## Abstract

We show that the global nonlinear stability threshold for convection with a thermal non-equilibrium model is exactly the same as the linear instability boundary. This result is shown to hold for the porous medium equations of Darcy, Forchheimer or Brinkman. This optimal result is important because it shows that linearized instability theory has captured completely the physics of the onset of convection. The equivalence of the linear instability and nonlinear stability boundaries is also demonstrated for thermal convection in a non-equilibrium model with the Darcy law, when the layer rotates with a constant angular velocity about an axis in the same direction as gravity.

## 1. Introduction

In recent papers Banu & Rees (2002) and Malashetty *et al*. (2004) have considered the linear instability problem for convection in a porous medium when the temperature of the fluid may differ from that of the solid pores. These are important papers which utilize the local thermal non-equilibrium theory given by Nield & Bejan (1999), p. 192, see also Nield & Kuznetsov (2001). Due to application of porous media theory in drying/freezing of foods and other mundane materials, e.g. Sanjuán *et al*. (1999), Gigler *et al*. (2000*a*,*b*), Martins & Silva (2004), Zorrilla & Rubiolo (2005*a*,*b*), and applications in everyday food technology such as microwave heating, e.g. Dincov *et al*. (2004), we believe the local thermal non-equilibrium theory will play a major role in future developments. Another area where local thermal non-equilibrium theory will feature strongly is in rapid heat transfer from e.g. computer chips via use of porous metal foams, e.g. Calmidi & Mahajan (2000), Zhao *et al*. (2004), and their use in heat pipes, e.g. Nield & Bejan (1999), pp. 449–451.

The paper of Banu & Rees (2002) deals with the onset of thermal convection when the porous medium is modelled using Darcy's law whereas Malashetty *et al*. (2004) allow for a Brinkman model. Rees (2002) is an important article which lucidly shows how one may compare convection in the theories of Darcy and of Brinkman. His asymptotic estimates are very useful, and he also shows how the Forchheimer theory enters the picture. The goal of this paper is to show that the results of Banu & Rees (2002) and of Malashetty *et al*. (2004) are very strong. Indeed, they are optimal in that we show here that the global nonlinear stability boundary one obtains from using local thermal non-equilibrium theory is exactly the same as the linear instability ones found by Banu & Rees (2002) and by Malashetty *et al*. (2004). We also consider the Forchheimer effect. The results of the current paper demonstrate that the work of Banu & Rees (2002) and of Malashetty *et al*. (2004) is complete in that their results completely capture the physics of the onset of thermal convection and no subcritical instabilities are possible. We also demonstrate the equivalence between the nonlinear stability and linear instability boundaries for local thermal non-equilibrium convection in a Darcy porous medium when the layer is undergoing a constant angular rotation about an axis in the same direction as gravity.

Consider now a layer of porous material saturated with fluid and contained between the planes *z*=0 and *z*=*d*. The temperatures of the solid, *T*_{s}, and fluid, *T*_{f}, are maintained at constants on the planes *z*=0 and *z*=*d* with(1.1)where *T*_{L}>*T*_{U} (If *T*_{U}≥*T*_{L} then we may demonstrate global nonlinear stability always holds.) The equations are, cf. Nield & Bejan (1999), Banu & Rees (2002), Malashetty *et al*. (2004),(1.2)(1.3)(1.4)(1.5)These equations hold in the domain , standard indicial notation holds, so *i*=1,2,3, * k*=(0,0,1) and Δ is the three-dimensional Laplacian. The variables

*v*

_{i},

*p*,

*T*

_{f}and

*T*

_{s}are the velocity, pressure and fluid and solid temperatures, respectively. The constants

*K*,

*μ*,

*g α*,

*γ*

_{1}, ,

*ϵ*,

*ρ*

_{α},

*c*

_{α},

*k*

_{α}(

*α*=f, s) are permeability, dynamic viscosity, gravity, thermal expansion coefficient, the Forchheimer coefficient, the Brinkman coefficient, porosity, density, specific heat, thermal diffusion coefficient (where

*α*=f, s denotes fluid or solid), (

*ρc*)

_{α}=

*ρ*

_{α}

*c*

_{α},

*α*=f, s, and

*h*is an interaction coefficient.

