres, cf. Straughan [3, ch. 8, 13]. An interesting article analysing various causes of LTNE situations is that of Virto *et al.* [14].

Continuum theories for LTNE effects on flow in porous materials appear to have started in the late 1990s (cf. the work of Nield [15], Minkowycz *et al.* [16] and Petit *et al.* [17]), and instability in thermal convection taking into account LTNE effects was addressed by Banu & Rees [18] and by Malashetty *et al.* [19]. Straughan [20] demonstrated that these instability results were really sharp by showing that they also represented a global nonlinear stability threshold, not just a linear instability one. Since then, many studies of thermal convection with LTNE have appeared, using a variety of geometries and incorporating various other effects, for example, rotation and double diffusion (see [21–24] and references therein).

In a separate development, second sound, the mechanism whereby temperature travels as a wave, is also a topic of increasing attention. In particular, as modern technology is creating smaller and smaller devices, the phenomenon of temperature travelling as a wave becomes increasingly important, especially in metallic-like solids. Pilgrim *et al.* [25] develop a mathematical model for finite-speed heat transport in semiconductor devices and they observe that …the ‘hyperbolic description will become increasingly important as device dimensions move even further into the deep sub-micron regime’ [25], p. 825. Many articles emphasizing the need to consider hyperbolic heat transport in nanowires and in thin films are reviewed in ch. 9 of the book by Straughan [4]. The same book, in general, analyses various occurrences of finite-speed heat transport. As Straughan [4] points out, most theoretical work involves the model of Cattaneo [26], and the history of Cattaneo theory is discussed in detail in Straughan [4, ch. 1].

Thermal convection in a fluid-saturated porous material allowing for a second sound effect has already been studied by Straughan [27], with an analysis for thermosolutal convection in Straughan [13], cf. Papanicolaou *et al.* [28]. Details of these developments may be found in the book by Straughan [4], ch. 8. However, this work uses only one temperature field and does not allow for the possibility of LTNE. Owing to the potential applications, our aim is to combine second sound and LTNE. However, because the second sound effect appears greater in solids, especially those involved in porous metallic foams, we restrict attention to this effect in solids, and retain the usual Fourier heat-transfer law in the fluid. We have not seen any work such as this before, and we find some novel results. We find that when an interaction coefficient is sufficiently large simultaneously with a relaxation time having sufficient magnitude, then commencement of convective motion according to linear instability theory is via oscillatory convection. It should be stressed that this is *not* the case with standard LTNE theory [18,19]. The linear instability analysis commences in a standard manner. However, one has to be careful because one does not find an equation directly for the Rayleigh number. Instead, one has to solve a quadratic and then minimize over the wavenumber. Care has to be taken with this process to ensure the correct root is chosen. In addition, we are able to determine a global nonlinear stability threshold. This is interesting because for the previous work on thermal convection, instability incorporating second sound did not appear to allow use of an energy method. The use of an energy method is novel because the system of partial differential equations involves Darcy's law, what appears to be a parabolic equation for the fluid temperature, and an essentially hyperbolic system for the solid temperature. In certain cases, the global stability threshold is close to the linear one, thereby greatly restricting the region where subcritical instabilities might arise.

## 2. Local thermal non-equilibrium model

The basic equations for thermal convection in a porous medium with LTNE effects are conveniently found in Straughan [20] or Straughan [29, pp. 172–177]. We rewrite these here, but modify the equation for the solid temperature to allow the heat flux to satisfy a Cattaneo law. Thus, our basic system of equations has the form
2.1In these equations, *x*_{i} and *t* denote space and time, *v*_{i},*p*,*T*^{s},*Q*_{i} and *T*^{f} denote fluid (pore averaged) velocity, pressure, solid temperature, heat flux in the solid and fluid temperature, respectively. The quantities *K*,*μ*,*g*,*α*,*ϵ*,*ρ*,*c*,*h*,*k*_{s},*k*_{f} and *τ*_{s}, denote permeability, fluid dynamic viscosity, gravity, fluid expansion coefficient, porosity, density, specific heat at constant pressure, a thermal interaction coefficient, thermal conductivity of the solid, thermal conductivity of the fluid and solid thermal relaxation time, respectively. A subscript or superscript ‘s’ or ‘f’ refers to the solid or fluid, Δ is the Laplace operator in three dimensions, standard indicial notation is used throughout, together with the Einstein summation convention, and **k**=(0,0,1).

