Thermal convection for a Darcy-Brinkman rotating anisotropic porous layer in local thermal non-equilibrium

The onset of natural convection in a fluid-saturated anisotropic porous layer, which rotates about the vertical axis, under the hypothesis of local thermal non-equilibrium, is analysed. Since the porosity of the medium is assumed to be high, the more suitable Darcy-Brinkman model is adopted. Linear instability analysis of the conduction solution is carried out. Nonlinear stability with respect to L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-norm is performed in order to prove the coincidence between the linear instability and the global nonlinear stability thresholds. The effect of both rotation and thermal and mechanical anisotropies on the critical Rayleigh number for the onset of instability is discussed.


Introduction
Thermal convection in fluid-saturated porous media is a research topic of great interest because of its several applications in real life problems, in engineering and geological context. Convection in porous media finds its remarkable attention in geothermal energy utilization, thermal insulation technology, cooling of electronic equipment, tube refrigerators, heat exchangers or more in general, problems on removal or storage of heat.
In industrial field, high-porosity materials, such as metal foams, are usually involved since they can be successfully used to manage heat transfer. Moreover, their light

Mathematical model
Let us take into account a horizontal highly porous medium, whose depth is d, saturated by an incompressible, homogeneous fluid at rest. The medium is assumed evenly warmed up from below and rotating about the upward vertical axis z with constant angular velocity . As consequence, in addition to the gravitational field, the Coriolis force acts on the medium. Moreover, we assume that the medium is in local thermal non-equilibrium so that the heat exchange between fluid and solid skeleton is allowed.
We denote by T L the temperature of the lower plane z = 0 and by T U the temperature of the upper plane z = d, while we refer to the fluid temperature with T f and to the solid temperature with T s . Then (3) where v, p, T f and T s are (seepage) velocity, reduced pressure, fluid phase temperature and solid phase temperature, respectively;μ, μ, ρ f , ρ s , c, g, α, , ε, h are effective and dynamic viscosity, fluid density, solid density, specific heat, gravity acceleration, thermal expansion coefficient, angular velocity, porosity and interaction coefficient, respectively.
The following boundary conditions are coupled to (3) T s = T f = T L on z = 0, being n the unit outward normal to planes z = 0, d. System (3) admits the following conduction solution m 0 : We are interested in studying the stability of the steady solution (5). Let us introduce the following perturbation fields {u, θ, φ, π} so as to obtain a new solution for (3) In order to study the stability of the null solution of (9), let us assume that perturbations are periodic in x and y directions with periods 2π a x and 2π a y , respectively. Let be the periodicity cell, we assume that perturbations belong to W 2,2 (V ), ∀t ∈ R + and they can be expanded as a Fourier series uniformly convergent in V .

Principle of exchange of stabilities
In order to perform a linear instability analysis, we rewrite (9) 1 in a more convenient form in which only relevant unknown fields appear. Hence, let us apply the double curl and the curl to (9) 1 and let us retain only the third component. The resulting equations are multiplied by ξ so as to obtain where (13) 2 is consequence of a further derivation with respect to z. A single equation is obtained once the operator (1 − ξ Da ) is applied to (13) 1 and (13) 2 is substituted in the resulting equation. This procedure allows us to write the linear version of (9) as follows Since (14) is autonomous, we look for solutions whose time dependence is separated from the temporal one, i.e.
By virtue of (15), (14) becomes Then we denote by (·, ·) and · the scalar product on the Hilbert space L 2 (V ), and the related norm, respectively. Let us multiply (16) 1 by w * ξ , (16) 2 by where the asterisks denote the complex conjugate. By virtue of boundary conditions (11), integrating over the periodicity cell V yields Every term in (18) is real, then necessarily σ is real as well. Thus, we have shown the validity of the strong form of the principle of exchange of stabilities. Therefore, convection can occur only through a steady motion.

