## Abstract

The stability of convection in a two-layer system in which a layer of fluid with a temperature-dependent viscosity overlies and saturates a highly porous material is studied. Owing to the difficulties associated with incorporating the nonlinear advection term in the Navier–Stokes equations into a stability analysis, previous literature on fluid/porous thermal convection has modelled the fluid using the linear Stokes equations. This paper derives global stability for the full nonlinear system, by utilizing a model proposed by Ladyzhenskaya. The nonlinear stability boundaries are shown to be sharp when compared with the linear instability thresholds.

## 1. Introduction

Thermal convection within a two-layer system constructed by a layer of fluid overlying a porous material saturated with the same fluid has numerous geophysical and industrial applications, such as the manufacturing of composite materials used in the aircraft and automobile industries, flow of water under the Earth’s surface, flow of oil in underground reservoirs and growing of compound films in thermal chemical vapour deposition reactors. A detailed review is given by Nield & Bejan (2006), with current highly relevant literature including Chen & Chen (1988), Ewing & Weekes (1998), Blest *et al.* (1999), Straughan (2002, 2008), Carr (2004), Chang (2004, 2005, 2006), Hirata *et al.* (2007), Hoppe *et al.* (2007), Mu & Xu (2007) and Hill & Straughan (2009).

Assessing the onset and type of convection is crucial in understanding and controlling these geophysical and industrial processes. This is achieved by analysing both the linear instability and nonlinear stability thresholds of the governing model. Comparing these thresholds allows the assessment of the suitability of linear theory to predict the physics of the onset of convection. The derivation of sharp unconditional stability thresholds is particularly physically useful due to the lack of restrictions on the initial data (Straughan 2004).

Nonlinear energy stability analyses of thermal fluid/porous systems are not widespread in the current literature, with the only previous work being that of Payne & Straughan (1998) and Hill & Straughan (2009). In both these papers, owing to the difficulties associated with incorporating the nonlinear **v**·∇**v** advection term in the Navier–Stokes equations into a stability analysis, the fluid is modelled using the linear Stokes equations.

This paper utilizes a model proposed by Ladyzhenskaya (1967, 1968, 1969; Straughan 2002, 2004, 2008), which is used as an alternative to Navier–Stokes. This allows for the development of an unconditional nonlinear energy stability analysis for thermal convection with temperature-dependent viscosity in a fluid/porous system, without the need to remove the nonlinear advection term **v**·∇**v**. It is important to note that the viscosity of a liquid is usually strongly dependent on temperature (cf. Capone & Gentile 1994, 1995; Galiano 2000). Convection problems for which the viscosity or conductivity is a function of temperature has received much recent attention in the literature (e.g. Payne & Straughan 2000; Manga *et al.* 2001; Shevtsova *et al.* 2001), making this work particularly timely.

The stability calculations required to construct the neutral curves involve determining eigenvalues and eigenfunctions, where the associated eigenvalue problems are not solvable analytically. The results are derived numerically using the Chebyshev tau-QZ method (Dongarra *et al.* 1996), which is a spectral method coupled with the QZ algorithm. All numerical results were checked by varying the number of polynomials to verify convergence. Standard indicial notation is employed throughout and **k**=(0,0,1).

## 2. Formation of the problem

Consider a fluid occupying the three-dimensional layer and saturating an underlying homogeneous porous medium The interface between the saturated porous medium and the fluid is at *z* = 0.

We assume that the dynamic viscosity *μ* has a linear temperature dependence of the form
for a constant *γ* > 0, where *T*, *μ*_{0} and *T*_{L} are temperature and reference viscosity and temperature values, respectively. Although we only consider liquids that have a viscosity which decreases with increasing temperature, the analysis can be easily generalized to a more general viscosity–temperature relationship. The governing model for the fluid layer we select is
2.1
(Antontsev *et al.* 2001; Straughan 2002, 2004) where *v*_{i}, *p*, *t* and *ρ*_{0} are velocity, pressure, time and reference density and κ_{f}, *g*, *c*_{p} and *α* are the thermal conductivity, acceleration due to gravity, specific heat at a constant pressure and coefficient of thermal expansion. A variation of this model was suggested by Ladyzhensakaya (1967, 1968, 1969) as an alternative to the Navier–Stokes equations and is a generalization of a well-known model in viscoelasticity (Antontsev *et al.* 2001). The parameter *μ*_{1}>0 is a constant, *D*_{ij}=(*v*_{i, j}+*v*_{j, i})/2 and The subscripts (or superscripts) f and m denote the fluid and porous layers, respectively.

In the porous medium, we assume a high porosity *ϕ* > 0.75, such that the governing equations are given by
2.2
where the variables *p*^{m}, *T*^{m} and *K* are the velocity, pressure, temperature and permeability, respectively. The starred quantities are defined in terms of the fluid and porous variables such that *S** = *ϕ**S*_{f} + (1 − *ϕ*)*S*_{m}, where *S** = κ* or (*ρ*_{0}*c*_{p})*. A comprehensive discussion of the variances and various physical attributes of modelling transport through porous media is given in Alazmi & Vafai (2000).