Equation (2.1)_{1} represents Darcy's law, (2.1)_{2} is conservation of mass, (2.1)_{3} is the energy balance equation in the solid, (2.1)_{4} is Cattaneo's law for the solid heat flux and (2.1)_{5} is the energy balance equation for the fluid. Equations (2.1) hold in the layer , with gravity acting in the negative *z*-direction. The boundary conditions considered are
2.2where *T*_{L}, *T*_{U} are constants with *T*_{L}>*T*_{U}. The steady solution whose stability we are interested in is
2.3where *β* is the temperature gradient given by
The steady pressure field, , follows from (2.1)_{1}.

To study the instability of solution (2.3), we let be perturbations to , i.e. we put
We then derive from equations (2.1) and (2.3) the equations governing the perturbation quantities. These are non-dimensionalized with the length, time, velocity, pressure and heat scales and *Q*^{s}=*k*_{s}*T*^{♯}/*d*, where *T*^{♯} is the temperature scale, We further introduce the Rayleigh number *Ra*=*R*^{2}, and the non-dimensional coefficients *H*,*A*,*γ* and by
where *κ*_{f}=*k*_{f}/*ρ*_{f}*d*^{2}. Then, the non-dimensional nonlinear system of perturbation equations is
2.4In these equations, subscripts ‘*t*’ and ‘*i*’ denote partial differentiation with respect to time and *x*_{i}, respectively, and *w*=*u*_{3}. Rees *et al.* [23] studied a problem of injection of a hot fluid into a porous medium where the solid and fluid temperatures were allowed to be different. This induces a thermal shock into the system, and they were able to show that in the *H*→0 limit, the system comprising the temperature equations becomes hyperbolic rather than parabolic. For system (2.4), when the system is essentially parabolic. However, we believe that the occurrence of oscillatory instabilities is due to increasing , and the system becomes effectively hyperbolic. A similar phenomenon was witnessed by Straughan [13] in thermohaline convection, although no thermal non-equilibrium effects were included.

Equations (2.4) hold on the domain . The boundary conditions are
2.5and satisfy a plane tiling periodicity in the (*x*,*y*)-plane.

## 3. Instability analysis

To investigate instability, we discard the nonlinear term in equation (2.4)_{3}, take curlcurl of equation (2.4)_{1} and retain the third component. We then seek a time dependence such as e^{σt}, and then using the boundary conditions (2.5), we may develop a sin *nπz* series solution. After some calculation, one may arrive at
3.1where *a* is a wavenumber and *Λ*=*n*^{2}*π*^{2}+*a*^{2}. This is a cubic equation in *σ*. By setting *σ*=0, one determines the equation from whence one derives the stationary convection boundary as
3.2The stationary convection boundary is found for fixed *γ* and *H* by minimizing in *a*^{2} with *n*=1.

To determine the effect of oscillatory convection, we put *σ*=i*σ*_{1}, , in (3.1) and then the real and imaginary parts of the resulting equation yield the equations
3.3After some manipulation, removing , one finds that *R*^{2}*a*^{2} satisfies the quadratic equation
3.4where
and
with *k*_{1} and *k*_{2} given by
Care must be taken with finding the critical Rayleigh number from (3.4) because a cavalier (or direct numerical) approach can lead to oscillating between the roots for *Ra*.