Linear instability analysis
Since our aim is to determine the critical Rayleigh number beyond which instability occurs, we focus on the marginal state in (16). By virtue of principle of exchange of stabilities, we set σ = 0 in (16) so as to obtain Because of periodicity of perturbation fields, accounting for boundary conditions (11) and since the sequence {sin nπ z} n∈N is a complete orthogonal system for L 2 ([0, 1]), we look for solution of (19) such that wheref n =f n (x, y) sin(nπ z) and where a is the wavenumber arising from spatial periodicity. Let us define the following operators Now let us apply L 1 and L 3 to (23) 2 so that By substituting (23) 1 -(23) 3 in (24), we get By splitting L 3 in the first term, we obtain By splitting L 2 in the first term in (26) Let us substitute (20) in (27) and retain only the n-th component.
being δ n = a 2 + n 2 π 2 . From (28) it follows that the critical Rayleigh number for the onset of steady convection is being f (n 2 , a 2 ) = ξ a 2 +n 2 π 2 +ξ 2 Da a 2 δ n +ξ Da n 2 π 2 δ n +ξ 2 T 2 n 2 π 2 +ξ Da δ 2 n +ξ 2 Da 2 δ 3 We can easily remark that the minimum with respect to n 2 is attained in n 2 = 1, since f (·, a 2 ) is a strictly increasing function. Hence Let us remark that f (1, a 2 ) is a strictly increasing function of T and η, therefore the stabilizing effect of rotation and fluid thermal conductivity has been proved. In particular, the effect of rotation on the onset of instability is expected since the Coriolis force acts in the horizontal direction, discouraging the motion in the vertical one.
Moreover, it is easy to note that if Da = 0, (31) coincides with the result found in [5]- [19]. While, for a low porous medium in absence of rotation, i.e. T = 0 and Da = 0, results coincide with the ones found in [6]. In addition, by assuming that the layer is isotropic, i.e. ξ = η = ζ = 1, the critical Rayleigh number (31) is the same as that one found in [18].

Nonlinear stability
In this section we want to perform a nonlinear stability analysis of conduction solution m 0 . The application of the energy method yields a loss of the rotation term. Therefore, we employ a differential constraint approach ( [19,20]) in order to capture the influence of rotation. Then, let us multiply (9) 3 by θ and (9) 4 by φ and integrate over the periodicity cell V . By defining the Lyapunov functional E(t) the production term I (t) and the dissipation function D(t) we find out that with The variational problem (36) is equivalent to where λ(x) is a Lagrange multiplier. By employing the Poincaré inequality in (34), we get being a = min{1, η} and b = min{1, ζ }. Hence, if R < R E , from (35) it turns out that where c = min 2a, 2b A . Equation (41) yields the exponential decay of temperature perturbation fields. Now let us remark that, by multiplying (9) 1 by u and by virtue of Cauchy-Schwartz, we obtain the exponential decay of u, i.e.
where ξ * = max{ξ, 1}. Thus, we have shown that the condition R < R E implies the global nonlinear and exponential stability of conduction solution m 0 . Now let us solve the variational problem (39) to determine the critical Rayleigh number R E . The Euler-Lagrange equations, together with the constraint equation, are Recalling the definitions in (22), (43) becomes By applying L 1 to (44) 3 and substituting (44) 2 and (44) 1 , we obtain The application of L 3 to (45) leads to the following equation which coincides with (24). As consequence, we have obtained the coincidence between the global nonlinear stability threshold R E and the linear instability threshold R S , implying the absence of subcritical instability region. This result allows us to claim that the condition R < R E = R S is a necessary and sufficient condition for the stability of m 0 , therefore in this respect the result is optimal.

