Modelling Bidispersive Local Thermal Non-Equilibrium Flow

Abstract

In this work, we present a system of equations which describes non-isothermal flow in a bidispersive porous medium under conditions of local thermal non-equilibrium. The porous medium consists of macro pores, and in the solid skeleton are cracks or fissures which give rise to micro pores. The temperatures in the solid skeleton and in the fluids in the macro and micro pores are all allowed to be independent. After presenting the general model, we derive a result of universal stability, which guarantees exponential decay of the solution for all initial data. We further present a concrete example by specializing the model to the problem of thermal convection in a layer heated from below.

1. Introduction

Fluid flow in a porous medium is a subject with a long history which is currently very active; for example, see Nield and Bejan [1], Straughan [2]. In particular, recent attention has often focussed on local thermal non-equilibrium, which is where there is a single porosity but the fluid and solid skeleton have different temperatures (e.g., Banu and Rees [3], Barletta and Rees [4,5], Eltayeb [6], Nield [7,8], Nield and Bejan [1], Nield and Kuznetsov [9,10], Nouri-Borujerdi et al. [11], Postelnicu and Rees [12], Rees [13,14], Rees and Bassom [15], Rees et al. [16], Straughan [17,18,19]).

In addition, another area which has separately attracted much attention is flow in a bidispersive (or double porosity) material. Here, the usual macro pores are present, but in addition, the solid skeleton contains cracks or fissures which give rise to micro pores. This is studied in detail in, for example, Falsaperla et al. [20], Nield [21], Nield and Bejan [1], Nield and Kuznetsov [9,10], Straughan [19,22].

The goal of this paper is to combine both approaches and develop a theory for a porous material with a double porosity structure, but one which allows for different temperatures in the solid skeleton, the fluid in the macro pores, and the fluid in the micro pores.

It is very important to realize that there are situations where the local thermal non-equilibrium theory predicts very different temperature evolution profiles from what is observed with a single temperature; see, for example, the very interesting analyses of Rees and Bassom [15] and Rees et al. [16]. Furthermore, in a very useful work, Rees [13,14] has shown how one may derive estimates for the interaction coefficients for the temperatures in a local thermal non-equilibrium theory, thus rendering this theory to be of practical value. In addition, David et al. [23], Homand-Etienne and Houpert [24], and Siratovich et al. [25] show where the inclusion of thermal effects can lead to stress-induced micro cracking in rocks such as granite, and so we believe that the inclusion of local thermal non-equilibrium effects in a bidispersive porous medium will be useful in real life.

We thus consider a porous body which has a porosity $\varphi$, i.e., the ratio of the volume of the macro pores to the total volume of the saturated porous material. The solid skeleton contains much smaller micro pores which may be cracks, or may even be due to a man-made structure (e.g., the picture in Nield and Kuznetsov [10], p. 3069). The micro pores give rise to a porosity $ϵ$, which is defined as being the ratio of the volume occupied by the micro pores to the volume of the porous body which remains after the macro pores are removed. This means that the fraction of volume occupied by the micro pores is $ϵ(1-\varphi )$ and the fraction occupied by the solid skeleton is $(1-ϵ)(1-\varphi )$.

It is worth pointing out that perhaps the major reason for the recent interest in double porosity materials is due to the many applications in real engineering and geophysics situations. For example, landslides and land movement due to thermal gradients (e.g., Hammond and Barr [26], Montrasio et al. [27]); provision of clean and safe drinking water (e.g., Ghasemizadeh et al. [28], Zuber and Motyka [29]); the controversial area of hydraulic fracturing for natural gas (e.g., Huang et al. [30], Kim and Moridis [31]); and many other applications may be found in Straughan [19].

2. Basic Model

As we have written in the introduction and reiterate now, the porous body is composed of a solid skeleton, macro pores, and micro pores. So, besides a macroporosity $\varphi$, there is a microporosity $ϵ$ such that $ϵ(1-\varphi )$ and $(1-ϵ)(1-\varphi )$ denote the fractions of the body occupied by the micro pores and the solid skeleton, respectively. We follow Nield and Kuznetsov [10] and denote by $U_i^f$ and $U_i^p$ the pore-averaged velocities in the macro and micro pores. In addition, Nield and Kuznetsov [10] denote by $T^f$ and $T^p$ the temperature in the macro and micro pores. The solid skeleton does not move, but we denote the temperature there by $T^s$.