The steady solution whose stability is under investigation is(1.6)where(1.7)is the temperature gradient and is a quadratic function found from equation (1.2).

## 2. Nonlinear stability

To investigate nonlinear stability we introduce perturbations *u*_{i}, *π*, *θ*, *ϕ* to and by(2.1)The perturbation equations are derived from (1.2)–(1.5) and are non-dimensionalized with velocity, pressure, temperature, time and length scales of *U*=*ϵk*_{f}/(*ρc*)_{f}*d*, *P*=*μ*d*U*/*K*, , , *L*=*d*. The Rayleigh number . The non-dimensional Forchheimer and Brinkman coefficients are *F*=*γ*_{1}*U*, , and *H*=*hd*^{2}/*ϵk*_{f} and *γ*=*ϵk*_{f}/(1−*ϵ*)*k*_{s} are the non-dimensional coefficients introduced by Banu & Rees (2002). One may then show that the non-dimensional perturbation equations have form(2.2)(2.3)(2.4)(2.5)where these equations hold on , *w*=*u*_{3} and *A*=*ρ*_{s}*c*_{s}*k*_{f}/*k*_{s}*ρ*_{f}*c*_{f} is a non-dimensional thermal inertia coefficient.

The boundary conditions to be satisfied are(2.6)if *λ*=0 (i.e. Darcy or Forchheimer flow), where *n*_{i} denotes the unit outward normal, whereas the boundary conditions to be satisfied are(2.7)if *λ*≠0 (i.e. Brinkman flow), together with *u*_{i}, *π*, *θ*, *ϕ* satisfying a plane tiling periodicity in *x*, *y*. Such forms are discussed in e.g. Chandrasekhar (1981), p. 43, and Straughan (2004), p. 51.

One may deduce the equivalence between the linear instability boundary and the nonlinear stability one by writing equations (2.2)–(2.7) as an abstract system of partial differential equations in a Hilbert space and then verifying that appropriate conditions hold, cf. the account of symmetry in Straughan (2004), §4.3. However, it is instructive and perhaps clearer to include a direct proof here. To this end let *V* be a three-dimensional period cell for the solution to equations (2.2)–(2.7) and let (.,.) and ∥.∥ denote the inner product and norm on *L*^{2}(*V*). Construct energy identities by multiplying equation (2.2) by *u*_{i}, equation (2.4) by *θ*, and equation (2.5) by *ϕ*/*γ* to obtain, after integration by parts and use of equation (2.3),(2.8)(2.9)(2.10)where ∥.∥_{3} denotes the *L*^{3}(*V*) norm. Now define *E*, *I*, *D* by(2.11)We add equations (2.8)–(2.10) and deduceand from this we may obtain(2.12)where(2.13)with being the space of admissible solutions, namely, are periodic over a plane tiling domain in *x* and *y*}. One may show from equation (2.12) (with the aid of Poincaré's inequality) that if *R*<*R*_{E} then *E*→0 exponentially in time, cf. similar details in e.g. Straughan (2004), pp. 11–12.

The exponential decay of *E* guarantees exponential decay of *θ* and *ϕ* (in *L*^{2}(*V*) norm). To obtain decay of * u* we note from equation (2.2) that one may showand so(2.14)Thus,

*R*<

*R*

_{E}also guarantees exponential decay of ∥

*∥ in Darcy theory, of ∥*

**u***∥*

**u**_{3}in the Forchheimer case, and of ∥∇

*∥ when the Brinkman model is employed.*

**u**Thus, *R*_{E} represents a global (i.e. for all initial data) nonlinear stability threshold. The quantity *R*_{E} is calculated from the Euler–Lagrange equations which we derive from equation (2.13), namely(2.15)where *ω* is a Lagrange multiplier.