Numerical analysis
In this section we would like to point out how parameters affect the onset of convection. Given the complex expression obtained in (31) for the critical Rayleigh number R S , it is not always easy to show analytically how parameters modify the occurrence of instability. That is why the expression for the critical Rayleigh number (31) is analysed numerically for different values of parameters with the aim of highlighting how they affect the onset of convection. In particular, we will show the dependence of R S with respect to the Taylor number, permeability, thermal conductivities and the Darcy number.
We would like to point out that results are reported as function of the inter-phase heat transfer coefficient H . This parameter is not easily measurable, as claimed in [5]. That is why we have decided to fix the range (10 −2 , 10 6 ) in which H can vary and to show the influence of parameters on R S for any H . Figure 1 shows the behaviour of the critical Rayleigh number R S for increasing values of T . The stabilizing effect of rotation on the onset of convection is clear, for any value of H , even though it is less remarkable when H → 0. This result is not surprising since we have previously pointed out that the derivative of R S in (31) with respect to T is strictly positive. Moreover, the stabilizing effect is expected from a physical point of view since rotation acts on the fluid in the horizontal direction, making the motion along the vertical axis more difficult.
We would like to underline that both for small and large values of H , the curves tend to become parallel to the x axis. The region where a plateau is reached represents the local thermal equilibrium situation and it will characterize the following figures, as well. Physically, if H → 0, the solid phase is separated from the fluid one and it ceases to affect the fluid thermal field. While, if H → ∞, solid and fluid temperature end up with being identical.
In Fig. 2, the stabilizing effect of the Darcy number Da on conduction is highlighted. Note that if Da = 0, results are valid for the classical Darcy model, as pointed out previously. It is well known that when porosity ε tends to 1, the classical Darcy model needs to be replaced by the Darcy-Brinkman model, for which Da = 0. Since the Darcy-Brinkman model is closer to a model describing the fluid motion in absence of porous medium (clear fluid), for which it is common knowledge that the critical Rayleigh number is greater than that one in a porous medium, result in Fig. 2 is consistent.
In Fig. 3a, the behaviour of R S with respect to permeability parameter ξ is shown. The destabilizing effect of permeability is evident, for any values of H , in agreement with findings of [5] and [6]. Recalling the definition of the Rayleigh number, this kind of behaviour is expected, since R is directly proportional to K z . Physically, increasing  Figure 3b shows how cell dimension changes for different values of permeability. In particular, when permeability in the horizontal direction K H grows, periodicity cells get wider. Now let us analyse how R S varies with respect to thermal conductivities. Figure 4a shows the stabilizing effect of solid thermal conductivity on conduction. The growth of R S with ζ means that solid thermal conductivity delays the onset of convection. Physically, the greater the solid thermal conductivity is the more easily the solid matrix absorbs heat from the fluid. The delaying effect is evident for large values of the interphase heat transfer coefficient H , while it is less remarkable for smaller values of H . This is not surprising since, as already pointed out, when H → 0 solid phase does not affect the fluid thermal field. This behaviour is evident also in Figure 4b, where it is highlighted how cell dimension varies with respect to ζ . In particular, for small values of H , the influence is negligible, while if H is great, ζ promotes wider periodicity cells.
Analogous result is obtained when looking at the effect of the fluid thermal conductivity parameter η on the periodicity cell dimension. Figure 5b shows that η promotes wider periodicity cells, as well.
The stabilizing effect of η is highlighted in Fig. 5a, where for any H , R S grows for increasing η. This behaviour is consistent with the analytical result for which the derivative of R S in (31) with respect to η is strictly positive. From a physical point of view, increasing κ f z implies that heat flows easier in the vertical direction within the fluid, fostering the onset of instability.

Conclusions
The linear and nonlinear stability analysis of the conduction solution in an anisotropic rotating porous medium with high porosity in local thermal non-equilibrium has been studied. Only steady convection is allowed, since the principle of exchanges of stability holds. A detailed proof of that has been performed. In order to study the global nonlinear stability, the energy method has been adopted. Coincidence between the linear instability and the global nonlinear stability thresholds is proved. This means that a necessary and sufficient condition for the onset of convection has been obtained.
Numerical simulations have been required in order to highlight the influence of parameters on the onset of convection. It has been pointed out that permeability has a destabilizing effect on conduction. This is because increasing permeability eases the fluid motion, implying an easier occurrence of instability. Whereas, thermal conductivities stabilize conduction, delaying the onset of convection. Moreover, it has been shown that, as expected, rotation and the Darcy number have a stabilizing effect on conduction, as well as the thermal conductivities' ratio γ .
Acknowledgements This paper has been performed under the auspices of the National Group of Mathematical Physics GNFM-INdAM. J. Gianfrani would like to thank Progetto Giovani GNFM 2020: "Problemi di convezione in nanofluidi e in mezzi porosi bidispersivi". The Authors would like to thank the anonymous reviewer for the helpful suggestions that led to improvement in the manuscript.

Author Contributions Not applicable.
Funding Open access funding provided by Università degli Studi di Napoli Federico II within the CRUI-CARE Agreement.
Data Availibility Statement Not applicable.

Conflicts of interest/Competing interests:
The authors declare that they have no conflicts of interest.
Code availability: Not applicable.
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