The momentum and continuity equations in the macro and micro pores are given by Nield and Kuznetsov [10] for a Brinkman porous body. We follow these writers, but employ a Darcy theory and so omit the Brinkman terms (cf. Straughan [19]). Thus, the momentum and continuity equations for the macro pores are:

$$-\frac{\mu }{K_f}{U}_i^f-\zeta (U_i^f-U_i^p)-p_{,i}^f-\frac{g_i{\rho }_0\alpha \varphi }{D}{T}^f-\frac{g_i{\rho }_0\alpha (1-\varphi )ϵ}{D}{T}^p=0,$$

(1)

$$U_{i,i}^f=0,$$

(2)

where $\mu$ is the dynamic viscosity, $K_f$, $p^f$ are the permeability and the pressure in the macro pores, $g_i$ is the gravity vector, $\alpha$ is the coefficient of thermal expansion in the fluid, and ${\rho }_0$ is a constant. The term $D=\varphi +(1-\varphi )ϵ$, $\zeta$ is an interaction coefficient, and standard indicial notation is used throughout with subscript $,i$ denoting $\partial /\partial x_i$.

In writing (1), we have followed Nield and Kuznetsov [10] and used a Boussinesq approximation to write the buoyancy term as linear in the weighted temperatures so that the buoyancy term has the form

$$g_i{\rho }_0\{1-\alpha (T-T_0)\},$$

(3)

where $T=\left(\varphi T^f+(1-\varphi )ϵT^p\right)/D$, $T_0$ is a reference temperature, and in (1) the constant terms in (3) have been absorbed in $p^f$.

In a similar manner, the momentum and continuity equations in the micro pores have the form

$$-\frac{\mu }{K_p}{U}_i^p-\zeta (U_i^p-U_i^f)-p_{,i}^p-\frac{g_i{\rho }_0\alpha \varphi }{D}{T}^f-\frac{g_i{\rho }_0\alpha (1-\varphi )ϵ}{D}{T}^p=0,$$

(4)

$$U_{i,i}^p=0.$$

(5)

To write the balance of energy equations for the temperature fields, we let $V_i^f$ and $V_i^p$ be the actual velocities in the macro and micro pores. These are connected to the pore-averaged velocities by $U_i^f=\varphi V_i^f$ and $U_i^p=ϵ(1-\varphi )V_i^p$. Thus, the equations for the temperature fields $T^s,T^f$, and $T^p$ are

$${ϵ}_1{(\rho c)}_s{T}_{,t}^s={ϵ}_1{\kappa }_s\Delta T^s+s_1(T^f-T^s)+s_2(T^p-T^s),$$

(6)

$$\varphi {(\rho c)}_f{T}_{,t}^f+\varphi {(\rho c)}_f{V}_i^f{T}_{,i}^f=\varphi {\kappa }_f\Delta T^f+h(T^p-T^f)+s_1(T^s-T^f),$$

(7)

$${ϵ}_2{(\rho c)}_p{T}_{,t}^p+{ϵ}_2{(\rho c)}_p{V}_i^p{T}_{,i}^p={ϵ}_2{\kappa }_p\Delta T^p+h(T^f-T^p)+s_2(T^s-T^p),$$

(8)

where

$${ϵ}_1=(1-ϵ)(1-\varphi ),{ϵ}_2=(1-\varphi )ϵ,$$

and where ${(\rho c)}_s$, ${(\rho c)}_f$, ${(\rho c)}_p$ are the products of the density and the specific heat at constant pressure in the solid, in the fluid in the macro pores, and in the fluid in the micro pores, respectively. The terms ${\kappa }_s$, ${\kappa }_f$, and ${\kappa }_p$ are thermal conductivities in the solid, and in the fluid in the macro and micro pores, respectively. We denote by s, f, and p the solid, the macro pores, and the micro pores. The terms h, $s_1$, and $s_2$ are interaction coefficients, and we have here assumed that the interactions are linear in the temperature differences.

The governing equations are (1), (2), and (4)–(8). These equations hold in a bounded regular domain $\Omega \subset {\mathbb{R}}^3$, for $t>0$.

Remark 1.

As we believe this is the first article to develop a local thermal non-equilibrium theory for a bidispersive porous material, we have followed the lead of Banu and Rees [3] and Nield and Kuznetsov [10] in ignoring fluid acceleration terms in Equations (1) and (4). For a Darcy porous material where we expect relatively slow flow, we believe that this is realistic. We could easily include acceleration (inertia) terms in (1) and (4), and very little would change in the rest of the article.

Remark 2.

