The onset of thermal convection in anisotropic and rotating bidisperse porous media

The onset of thermal convection in anisotropic rotating bidisperse porous media is investigated. The optimal result concerning the coincidence between linear instability and nonlinear stability thresholds with respect 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.


Introduction
The onset of thermal convection for both clear fluids (see [7,20]) and fluids saturating porous materials (see [19,25]) has been widely studied by many scientists-in the past as nowadays-due to its relevant applications in different fields, such as geophysics (geothermal reservoirs, geological storage of carbon dioxide), astrophysics (pore water convection within carbonaceous chondrite parent bodies), engineering and industrial process (water treatment process, nuclear waste disposal, chemical reactor engineering, and the storage of heat-generating materials such as grain and coal) (see [14] and references therein). In [1,19] numerical studies of fluid flows through porous media have been performed; then in [5,6,12] the onset of convection in anisotropic porous media under various assumptions has been analyzed. Moreover, [13,31] investigated clear fluid flows with pressure dependent viscosity, while [21][22][23] deal with the same problem but through porous media. Finally, [11,20,24,30] analyze the onset of convection of a binary mixture in a porous medium, of a non-Newtonian fluid between two vertical plate, of a fluid saturating a porous medium under LTNE assumption and of fluids saturating rotating porous media, respectively.
However, in recent times, double-porosity materials have attracted the attention of a large number of researchers. A double-porosity material is also referred to as bidisperse porous medium (BDPM): it has the normal pore structure, but the solid skeleton has cracks or fissures in it. In particular, a BDPM is composed of clusters of large particles that are agglomerations of small particles, there are macropores between the clusters and micropores within them, and in particular the macropores are referred to as f-phase, while the remainder of the structure is referred to as p-phase. The reason of this nomenclature is that one can think of the f-phase as being a fracture phase, the p-phase as a porous phase [17].
Let ϕ be the porosity of the macropores and be the porosity of the micropores, thus (1 − ϕ) is the fraction of volume occupied by the micropores, ϕ + (1 − ϕ) is the fraction of volume occupied by the fluid, (1 − )(1 − ϕ) is the fraction of volume occupied by the solid skeleton.
The fundamental theory for thermal convection in bidisperse porous media can be found in [15][16][17].
The study of bidisperse convection has a large number of practical applications, such as industrial ones, for example in order to design heat pipes (as reported in [14], since the bidisperse wick structure significantly increases the area available for liquid film evaporation, it has been proposed for use in the evaporator of heat pipes), or medical ones, in fact brain and human bones may be modeled as bidisperse where are the reduced pressures, x = (x, y, z), v s = seepage velocity for s = {f, p}, ζ = interaction coefficient between the f-phase and the p-phase, g = −gk = gravity, μ = fluid viscosity, F = reference constant density, α = thermal expansion coefficient, c = specific heat, c p = specific heat at a constant pressure, 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 steady state (conduction solution): Defining {u f , u p , θ, π f , π p } a perturbation to the steady solution, the evolution equations for the perturbation fields are where u f = (u f , v f , w f ), u p = (u p , v p , w p ). Using the following non-dimensional parameters where the scales are given bỹ and introducing the Taylor number T and the thermal Rayleigh number R, respectively, given by the resulting non-dimensional perturbation equations, omitting all the asterisks, are under the initial conditions , with ∇ · u s 0 = 0, for s = {f, p}, and the boundary conditions w f = w p = θ = 0 on z = 0, 1.
In the sequel, we will suppose that the perturbation fields are periodic in the x and y directions of period 2π l and 2π m , respectively, and we will denote by the periodicity cell.

Instability analysis
In this section we will perform linear instability analysis of the basic solution, to this aim let us first linearise the perturbation Eq. (4), i.e., Since the system (6) is autonomous, we seek solutions which have time-dependence like e σt , i.e., solutions of form with σ ∈ C and s = {f, p}. By virtue of (7), (6) becomes Let * be the complex conjugate of a field. Multiplying where M f = (K f ) −1 and M p = (K p ) −1 , while (·, ·) and · are inner product and norm on the Hilbert space L 2 (V ), respectively. Setting σ = σ r + iσ i , the imaginary part of equation (9) is Applying the same procedure to the complex conjugate of (8), multiplying by u f , u p , θ one gets Adding (10) and (11), one obtains and hence the strong version of the principle of exchange of stabilities holds: if the convection sets in, it arises necessary via a stationary motion (steady convection).

