Effect of anisotropy on the onset of convection in rotating bi-disperse Brinkman porous media

Thermal convection in a horizontally isotropic bi-disperse porous medium (BDPM) uniformly heated from below is analysed. The combined effects of uniform vertical rotation and Brinkman law on the stability of the steady state of the momentum equations in a BDPM are investigated. Linear and nonlinear stability analysis of the conduction solution is performed, and the coincidence between linear instability and nonlinear stability thresholds in the 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 obtained.

the fraction of volume occupied by the solid skeleton. Double porosity materials are employed in engineering, medicine, chemistry, since brain, bones or heat pipes can be modelled as bi-disperse porous media (see [15,26] and references therein). In particular, the use of anisotropic bi-disperse porous materials may have much more potential, compared to the anisotropic single porous media, due to many possibilities to design man-made materials for heat transfer or insulation problems (see [7,8,10,[26][27][28] and references therein).
The study of fluid flow in rotating porous media is motivated by its theoretical significance and practical applications (see [3,[5][6][7][8]14,29] and the references therein) like, for example, in geophysics and in engineering (food process industry, chemical process industry, centrifugal filtration processes, rotating machinery). For instance, in [6], Capone et al. analyse the influence of vertical rotation on the onset of convection in a single temperature BDPM, according to Darcy's law, while in [8], the authors investigate the onset of convection in a horizontal, vertical rotating, anisotropic porous layer assuming a local thermal non-equilibrium.
The goal of the present paper is to study the onset of thermal convection in an anisotropic bi-disperse porous medium uniformly rotating about a vertical axis, assuming the validity of the Brinkman law both for the micropores and the macropores. In particular, the use of Brinkman law in both micropores and macropores for a bi-disperse porous medium has been introduced by Nield and Kuznetsov [21] accounting for the discussion on the dispersion in a BDPM made by Moutsopolous and Koch in [19]. In fact, in [19] the authors prove a good agreement between theoretical predictions end experimental measurements when one considers a dilute array of large spheres in a Brinkman medium and the flow around the large spheres is modelled using Brinkman's equation.
The plan of the paper is as follows. In Sect. 2, we introduce the mathematical model and, in order to study the stability of the conduction solution, we introduce the dimensionless equations for the evolution (in time) of the perturbation fields. In Sect. 3, we perform the instability analysis of the conduction solution and we prove the validity of the strong form of the principle of exchange of stabilities, which means that if the convection sets in, it arises via a stationary state. Section 4 deals with the nonlinear stability analysis of the conduction solution, and the coincidence between instability and (global) nonlinear stability thresholds in the L 2 -norm is proved. Numerical simulations concerning the asymptotic behaviour of the instability threshold with respect to the meaningful parameters of the model are performed in Sect. 5. The paper ends with a concluding section (Sect. 6), in which all the obtained results are collected. where is the reduced pressure, with x = (x, y, z), v s = seepage velocity, P s = pressure, = density, ζ = interaction coefficient between the f-phase and the p-phase, g = −gk = gravity, μ = fluid viscosity,μ s = effective viscosity, F = reference constant density, α = thermal expansion coefficient, c = specific heat, c p = specific heat at a constant pressure, thermal conductivity (The subscript sol refers to the solid skeleton). To (1), the following boundary conditions are appended: where n is the unit outward normal to the impermeable horizontal planes delimiting the layer and T L > T U . The problem (1)-(2) admits the stationary conduction solution: Denoting by {u f , u p , θ, π f , π p } a perturbation to the steady solution, one recovers that the evolutionary system governing the perturbation fields is given by where Introducing the non-dimensional parameters where the scales are given bỹ and setting which are the Taylor number T , the Darcy number Da f , and the thermal Rayleigh number R, respectively, and the resulting non-dimensional perturbation equations, dropping all the asterisks, are under the initial conditions with ∇ · u s 0 = 0, s = { f, p}, and the stress-free boundary conditions [12] Moreover, according to experimental results, let us assume that the perturbation fields are periodic functions in the x, y directions and denote by the periodicity cell.