Thus, the oscillatory convection boundary is determined from
3.5From (3.3), is given by
3.6The oscillatory convection threshold is found by minimizing in *a*^{2}, comparing with the minimum of , and checking with (3.6) whether or not.

Numerical results are reported in §5.

## 4. Global nonlinear stability

We use the energy method, cf. Straughan [30]. While this method is well known for standard fluid mechanics problems, the application to system (2.4) is non-standard, and so we supply brief details. We stress that with other systems involving thermal waves in convection, we have not seen how to derive global nonlinear stability bounds, cf. Straughan [4,13,27]. (This point is discussed in more detail in Straughan [4], pp. 193–195.) Thus, we believe this analysis is of interest in its own right.

The difficulty with Cattaneo systems is the lack of dissipation in the temperature field. Let *V* be a period cell for the solution to (2.4) and (2.5), and let ∥⋅∥ and (⋅,⋅) denote the norm and inner product on *L*^{2}(*V*). We commence by multiplying (2.4)_{4} by *ϕ*, (2.4)_{5} by , and integrating each over *V* . We add the results, observing that it is essential to do this in order to remove the terms, otherwise we cannot progress owing to the lack of dissipation in *ϕ*. Thus, we derive the following equation:
4.1We next multiply each of (2.4)_{1} and (2.4)_{3} by *u*_{i} and *θ*, respectively. This yields
4.2and
4.3For coupling parameters, λ_{1}, λ_{2}>0 to be determined, we now form (4.3)+λ_{1}(4.2)+λ_{2}(4.1) to obtain an energy equation
4.4where
4.5the dissipation function *D* is
4.6and the production term *I* has the form
4.7

Now put
4.8where is the space of admissible solutions. Then, if we require , from (4.4), we may obtain
4.9Next, using Poincaré's inequality and the definition of *D*, we see that
4.10By combining (4.10) and (4.9), one shows there is a constant *k*>0 such that
and *E* decays exponentially, yielding global stability in *θ*,*ϕ* and . Decay and global stability of *u*_{i} then follow from (4.2), which, after use of the arithmetic–geometric mean inequality on the right-hand side, yields

To obtain a global (for all initial data) nonlinear stability threshold, we must resolve the variational problem (4.8). Thus, take *R*_{E}=1, the threshold value, and then (4.8) leads to the Euler–Lagrange equations
4.11where *ζ*(**x**) is a Lagrange multiplier. Next, take curlcurl of (4.11)_{1} and retain the third component. Eliminate *w* and *ϕ* from the resulting system to obtain a fourth-order equation in *θ*. Use the boundary conditions to find that with *Λ*=*π*^{2}+*a*^{2}, the energy Rayleigh number *R*^{2} is then given by
4.12

We require to find
It is easy to see that the maximum of *R*^{2} in λ_{1} is achieved when λ_{1}=1. Then, a similar calculation involving λ_{2} yields the maximum with λ_{2}=1/*γ*. Thus, we find
which yields the energy Rayleigh number maximum as .

## 5. Numerical results and conclusions

Values for the parameters *A* and *γ* are easy to find for real materials. However, values for the interaction coefficient *H* and the non-dimensional relaxation time are more elusive. Rees [21,31] gives very interesting analyses where he develops a possible means to calculate *H* for various types of porous materials. In particular, he uses porous media composed of one-dimensional stripes of fluid between the solid, randomly striped one-dimensional porous media, two-dimensional media where the fluid occupies a checkerboard pattern, box-type configurations and random networks. His calculations indicate there is a strong correlation between the porosity and the thermal conductivities of the fluid and solid components, but there is also a major effect owing to a geometrical factor. This is a very interesting calculation, and will be useful when dealing with a known geometrical pattern of porous media and given solid and fluid components. We have computed many numerical solutions and report computations for the case where the solid skeleton is copper oxide, CuO, and aluminium oxide, Al_{2}O_{3}, both materials that have practical use in heat exchangers. We find oscillatory convection is possible in these cases, but for relatively high *H* values. Thus, we also report computations for the case where *A*=1 and *γ*=1.