Banu & Rees (2002) and Malashetty *et al*. (2004) show that the strong form of the principle of exchange of stabilities holds for the linearized version of equations (2.2)–(2.7), i.e. they show one may take the growth rate equal to zero. The key thing now is to observe that system (2.15) is identically the same eigenvalue problem as the linearized one from equations (2.2)–(2.7) with the growth rate *σ* (which arises from a time dependence like e^{σt}) equal to zero. Thus, the linear instability eigenvalues of Banu & Rees (2002) and of Malashetty *et al*. (2004), are exactly the same as the ones for global nonlinear stability, . What this means is that if there is instability of solution (1.6); this is true also for the nonlinear equations due to Sattinger's instability theory, Sattinger (1970), p. 813. If, however, there is definitely nonlinear asymptotic stability of solution (1.6). If there is stability since . Since *R*_{L}≡*R*_{E} this means no subcritical instabilities can arise. This is not true for all convection problems, cf. Proctor (1981).

We note that the Forchheimer term *F* plays no role in this analysis. When the basic velocity field is not zero the Forchheimer term may exert a stabilizing effect. For example, Payne *et al*. (1999), Payne & Straughan (1999), Payne & Straughan (2000) and Franchi & Straughan (2003) show how its effect is important in thermal convection when the viscosity depends on temperature.

## 3. Nonlinear stability in a rotating layer

We now consider the problem of convection in a horizontal porous layer which is rotating with a constant angular speed *Ω* about a vertical axis in the *z*-direction. We restrict attention to Darcy's law, but still allow for the local thermal non-equilibrium model. The key modification is to include a term representing the Coriolis force.

The equations for a perturbation about the steady solution (1.6), (1.7) may be derived using the technique of Chandrasekhar (1981), pp. 80–83, and Vadasz (1998), allowing for the local thermal non-equilibrium effect of Banu & Rees (2002). Equations (2.3)–(2.6) remain the same but equation (2.2) becomes (here *F*=0, *λ*=0, and *T*^{2} is the Taylor number which measures rotation rate)(3.1)Thus, equation (3.1) together with equations (2.3)–(2.6) represent the Vadasz model taking into account local thermal non-equilibrium effects.

We first consider the linearized theory in three space dimensions. With , and a similar form for *u*_{i} and *ϕ*, equations (2.4) and (2.5) yield(3.2)The variables in these equations are treated as complex at the outset. Denoted by Δ^{*} the operator . We multiply equation (3.2)_{1} by −Δ^{*}*θ*^{*} (complex conjugate), and integrate over *V*, and next, multiply equation (3.2)_{2} by −Δ^{*}*θ*^{*}/*γ* and likewise integrate over *V*, together with integrations by parts. Thus,(3.3)(3.4)where , and where the norm and inner product are momentarily on the complex space *L*^{2}(*V*). Equation (2.7) of Straughan (2001) still holds (this equation follows from equation (3.1)) and is(3.5)We add equations (3.3)–(3.5) to derive(3.6)The right-hand side of equation (3.6) is real as is the coefficient of σ. Thus, the eigenvalues . The linear instability eigenvalue problem is then governed by the linearized equations (2.3)–(2.6), together with equation (3.1), with the growth rate set equal to zero. Thus, the relevant equations are(3.7)(3.8)(3.9)where If we seek a solution of form , where *f* is a planform satisfying , *a* being a wavenumber, then equations (3.7)–(3.9) become(3.10)where , with *D*=d/d*z*. The variables *Θ* and *Φ* are eliminated to obtain the equation(3.11)The Fourier modes have form *W*=sin*nπz* and then we find(3.12)By differentiation one shows that *n*=1 minimizes *R*^{2} as a function of *n*, then(3.13)To determine the critical linear Rayleigh number one minimizes equation (3.13) in *a*^{2} This equation is studied (numerically) for *T*=0 in detail by Banu & Rees (2002). We do not perform such a study here for *T*≠0 since our goal is to show the nonlinear stability problem has the same Rayleigh number, although we do observe that the stabilizing effect of rotation is evident from equation (3.13).