In this work, we treat the thermal coupling terms $h,s_1$ and $s_2$ as constants. This is in line with the treatment for a local thermal non-equilibrium single porosity model by Banu and Rees [3], and also the interaction coefficients are treated as constant by Nield and Kuznetsov [10] in their development of a bidispersive porous model. For the present scenario, it is conceivable that the thermal interaction coefficients may depend on the solution, but this would lead to an extremely complicated model. In connection with this, we point out that Franchi et al. [32] study continuous dependence on the interaction parameters in a single-temperature bidispersive porous medium. We believe such a continuous dependence (or structural stability) result could be established for the model presented here, but the calculations will be technically involved and this is deferred to a later article.

3. Universal Stability

In this section we define the governing equations on $\Omega \times \{t>0\}$ with conditions on the boundary $\mathsf{\Gamma }$ of $\Omega$. Suppose on $\mathsf{\Gamma }\times \{t>0\}$

$$U_i^f{n}_i=0,U_i^p{n}_i=0,T^s=T_1(\mathbf{x},t),T^f=T_2(\mathbf{x},t),T^p=T_3(\mathbf{x},t),\mathbf{x}\in \mathsf{\Gamma },t>0,$$

(9)

for given functions $T_1$, $T_2$, $T_3$, where $\mathbf{n}$ is the unit outward normal to $\mathsf{\Gamma }$.

The initial conditions are

$$T^s(\mathbf{x},0)=T_0^s(\mathbf{x}),T^f(\mathbf{x},0)=T_0^f(\mathbf{x}),T^p(\mathbf{x},0)=T_0^p(\mathbf{x}),$$

(10)

for prescribed functions $T_0^s,T_0^f,T_0^p$. Let the boundary-initial value problem comprising (1), (2), and (4)–(10) be denoted by $\mathcal{P}$.

We let ${\overline{U}}_i^f$, ${\overline{U}}_i^p$, ${\overline{p}}^f$, ${\overline{p}}^p$, ${\overline{T}}^s$, ${\overline{T}}^f$, and ${\overline{T}}^p$ be a basic solution to $\mathcal{P}$. We wish to analyze the stability of this basic solution. Hence, we define perturbations $u_i^f$, $u_i^p$, ${\pi }^f$, ${\pi }^p$, ${\theta }^s$, ${\theta }^f$, and ${\theta }^p$ by

$$\begin{array}{c}{U}_i^f={\overline{U}}_i^f+u_i^f,U_i^p={\overline{U}}_i^p+u_i^p,\hfill \\ p^f={\overline{p}}^f+{\pi }^f,p^p={\overline{p}}^p+{\pi }^p,\hfill \\ T^s={\overline{T}}^s+{\theta }^s,T^f={\overline{T}}^f+{\theta }^f,T^p={\overline{T}}^p+{\theta }^p.\hfill \end{array}$$

We now non-dimensionalize the governing equations for the perturbations $u_i^f$, $u_i^p$, ${\pi }^f$, ${\pi }^p$, ${\theta }^s$, ${\theta }^f$, and ${\theta }^p$. We choose length and time scales L and $\mathcal{T}=\varphi {(\rho c)}_f/h$, and we define non-dimensional numbers ${\mathsf{\Lambda }}_h$, ${\mathsf{\Lambda }}_p$, and ${\mathsf{\Lambda }}_s$ by

$${\mathsf{\Lambda }}_h=\frac{{(\rho c)}_p}{{(\rho c)}_f},{\mathsf{\Lambda }}_p=\frac{{(\rho c)}_p{ϵ}_2}{{(\rho c)}_f\varphi },{\mathsf{\Lambda }}_s=\frac{{(\rho c)}_s{ϵ}_1}{{(\rho c)}_f\varphi }.$$

We further define non-dimensional parameters $R_1$, $R_2$, ${\mu }_1$, ${\mu }_2$, ${\varphi }_1$, ${\varphi }_2$, $S_1$, $S_2$, ${\mathsf{\Gamma }}_s$, ${\mathsf{\Gamma }}_p$, ${\mathsf{\Gamma }}_f$ by

$$R_1=\frac{{(\rho c)}_{f}U}{Lh}{R}_2=\frac{\alpha {\rho }_0{T}^{♯}}{\zeta U}{\mu }_1=\frac{\mu }{\zeta K_f},{\mu }_2=\frac{\mu }{\zeta K_p},$$

$${\varphi }_1=\frac{\varphi }{D},{\varphi }_2=\frac{(1-\varphi )ϵ}{D},S_1=\frac{s_1}{h},S_2=\frac{s_2}{h},$$

$${\mathsf{\Gamma }}_s=\frac{{ϵ}_1{\kappa }_s}{L^{2}h},{\mathsf{\Gamma }}_f=\frac{\varphi {\kappa }_f}{L^{2}h},{\mathsf{\Gamma }}_p=\frac{{ϵ}_2{\kappa }_p}{L^{2}h},$$

where U and $T^{♯}$ are velocity and temperature scales. The quantities L, U, and $T^{♯}$ are specified exactly when one considers a particular problem of thermal flow, as for example in Section 4.