For CuO and Al_{2}O_{3}, values for *A* and *γ* are (cf. Straughan [3])
and
The stationary convection boundary is found by minimizing in (3.2) in *a*^{2}, whereas the oscillatory convection boundary minimizes in (3.5), ensuring also that , as given by (3.6). While we have not been able to prove *n*=1 yields the minimum for with (3.5), all of our computations have verified this.

The behaviour we find for all three cases we have studied is exemplified by figure 1, which is for *A*=1, *γ*=1. This shows that the stationary convection curve increases with increasing *H*, but in a less than linear manner. Once *H* is sufficiently large, oscillatory convection dominates, as shown for , curve c; , curve b; and , curve a. Our computations did not find oscillatory convection for much less than 0.5 for each of the three cases studied. Thus, it would appear that oscillatory convection is possible, with a very strong stabilizing effect, but only when the relaxation time is sufficiently high. In figure 1, the global nonlinear stability curve is marked as E. Thus, the solution is definitely unstable above the appropriate curves S/a, S/b or S/c and definitely globally stable below curve E. This yields a band, between the curves S/c, S/b or S/a and E where possible subcritical instabilities may arise. (In ongoing three-dimensional computations on a triply penetrative convection problem, cf. Straughan [32], we definitely find the presence of subcritical instabilities, especially when oscillatory convection occurs.) As , our computations indicate that the curves a, b and c asymptote to E. If one uses the analysis of Rees [21] to estimate *H* for a high porosity material such as a metallic foam, then with the solid CuO or Al_{2}O_{3} and the fluid water, we find with *ϵ*=0.9, *k*_{f}=0.6, *k*_{s}=30 W m K^{−1}, then *H*≈1.23*C*, where *C* is a geometrical factor depending on the structure of the porous medium. For example, for a one-dimensional stripe of fluid, Rees [21] calculates *C*=12, and for *N* stripes, *C*=3*N*^{2}. Thus, the *H* values in tables 1–3, and in figures 1–3 are consistent with what we expect to find in practice, and one may expect oscillatory convection for sufficiently large.

Close bounds for the transition values from stationary to oscillatory convection are given in table 1. It should be observed that the pattern of behaviour shown in figure 1 is also found for the cases of CuO and Al_{2}O_{3}. Tables 2 and 3 give stationary convection and oscillatory convection values for Al_{2}O_{3} with and . Figures 2 and 3 pertain to a porous skeleton of Al_{2}O_{3} with . We see clearly in figure 2 that once oscillatory convection dominates, it has a very strong stabilizing effect. This behaviour is found in all cases we have studied. Likewise, figure 3 for the wavenumber is typical of all cases we have investigated. Before the onset of oscillatory convection, the wavenumber increases, meaning the convection cells decrease in size in the (*x*,*y*)-plane, i.e. they become narrower. At the transition to oscillatory convection, the cell size changes by a large amount to a much wider cell. This initially decreases in width relatively rapidly for increasing *H* and gradually settles to a less wide convection cell.

We have proposed a model for thermal convection in a fluid-saturated porous material where LTNE effects are present. The novelty is that we allow for temperature waves in the solid via a Cattaneo-like heat flux theory. Thermal convection in this model is analysed, and we discover novel consequences. If the thermal relaxation time is sufficiently great, then oscillatory convection is found to be the dominant mechanism. This is very different from the standard LTNE theory (cf. [18–20]). We are also able to provide a nonlinear global stability threshold that has not been seen in previous work dealing with thermal convection in the presence of temperature waves.

## Funding statement

This work was supported by the Leverhulme Research grant ‘Tipping points, mathematics, metaphors and meanings’.

## Acknowledgements

I thank two anonymous referees for helpful comments that have led to improvements in this article.

- Received March 21, 2013.
- Accepted June 11, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.