Onset of convection
To analyse the onset of convection, i.e., to find the linear instability threshold of the conduction solution, we consider the linear version of (4) and seek for solutions in which u f , u p , θ, π f , π p have time dependence like Let (·, ·) and · denote inner product and norm on the complex Hilbert space L 2 (V ), respectively, and let * denote the complex conjugate of a field. Multiplying (6) 1 by u f * , (6) 2 by u p * and (6) 3 by θ * , the integration over V leads to: If σ = σ r + iσ i , the imaginary part of (7) is On the other hand, applying the same procedure to the complex conjugate of (6) and multiplying by u f , u p , θ , one gets: Adding (8) and (9), one obtains: and hence necessarily σ i = 0, i.e., σ ∈ R and the strong form of the principle of exchange of stability holds, i.e., the oscillatory convection cannot arise.
Denoting by and defining the operators the third components of curl and of double curl of (4) 1,2 are, respectively, given by and Applying the operator B to (12) 1 , by virtue of (12) 2 , one has This equation, together with that one obtained by applying the operator Ψ to (12) 2 , leads to Applying the operator Ψ to (13) 1 and Ψ B to (13) In view of (14), (15) can be written as Therefore, to find the linear instability threshold, we consider (6) 3 , (16) 1 and (16) Employing normal modes in (17), i.e. assuming the following representation [12] w with W f 0 , W p 0 , 0 real constants, setting a 2 = l 2 + m 2 and n = a 2 + n 2 π 2 , from (17), it turns out that where Setting h 11 = n e(A 1 M + σ f n 2 π 2 ) + 2 n e(Meσ + B 1 ) + f bn 2 π 2 + B 1 M + e 2 3 n A 1 + e 3 σ 4 n , n + η f n 2 π 2 , h 13 =B 1 + e n A 1 + e 2 σ 2 n , can be written as The smallest value of R 2 which vanishes the determinant of (21) is the critical Rayleigh number for the onset of instability, i.e. with We have proved that: (i) both numerator and denominator of (23) are strictly positive; (ii) the minimum of f 2 L with respect to n, by using numerical computations, is attained at n = 1. Hence, The minimum of f 2 L (1, a 2 ) with respect to a 2 is analysed in Sect. 5.
Remark 1 Let us observe that: (i) if one assumes the validity of the Darcy's law (Da f = 0), from (22) one gets: which coincides with the critical threshold found in [7]; (ii) if Da f = 0 and h = k = 1 (isotropic case), from (22) one obtains the critical Rayleigh number found in [6], i.e.
and (26) coincides with the critical threshold found in [16].

Nonlinear stability
In order to study the nonlinear stability of the conduction solution, by virtue of (11), since and (4) 5 can be written as: where the following differential operators have been defined: To determine the nonlinear stability threshold, in order to evaluate the rotation effect, it is not possible to proceed via a standard energy analysis, since the Coriolis terms in the momentum equations are antisymmetric. For this reason, we employ the differential constraint approach (see [8,17,25]) and set Retaining (27) 1,2 as constraints, multiplying (27) 3 by θ and integrating over the periodicity cell, one gets and is the space of kinematically admissible solutions. The variational problem associated with the maximum problem (30) is equivalent to where λ (x) and λ (x) are Lagrange multipliers and with periods 2π/l, 2π/m, respectively; D < ∞}.
Applying the Poincaré inequality, one obtains that D ≥ π 2 θ 2 , and hence, from (29) it follows that the condition R < R E implies E(t) → 0 at least exponentially.
Remark 2 Multiplying (4) 1 by u f , (4) 2 by u p , integrating over the period cell V and adding the resulting equations, one finds (32) Settingk = max(k, 1) andĥ = max(h, 1), by virtue of generalized Cauchy inequality and Poincaré-like inequality, from (32) one obtains where c 1 , c 2 are positive constants depending on the domain V . By virtue of (33), the condition R < R E also guarantees the exponential decay of u f and u p .
In order to determine the critical Rayleigh number R 2 E by solving the variational problem (31), we need to solve the associated Euler-Lagrange equations given by Eliminating the variable θ in (34), it turns out that By using (18) 1 and (18) 2 and choosing [17] λ = λ 0 sin(nπ z)e i(lx+my) , Requiring a zero determinant for (37), we get hence, we have the coincidence between the instability threshold and the global nonlinear stability threshold with respect to the L 2 -norm (subcritical instabilities do not exist).

Numerical results
In this section, we numerically analyse the asymptotic behaviour of R 2 L with respect to the parameters h, k, T 2 , Da f in order to study the influence of permeability, rotation and Brinkman law on the onset of convection. As pointed out in Sect. 3, in all the performed computations we set n = 1.
Tables 1 and 2 show that as h increases with k fixed, both Rayleigh and wave numbers decrease, so it is easier for convection to set in and the convection cells become wider; k increasing with h fixed leads to a similar trend, but in this case the Rayleigh number decreases more slowly, as we can also see from Fig. 1. These numerical simulations show that h and k have a destabilizing effect on the onset of convection (see also Fig. 4).    Table 3 and Fig. 2 show an expected behaviour: the Brinkman term has a stabilizing effect on the onset of convection, i.e. as Da f increases, the Rayleigh number increases and the system becomes more stable. Also, comparing the critical Rayleigh number R 2 L in Table 4a for Da f = 0.001 with the one in Table 4b for Da f = 1, the stabilizing effect of Da f becomes apparent.
Tables 4 and 5 and Fig. 3 display a similar trend, as the Taylor number T 2 increases, the critical Rayleigh number increases, so the heat transfer due to convection is inhibited and rotation has a stabilizing effect on the onset of convection, as we expected. As T 2 increases, the wavenumber a 2 also increases and this means that the convection cells become narrower.

Conclusions
The onset of thermal convection in an anisotropic BDPM, uniformly rotating about a vertical axis and uniformly heated from below, has been analysed according to the Brinkman law in both micropores and macropores. In particular, it has been proved that: -the strong form of the principle of exchange of stabilities holds and hence, when convection occurs, it sets in through a stationary motion; -the linear instability threshold and the global nonlinear stability threshold in the L 2 -norm coincide: this is an optimal result since the stability threshold furnishes a necessary and sufficient condition to guarantee the global (i.e., for all initial data) nonlinear stability; -the critical Rayleigh number for the onset of convection increases with the Taylor number, i.e., rotation has a stabilizing effect on the onset of convection; -the critical Rayleigh number for the onset of convection increases with the Darcy number, i.e., the Brinkman law has a stabilizing effect.