Hence, the non-dimensional perturbation equations may be shown to have the form

$$\begin{array}{ccc}\hfill {\mu }_1{u}_i^f+(u_i^f-u_i^p)& =& -{\pi }_{,i}^f-R_2{\varphi }_1{g}_i{\theta }^f-R_2{\varphi }_2{g}_i{\theta }^p,\hfill \end{array}$$

(11a)

$$\begin{array}{ccc}\hfill u_{i,i}^f& =& 0,\hfill \end{array}$$

(11b)

$$\begin{array}{ccc}\hfill {\mu }_2{u}_i^p+(u_i^p-u_i^f)& =& -{\pi }_{,i}^p-R_2{\varphi }_1{g}_i{\theta }^f-R_2{\varphi }_2{g}_i{\theta }^p,\hfill \end{array}$$

(11c)

$$\begin{array}{ccc}\hfill u_{i,i}^p& =& 0,\hfill \end{array}$$

(11d)

$$\begin{array}{ccc}\hfill {\mathsf{\Lambda }}_s{\theta }_{,t}^s& =& {\mathsf{\Gamma }}_s\mathsf{\Delta }{\theta }^s+S_1({\theta }^f-{\theta }^s)+S_2({\theta }^p-{\theta }^s),\hfill \end{array}$$

(11e)

$$\begin{array}{ccc}\hfill {\theta }_{,t}^f+R_1({\overline{U}}_i^f{\theta }_{,i}^f+{\overline{T}}_{,i}^f{u}_i^f+u_i^f{\theta }_{,i}^f)& =& {\mathsf{\Gamma }}_f\mathsf{\Delta }{\theta }^f+({\theta }^p-{\theta }^f)+S_1({\theta }^s-{\theta }^f),\hfill \end{array}$$

(11f)

$$\begin{array}{ccc}\hfill {\mathsf{\Lambda }}_p{\theta }_{,t}^p+{\mathsf{\Lambda }}_p{R}_1({\overline{U}}_i^p{\theta }_{,i}^p+{\overline{T}}_{,i}^p{u}_i^p+u_i^p{\theta }_{,i}^p)& =& {\mathsf{\Gamma }}_p\mathsf{\Delta }{\theta }^p+({\theta }^f-{\theta }^p)+S_2({\theta }^s-{\theta }^p).\hfill \end{array}$$

(11g)

From the definition of the perturbations and the conditions (9), the boundary conditions are

$$u_i^f{n}_i=0,u_i^p{n}_i=0,{\theta }^s=0,{\theta }^f=0,{\theta }^p=0,\mathrm{on}\mathsf{\Gamma }\times \{t>0\}.$$

(12)

The initial conditions are of form

$${\theta }^s(\mathbf{x},0)={\theta }_0^s(\mathbf{x}),{\theta }^f(\mathbf{x},0)={\theta }_0^f(\mathbf{x}),{\theta }^p(\mathbf{x},0)={\theta }_0^p(\mathbf{x}),\mathbf{x}\in \Omega ,$$

for prescribed functions ${\theta }_0^s$, ${\theta }_0^f$, ${\theta }_0^p$.

Our goal is to now develop a result of universal stability in the sense of Serrin [33]. That is, we find conditions on the base flow to ensure $u_i^f$, $u_i^p$, ${\theta }^s$, ${\theta }^f$, and ${\theta }^p$ decay exponentially in a suitable measure for all initial data (i.e., a global stability estimate).

Let $\parallel ·\parallel$ and $(·,·)$ be the norm and inner product on $L^2(\Omega )$. To achieve the above aim, we commence by multiplying Equation (11a) by $u_i^f$ and we integrate over $\Omega$. Further, multiply (11c) by $u_i^p$ and integrate over $\Omega$. Use integration by parts and the boundary conditions (12), and then add the resulting equations to obtain

$${\mu }_1\parallel {\mathbf{u}}^f{\parallel }^2+{\mu }_2\parallel {\mathbf{u}}^p{\parallel }^2+{\parallel {\mathbf{u}}^f-{\mathbf{u}}^p\parallel }^2=-R_2{\varphi }_1(g_i{\theta }^f,u_i^f+u_i^p)-R_2{\varphi }_2(g_i{\theta }^p,u_i^f+u_i^p).$$