The temperatures at the upper and lower boundaries are held fixed at *T*_{U} and *T*_{L}, respectively, with continuity of temperature, velocity and heat flux at the interface *z* = 0. The remaining boundary conditions at *z* = 0 are the continuity of normal stresses
2.3
and tangential stresses
2.4
for *β* = 1, 2. The derivation of appropriate boundary conditions at the fluid/porous interface is non-trivial (cf. Vafai & Thiyagaraja 1987; Alazmi & Vafai 2001; Vafai 2005).

Under these boundary conditions, the governing equations (2.1) and (2.2) admit a steady-state solution in which the velocity field is zero and
where ϵ_{T} = τ_{f}/τ_{m}, τ_{f} = κ_{f}/(*ρ*_{0}*c*_{p})_{f}, τ_{m} = κ*/(*ρ*_{0}*c*_{p})_{f} and with the cap denoting the steady state. To study the stability of the steady state, we introduce the perturbations where *d*_{ij}=(*u*_{i, j}+*u*_{j, i})/2, and non-dimensionalisize with the scalings
where *R*_{a} = *R*^{2} is the fluid Rayleigh number. By replacing *d* and τ_{f} by *d*_{m} and τ_{m}, respectively, the porous layer scalings follow analogously, where is the porous Rayleigh number. This yields the non-dimensional perturbation equations
2.5
in with *f*_{1} = 1 + *Γ*(*M*_{2}+*M*_{1}*z*), *d*_{ij}=(*u*_{i, j}+*u*_{j, i})/2 and
2.6
in with *f*_{2} = 1 + *Γ**M*_{2}(1 + *z*), . The remaning parameters are the Prandtl number *P**r* = *μ*_{0}/(κ_{f}*ρ*_{0}), Darcy number *ω* = *μ*_{1}/(*ρ*_{0}*d*^{2}), *Γ* = *γ*(*T*_{L}−*T*_{U}), *G*_{m} = (ρ_{0}*c*_{p})*/(ρ_{0}*c*_{p})_{f}, and

## 3. Linear instability analysis

To proceed with a linear analysis, the nonlinear terms from equations (2.5) and (2.6) are discarded. We assume normal modes of the form
with analogous definitions in the porous medium. Taking the double curls of equations (2.5)_{1} and (2.6)_{1} to remove the pressure terms, where the third component is chosen, leads to the linearized equations
where *D* = d/d*z*, and The boundary conditions for the 12th order system at *z* = 1 are
at *z* = −1. On the interface *z* = 0, we have
and
The numerical results are presented in §5.

## 4. Nonlinear stability analysis

Let us define *Ω*_{f} and *Ω*_{m} to represent the period cells in the fluid and porous layers respectively, and introduce the notation of norm and inner product on the spaces *L*^{2}(*Ω*_{f}) and *L*^{2}(*Ω*_{m}), where
To obtain global nonlinear stability bounds in the stability measure *L*^{2}(*Ω*_{f}), we multiply equations (2.5)_{1} and (2.5)_{3} by *u*_{i} and *θ*, respectively, and integrate over the period cell. An analogous process is applied to equations (2.6)_{1} and (2.6)_{3}. We may now define the functional *E*(*t*) by
for coupling parameters *λ*_{1}, *λ*_{2} and *λ*_{3}>0, such that
4.1
Utilizing a similar approach to Hill & Straughan (2009), the first and third terms on the right-hand side of equation (4.1) are integrated by parts, and the non-dimensionalized versions of boundary condtions (2.3) and (2.4) are employed to yield
where and *Λ* represents the fluid/porous interface at *z* = 0. Similarly, by integrating by parts and utilizing the non-dimensionalized boundary conditions
where *λ*_{1} = *λ* and

Combining these definitions, it follows that 4.2

To address the cubic nonlinearities in equation (4.2), we introduce the *L*^{3} norm ∥ · ∥_{3}. Multiplying equations (2.5)_{3} and (2.6)_{3} by *θ*^{2} and (*θ*^{m})^{2}, respectively, integrating over the period cell, and using Poincaré’s inequality, we find
4.3
where to ensure the removal of the boundary integrals (Hill & Straughan 2009). We now use Young’s inequality on the cubic integral terms in both equations (4.2) and (4.3), such that
for *c* > 0, where *Q*_{1} ≠ *Q*_{2}.

Letting
and combining equations (4.2) and (4.3), we now have
4.4
where *α*_{i}, *β*_{j} are positive constants for *i*=1,2,3; *j* = 1,2, introduced by using Young’s inequality on the cubic terms. Before we choose the coefficients to bind the cubic **d**, *θ*, **d**^{m} and *θ*^{m} integrals, we must address the boundary integrals and cubic **u** and **u**^{m} terms in equation (4.4).