To study the nonlinear convection problem we observe that from equation (3.1) there follows(3.14)Equations (2.9) and (2.10) continue to hold. Thus, we define *E*, *I* and *D* now by(3.15)(3.16)(3.17)Hence, from equations (2.9) and (2.10) we deduce(3.18)where(3.19)If *Λ*^{−1}<1, then using the Poincaré inequality we see that and so from inequality (3.18) we deduce that *E*(*t*)→0 at least exponentially. From equation (3.1) one showsHence,(3.20)Thus, *Λ*^{−1}<1 also guarantees exponential decay of ∥* u*(

*t*)∥. One may show that

*Λ*

^{−1}<1 is equivalent to

*R*<

*R*

_{E}, where

*R*

_{E}is the value of

*R*for which

*Λ*=1. This value of

*R*

_{E}is our nonlinear stability threshold.

To solve the maximum problem (3.19) we regard equation (3.14) as a constraint, cf. van Duijn *et al*. (2002), Pieters (2004). Thus we solve the maximum problem(3.21)where is a Lagrange multiplier. The Euler–Lagrange equations which arise may be conveniently written in terms of the operators and *M* defined as , *I* being the identity operator. The Euler–Lagrange equations are(3.22)We eliminate *ℓ* to find *θ*, *ϕ* satisfy the equations(3.23)Of course, equation (3.14) still holds. We observe that equations (3.14) and (3.23) govern the nonlinear stability problem and are the same equations as those which follow from equations (3.7)–(3.9) for the linear instability problem. Thus, the critical value of *R*^{2} for global nonlinear stability, , is the same as the linear instability one . This is the result we set out to prove.

In fact, if we eliminate *w*, *ϕ* from equations (3.14) and (3.23) we find is an eigenvalue of the problem(3.24)where the differential operators *A* and *C* are given by(3.25)One may ask why the nonlinear stability result holds for all perturbations. The question of allowing an arbitrary perturbation, which is connected with the completeness of the function space on which the solution is defined, is addressed in many places, e.g. Galdi & Straughan (1985), Steen (1986), Graham *et al*. (1993), Mielke (1997), Kaiser & Xu (1998), Ly & Titi (1999), Oliver & Titi (2000), Kaiser & Schmitt (2001) and Kaiser & Mulone (2005). To establish that nonlinear stability is with respect to all (periodic) perturbations one can follow the analysis of Kaiser & Xu (1998) (who study rotating convection in a fluid, where there is no equation for *ϕ* and a different equation for *u*_{i}).

The operators *A* and *C* in equation (3.24) have counterparts in Kaiser & Xu (1998), p. 294, although *A* and *C* given by equation (3.25) are less involved than the block operators of Kaiser & Xu (1998). Nevertheless we may follow their analysis, *mutatis mutandis*. Since *A* is a strictly positive, self-adjoint operator, and *C* is a positive Hermitian operator, then effectively their method yields(3.26)where with *α*, *β*>0 being the side lengths of a periodicity rectangle (for example). (Other periodicity shapes may be accommodated.) We now let *α*, *β* vary over ^{+}. In this way we find satisfies the same minimization problem as equation (3.13). However, the analysis is valid for all (periodic) perturbations. (Essentially the idea is to writeThe function *θ* may then be an arbitrary function in .)

## Acknowledgments

I am indebted to three anonymous referees for constructive remarks which have led to improvements in the manuscript.

## Footnotes

- Received January 4, 2005.
- Accepted August 1, 2005.

- © 2005 The Royal Society