(13)

We use the arithmetic-geometric mean inequality on the right hand side of this identity, and one may thus show

$$\frac{{\mu }_1}{2}\parallel {\mathbf{u}}^f{\parallel }^2+\frac{{\mu }_2}{2}\parallel {\mathbf{u}}^p{\parallel }^2\le {\kappa }_1\parallel {\theta }^f{\parallel }^2+{\kappa }_2{\parallel {\theta }^p\parallel }^2,$$

(14)

where ${\kappa }_1$, ${\kappa }_2$ are given by

$${\kappa }_1={\varphi }_1\ell ,{\kappa }_2={\varphi }_2\ell ,$$

with

$$\ell ={\left|g\right|}^2{R}_2^2\left(\frac{{\varphi }_1+{\varphi }_2}{2}\right)\left(\frac{1}{{\mu }_1}+\frac{1}{{\mu }_2}\right).$$

The next step is to multiply (11e) by ${\theta }^s$, (11f) by ${\theta }^f$, and (11g) by ${\theta }^p$, integrate each over $\Omega$, use integration by parts and (12), and then add the results to find

$$\begin{array}{c}\frac{d}{dt}\left(\frac{{\mathsf{\Lambda }}_s}{2}\parallel {\theta }^s{\parallel }^2+\frac{1}{2}\parallel {\theta }^f{\parallel }^2+\frac{{\mathsf{\Lambda }}_p}{2}{\parallel {\theta }^p\parallel }^2\right)+{\mathsf{\Gamma }}_s\parallel \nabla {\theta }^s{\parallel }^2+{\mathsf{\Gamma }}_f\parallel \nabla {\theta }^f{\parallel }^2+{\mathsf{\Gamma }}_p\parallel \nabla {\theta }^p{\parallel }^2+{\parallel {\theta }^f-{\theta }^p\parallel }^2\hfill \\ +S_1\parallel {\theta }^f-{\theta }^s{\parallel }^2+S_2{\parallel {\theta }^p-{\theta }^s\parallel }^2=R_1({\overline{T}}^f{u}_i^f,{\theta }_{,i}^f)+{\mathsf{\Lambda }}_p{R}_1({\overline{T}}^p{u}_i^p,{\theta }_{,i}^p).\hfill \end{array}$$

(15)

Denote by ${\mathcal{I}}^f$ and ${\mathcal{I}}^p$

$${\mathcal{I}}^f=\underset{\Omega \times \{t>0\}}{max}|{\overline{T}}^f|,{\mathcal{I}}^p=\underset{\Omega \times \{t>0\}}{max}\left|{\overline{T}}^p\right|,$$

(16)

and then we may employ the arithmetic-geometric mean inequality on the right of (15) to show that

$$\begin{array}{c}\frac{d}{dt}\left(\frac{{\mathsf{\Lambda }}_s}{2}\parallel {\theta }^s{\parallel }^2+\frac{1}{2}\parallel {\theta }^f{\parallel }^2+\frac{{\mathsf{\Lambda }}_p}{2}{\parallel {\theta }^p\parallel }^2\right)+{\mathsf{\Gamma }}_s\parallel \nabla {\theta }^s{\parallel }^2+\frac{{\mathsf{\Gamma }}_f}{2}\parallel \nabla {\theta }^f{\parallel }^2+\frac{{\mathsf{\Gamma }}_p}{2}{\parallel \nabla {\theta }^p\parallel }^2\hfill \\ \le \frac{R_1^2{\mathcal{I}}_f^2}{2{\mathsf{\Gamma }}_f}\parallel {\mathbf{u}}^f{\parallel }^2+\frac{{\mathsf{\Lambda }}_p^2{R}_1^2{\mathcal{I}}_p^2}{2{\mathsf{\Gamma }}_p}{\parallel {\mathbf{u}}^p\parallel }^2.\hfill \end{array}$$

(17)

We now employ (14) to estimate the right hand side of (17), and thus we arrive at

$$\begin{array}{c}\frac{d}{dt}\left(\frac{{\mathsf{\Lambda }}_s}{2}\parallel {\theta }^s{\parallel }^2+\frac{1}{2}\parallel {\theta }^f{\parallel }^2+\frac{{\mathsf{\Lambda }}_p}{2}{\parallel {\theta }^p\parallel }^2\right)+{\mathsf{\Gamma }}_s{\lambda }_1{\parallel {\theta }_s\parallel }^2\hfill \\ +\left(\frac{{\mathsf{\Gamma }}_f{\lambda }_1}{2}-{\kappa }_1{R}_1^{2}Q\right)\parallel {\theta }^f{\parallel }^2+\left(\frac{{\mathsf{\Gamma }}_p{\lambda }_1}{2}-{\kappa }_2{R}_1^{2}Q\right){\parallel {\theta }^p\parallel }^2\le 0,\hfill \end{array}$$