To achieve this, we use the following Poincaré-like inequalities:
4.5
4.6
where *c*_{1}, *c*_{2} and *c*_{3} > 0. Similar inequalities follow, in the porous case, for constants and . A proof of these inequalities and the definitions of the constants are given in appendix A. Applying equations (4.5) and (4.6) to equation (4.4) yields
4.7
Now put *λ*_{4} = *λ*_{4} ′ + *k*ε, and let
which is satisfied for
We now minimize
with respect to *α*_{1} and *β*_{1} to yield
and choose *α*_{3} to minimize
From equation (4.7), choosing *k* = 27/(8π^{2}), we can now deduce
where we require
4.8
Defining
it follows that
where
4.9
Utilizing Poincaré like inequalities (cf. Payne & Straughan 1998) allows us to deduce that
where *m* > 0. Integrating, we have *E*_{1}(*t*) ≤ *E*_{1}(0)*e*^{−mt} → 0 as where convergence is at least exponential, so we have established unconditional nonlinear stability provided equations (4.8) and (4.9) hold.

The corresponding Euler–Lagrange equations that arise at the sharpest threshold *R*_{E}=1 are
4.10
in the fluid layer, and
4.11
in the porous layer, where *L* and *L*^{m} are Lagrange multipliers.

By taking the double *curl* of equations (4.10)_{1} and (4.11)_{1} and adopting normal mode representations, the 12th-order eigenvalue problem (4.10) and (4.11) can be used to locate the critical nonlinear Rayleigh number *R**a*_{E}, which is given by
Numerical results for the nonlinear energy approach are presented in §5.

## 5. Results and conclusions

The first key result that can be derived is for the case *Γ* = 0, which corresponds to the viscosity of the fluid being constant with respect to temperature. Under this condition, the equations for linear instability and nonlinear stability are identical to those of Hill & Straughan (2009), for which excellent agreement was shown between the two. It is important to note, though, that in this paper the nonlinear advection term **v** · ∇**v** is included in the analysis, whereas the analysis of Hill & Straughan (2009) is limited to the nonlinear Stokes problem.

For *Γ* ≠ 0, we now solve the eigenvalue problem (4.10) and (4.11) by means of a *D*^{2} Chebyshev tau method. The details are similar to those given by Dongarra *et al.* (1996). The parameters, unless stated otherwise, are fixed at δ = 5 × 10^{−6}, *G*_{m} = 10, *P**r* = 6 and ϵ_{T} = 0.7. The porous material is assumed to be that of a Foametal (which is used extensively in industrial applications such as heat exchangers, chemical reactors and fluid filters), with physical values of permeability and porosity of 8.19 × 10^{−8} m^{2} and 0.79, respectively (cf. Straughan 2002; Goyeau *et al.* 2003).

Figure 1 shows the neutral curves for a variation of *Γ* values, where the linear instability and nonlinear stability thresholds are represented by solid and dashed lines, respectively.

It is clear that an increase in *Γ* causes the system to become more stable. Assuming that the temperature at the boundaries remains fixed, this corresponds to the strength of the linear dependence of viscosity on temperature increasing. As the viscosity decreases with an increase in temperature, this physcially makes sense. An interesting result is that the bimodal nature of the neutral curve is unaffected by the change in *Γ*.

As the linear instability and nonlinear stability results clearly show excellent agreement, we can conclude that the linear theory accurately encapsulates the physics of the onset of convection.

## Appendix A. A proof of inequalities (eqn4.5) and (eqn4.6)

Let *Ω*_{f} represent the fluid period cell, and *u*_{i} ∈ *C*^{1} be a solenoiodal function satisfying the boundary condition *u*_{i} = 0 on *Λ*_{0}, where the boundary of *Ω*_{f} is given by ∂*Ω*_{f}=*Λ* + *Λ*_{0}. In the same vein as in §4, *Λ* represents the fluid/porous interface at *z* = 0. For a *C*^{1} function *f*_{i} to be chosen at our discretion, we observe that
By letting
where *p*_{1} and *p*_{2} are constants to be chosen at our discretion, it follows that
The arithmetic-geometric mean inequality leads us to
where *α* > 0 is a constant, and Employing this inequality yields
A1
Letting *α* = 1/2 and *p*_{2} = 3*p*_{1}*d*_{1}/4 to remove the boundary integrals, we now have
By using the Cauchy–Schwartz inequality on the right-hand side, we find
which is inequality (4.5).

To derive inequality (4.6), we return to equation (A1). By letting
and using equation (4.5), it follows that
as required, where *α* is a constant to be chosen at our discretion.

## Footnotes

- Received June 18, 2009.
- Accepted July 31, 2009.

- © 2009 The Royal Society