(18)

where

$$Q=\frac{{\mathcal{I}}_f^2}{{\mu }_1{\mathsf{\Gamma }}_f}+\frac{{\mathsf{\Lambda }}_p^2{\mathcal{I}}_p^2}{{\mu }_2{\mathsf{\Gamma }}_p},$$

where we have also employed Poincaré’s inequality and ${\lambda }_1$ is the first eigenvalue in the membrane problem for $\Omega$.

From inequality (18), we see that a condition for universal stability is that the coefficients of the last two terms are positive. If we return to the definition of ${\kappa }_1$ and ${\kappa }_2$, then we find that a sufficient condition for universal stability is that

$$\frac{{\mathsf{\Gamma }}_f{\lambda }_1}{2}>{\varphi }_{1}H{R}_1^2{R}_2^2\left[\frac{{\mathcal{I}}_f^2}{{\mu }_1{\mathsf{\Gamma }}_f}+\frac{{\mathsf{\Lambda }}_p^2{\mathcal{I}}_p^2}{{\mu }_2{\mathsf{\Gamma }}_p}\right]$$

and

$$\frac{{\mathsf{\Gamma }}_p{\lambda }_1}{2}>{\varphi }_{2}H{R}_1^2{R}_2^2\left[\frac{{\mathcal{I}}_f^2}{{\mu }_1{\mathsf{\Gamma }}_f}+\frac{{\mathsf{\Lambda }}_p^2{\mathcal{I}}_p^2}{{\mu }_2{\mathsf{\Gamma }}_p}\right]$$

where

$$H=\frac{{\left|\mathbf{g}\right|}^2({\varphi }_1+{\varphi }_2)}{2}\left(\frac{1}{{\mu }_1}+\frac{1}{{\mu }_2}\right).$$

It is likely that ${\mathcal{I}}_f$ and ${\mathcal{I}}_p$ are measurable quantities in a given situation and the numbers $R_1$ and $R_2$ may be calculated. Thus, this universal stability estimate should be of use. However, we do not expect it to be optimum, and for a particular flow situation it is better to return to the precise equations at hand. In the next section we present the appropriate equations for thermal convection in a plane layer.

4. Thermal Convection

We now specialize Equation (11) to the case of thermal convection. Suppose that the saturated porous medium is contained in the layer

$$\{(x,y)\in {\mathbb{R}}^2\}\times \{0<z<d\}$$

(19)

with the temperatures

$$\begin{array}{c}{T}^s=T^f=T^p=T_L,\mathrm{at}z=0,\end{array}$$

(20)

$$\begin{array}{c}{T}^s=T^f=T^p=T_U,\mathrm{at}z=d,\end{array}$$

(21)

where $T_L$, $T_U$ are constants with $T_L>T_U$. The velocity boundary conditions are $U_i^f{n}_i=0$, $U_i^p{n}_i=0$ at $z=0,d$. The steady solution in whose stability we are interested has form

$${\overline{T}}^s={\overline{T}}^f={\overline{T}}^p=T_L-\beta z=0;{\overline{U}}_i^f\equiv 0,{\overline{U}}_i^p\equiv 0,$$

where

$$\beta =\frac{T_L-T_U}{d}$$

is the temperature gradient.

If $u_i^f,u_i^p,{\pi }^f,{\pi }^p,{\theta }^s,{\theta }^f,{\theta }^p$ denote perturbations to the steady solution, then one verifies that these quantities satisfy the equations

$$\begin{array}{c}\frac{\mu }{K_f}{u}_i^f+\zeta (u_i^f-u_i^p)=-{\pi }_{,i}^f+\frac{g{\rho }_0\alpha \varphi }{D}{\theta }^f{k}_i+\frac{g{\rho }_0\alpha (1-\varphi )ϵ}{D}{\theta }^p{k}_i,\hfill \\ \frac{\mu }{K_p}{u}_i^p+\zeta (u_i^p-u_i^f)=-{\pi }_{,i}^p+\frac{g{\rho }_0\alpha \varphi }{D}{\theta }^f{k}_i+\frac{g{\rho }_0\alpha (1-\varphi )ϵ}{D}{\theta }^p{k}_i,\hfill \\ {ϵ}_1{(\rho c)}_s{\theta }_{,t}^s={ϵ}_1{\kappa }_s\Delta {\theta }^s+s_1({\theta }^f-{\theta }^s)+s_2({\theta }^p-{\theta }^s),\hfill \\ \varphi {(\rho c)}_f{\theta }_{,t}^f+{(\rho c)}_f{u}_i^f{\theta }_{,i}^f={(\rho c)}_f\beta w^f+\varphi {\kappa }_f\Delta {\theta }^f+h({\theta }^p-{\theta }^f)+s_1({\theta }^s-{\theta }^f),\hfill \\ {ϵ}_2{(\rho c)}_p{\theta }_{,t}^p+{(\rho c)}_p{u}_i^p{\theta }_{,i}^p={(\rho c)}_p\beta w^p+{ϵ}_2{\kappa }_p\Delta {\theta }^p+h({\theta }^f-{\theta }^p)+s_2({\theta }^s-{\theta }^p),\hfill \end{array}$$

(22)

where g is the gravity constant, $k=(0,0,1)$, $w^f=u_3^f$, and $w^p=u_3^p$.

We non-dimensionalize (22) with the time, length, pressure, and velocity scales ${(\rho c)}_f{d}^2/{\kappa }_f$, d, $d\zeta U$, ${\kappa }_f/{(\rho c)}_{f}d$; the temperature scale is

$$T^{♯}=U\sqrt{\frac{\zeta {(\rho c)}_f\beta d^2}{\varphi {\kappa }_{f}g{\rho }_0\alpha }}$$

(23)

and the Rayleigh number $R_a$ is

$$R_a=R^2=\frac{{(\rho c)}_f\beta d^{2}g\alpha {\rho }_0}{\zeta {\kappa }_f\varphi }.$$

Define ${\mu }_1,{\mu }_2,{\mathsf{\Lambda }}_s,{\mathsf{\Lambda }}_p,{\mathsf{\Lambda }}_{Tp},{\mathsf{\Lambda }}_{Ts},S_1,S_2$, and H by

$$\begin{array}{c}{\mu }_1=\frac{\mu }{\zeta K_f},{\mu }_2=\frac{\mu }{\zeta K_p},{\mathsf{\Lambda }}_s=\frac{{(\rho c)}_s}{{(\rho c)}_f},{\mathsf{\Lambda }}_p=\frac{{(\rho c)}_p}{{(\rho c)}_f},\hfill \\ {\mathsf{\Lambda }}_{Tp}={\mathsf{\Lambda }}_p\frac{{ϵ}_2}{\varphi },{\mathsf{\Lambda }}_{Ts}={\mathsf{\Lambda }}_s\frac{{ϵ}_1}{\varphi },S_1=\frac{s_1{d}^2}{\varphi {\kappa }_f},S_2=\frac{s_2{d}^2}{\varphi {\kappa }_f},H=\frac{h{d}^2}{\varphi {\kappa }_f}.\hfill \end{array}$$

Then, (22) may be rewritten in the non-dimensional form

$${\mu }_1{u}_i^f+(u_i^f-u_i^p)=-{\pi }_{,i}^f+R\frac{\varphi }{D}{\theta }^f{k}_i+R\frac{(1-\varphi )ϵ}{D}{\theta }^p{k}_i,$$

(24a)

$${\mu }_2{u}_i^p+(u_i^p-u_i^f)=-{\pi }_{,i}^p+R\frac{\varphi }{D}{\theta }^f{k}_i+R\frac{(1-\varphi )ϵ}{D}{\theta }^p{k}_i,$$

(24b)

$${\mathsf{\Lambda }}_{Ts}{\theta }_{,t}^s=K_{SV}\mathsf{\Delta }{\theta }^s+S_1({\theta }^f-{\theta }^s)+S_2({\theta }^p-{\theta }^s),$$

(24c)

$${\theta }_{,t}^f+\frac{1}{\varphi }{u}_i^f{\theta }_{,i}^f=R{w}^f+\mathsf{\Delta }{\theta }^f+H({\theta }^p-{\theta }^f)+S_1({\theta }^s-{\theta }^f),$$

(24d)

$${\mathsf{\Lambda }}_{Tp}{\theta }_{,t}^p+\frac{{\mathsf{\Lambda }}_p}{\varphi }{u}_i^p{\theta }_{,i}^p=R{\mathsf{\Lambda }}_p{w}^p+K_{PV}\mathsf{\Delta }{\theta }^p+H({\theta }^f-{\theta }^p)+S_2({\theta }^s-{\theta }^p),$$

(24e)

where $K_{SV}={ϵ}_1{\kappa }^s/\varphi {\kappa }^f$ and $K_{PV}={ϵ}_2{\kappa }_p/\varphi {\kappa }_f$, and $u_{i,i}^f=0,u_{i,i}^p=0$.

Equations (24) are defined on ${\mathbb{R}}^2\times \{z\in (0,1)\}\times \{t>0\}$, with

$$u_i^f{n}_i=0,u_i^p{n}_i=0,{\theta }^s=0,{\theta }^f=0,{\theta }^p=0.$$

on $z=0,1$.

In addition, we suppose $u_i^f,u_i^p,{\pi }^f,{\pi }^p,{\theta }^s,{\theta }^f,{\theta }^p$ satisfy a plane tiling periodicity in the $(x,y)$ directions (cf. Straughan [19], p. 189) for a typical hexagonal cell structure.

We do not attempt a complete analysis of the nonlinear stability and linear instability of system (24) here. While Straughan [22] has numerically verified that the linear instability results of Nield and Kuznetsov [10]—for the case where ${\theta }^s$ is neglected and the Brinkman equations are considered—are close to the nonlinear stability ones, the same is not true when the Darcy equations are taken into consideration (see Straughan [19]). Since we might expect oscillatory instabilities with ${\theta }^s\equiv 0$, we believe a careful instability analysis of (24) will require substantial numerical computation. This, together with a careful nonlinear energy stability analysis, will be a lengthy article, and we defer this to the future. However, we may commence a nonlinear energy stability here to give a view as to how things will behave.

Let the period cell of the solution be V, and let $\parallel ·\parallel$ and $(·,·)$ denote the norm and inner product on $L^2(V)$. To develop a fully nonlinear energy stability analysis, we multiply (24a) by $u_i^f$, (24b) by $u_i^p$, and integrate each over V. We likewise multiply (24c) by ${\theta }^s$, (24d) by ${\theta }^f$, (24e) by ${\theta }^p$, and integrate each over V. After addition of the resulting identities and use of the boundary conditions, one may show that

$$\frac{dE}{dt}=RI-D,$$

(25)

where

$$E(t)=\frac{1}{2}{\mathsf{\Lambda }}_{Ts}\parallel {\theta }^s{\parallel }^2+\frac{1}{2}\parallel {\theta }^f{\parallel }^2+\frac{1}{2}{\mathsf{\Lambda }}_{Tp}{\parallel {\theta }^p\parallel }^2,$$

$$\begin{array}{cc}\hfill I(t)=& \left(1+\frac{\varphi }{D}\right)(w^f,{\theta }^f)+\frac{(1-\varphi )ϵ}{D}({\theta }^p,w^f)\hfill \\ & +\frac{\varphi }{D}({\theta }^f,w^p)+\left[{\mathsf{\Lambda }}_p+\frac{ϵ(1-\varphi )}{D}\right](w^p,{\theta }^p),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill D(t)=& K_{SV}\parallel \nabla {\theta }^s{\parallel }^2+\parallel \nabla {\theta }^f{\parallel }^2+K_{PV}\parallel \nabla {\theta }^p{\parallel }^2+S_1\parallel {\theta }^f-{\theta }^s{\parallel }^2+S_2\parallel {\theta }^p-{\theta }^s{\parallel }^2+H{\parallel {\theta }^p-{\theta }^f\parallel }^2\hfill \\ & +{\mu }_1\parallel {\mathbf{u}}^f{\parallel }^2+{\mu }_2\parallel {\mathbf{u}}^p{\parallel }^2+{\parallel {\mathbf{u}}^f-{\mathbf{u}}^p\parallel }^2.\hfill \end{array}$$

The global nonlinear stability threshold follows by computing $\underset{\mathcal{H}}{max}I/D$ over a suitable space $\mathcal{H}$ (cf. Straughan [19]), and is then compared to the equivalent linear instability threshold.

5. Conclusions

We have produced a system of equations for a saturated double porosity body under conditions of local thermal non-equilibrium. This is effectively a generalization of the bidispersive theory of Nield and Kuznetsov [9,10], and the local thermal non-equilibrium theory for a single porosity material of Nield [7,8] and Banu and Rees [3].

We have shown that one may derive a universal stability estimate when a general base flow is considered. This is in the spirit of the original universal stability estimate for the Navier–Stokes equations by Serrin [33]. When one requires accurate quantitative stability estimates, then one must address a specific flow problem as we show in Section 4 when we introduce thermal convection.