Nonlinear stability
In order to study the influence of rotation on the nonlinear stability of the conduction solution, since the Coriolis terms in momentum equations are antisymmetric, instead of applying the standard energy method, let us apply the differential constraint approach (see [5,11,24]). To this end, let us set and by virtue of (4) 5 , one obtains with periodic in x, y with periods 2π/l, 2π/m; D < ∞; verifying (15) 1,2 } the space of kinematically admissible solutions. The variational problem (21) is equivalent to the following variational problem: where λ(x) and ψ(x) are Lagrange multipliers and periodic in x, y with periods 2π/l, 2π/m; D < ∞}.
By virtue of Poincaré inequality, since from (20) one obtains that condition R < R E guarantees the global nonlinear stability of the conduction solution with respect to the L 2 -norm, according to the following inequality

Remark 1.
Multiplying (8) 1 by u f , (8) 2 by u p , integrating over the period cell V and adding the resulting equations, one finds Settingk = max(k 1 , k 2 , 1) andĥ = max(h 1 , h 2 , 1) and using the generalized Cauchy inequality on the right-hand side of (24), one obtains and hence condition R < R E guarantees that u f 2 → 0 and u p 2 → 0 as t → ∞, too.
In order to solve the variational problem (22), let us consider the associated Euler-Lagrange equations: Defining the operators F. Capone, M. Gentile and G. Massa ZAMP and taking 2Δ of (26) 2,3,4,5 , the Euler-Lagrange equations become Eliminating variable θ and setting By employing normal modes and choosing [6,11] λ = λ 0 sin(nπz)e i(lx+my) , from (28) Requiring a zero determinant for (31) we find R 2 E = R 2 L , and hence the global nonlinear stability threshold and the linear instability threshold coincide and subcritical instabilities do not exist.

Numerical simulations
We now present numerical results to solve (18), in order to analyze the asymptotic behavior of R 2 L with respect to T , h i , k i , for i = 1, 2, i.e., to study the influence of rotation and anisotropic permeability on the onset of convection. As regards the physical parameters, in all numerical simulations, we have chosen a set of values analogous to those ones fixed in [27], in order to compare our results with those ones obtained in [27], to stress the influence of rotation and anisotropy on the onset of convection.
In all the computations, we have performed, the minimum of R 2 L with respect to n is attained at n = 1. Each of the following tables and figures show the stabilizing effect of rotation on the onset of convection. Table 1 shows that for large values of the Taylor number T and when h 1 >> k 1 , m becomes zero, this means that, as rotation increases, the convection cells become rolls with the axis in the y-direction. Table 2 shows a transition from convection patterns as rolls along y-axis (m = 0 for very small T ) to convection patterns as rolls along x-axis (l = 0), as the rotation increases and for h 1 << k 1 . For these physical values, the asymptotic behavior of R 2 L with respect to T is shown in Fig. 1. We can also observe that, as T increases, R 2 L increases more slowly when h 1 << k 1 then h 1 >> k 1 . Let us point out that bi-dimensional convection cells (rolls along x-axis for l = 0 and rolls along y-axis for m = 0) were already found in [27] as an effect of anisotropic macropermeability and micropermeability in absence of rotation.
From Table 3, we numerically find out that for parameters {h 1 = 1, h 2 = 0.1, k 1 = 1, k 2 = 10, η = 0.2, γ 1 = 2, γ 2 = 0.2} the critical value of m is mainly zero, except for very small values of the Taylor number T ∈ [0, 3), for which l and m are both nonzero, i.e., for very little rotation of the layer, threedimensional convection cells are expected.
As a matter of fact, the wavelengths in the x and y directions arex = 2π l andŷ = 2π m . The condition y/x = 0 implies l = 0, this means that the convective fluid motion occurs in the y and z directions     Table a: h 2 = 0.9, k 1 = 0.2, k 2 = 1.1, η = 0.2, γ 1 = 0.9, γ 2 = 1.8, T = 10. (the solution is a function of y and z), i.e., the convection cells are rolls in the x-direction. Instead, the conditionŷ/x → ∞ implies m = 0 and the convective fluid motion occurs in the x and z directions, so the cells are rolls in the y-direction [27].
Tables 4 and 5 exhibit the influence of anisotropy parameters for both macropores and micropores on the onset of convection, and the values of h 1 , h 2 , k 1 , k 2 are fixed such that the permeability ratios in the macropores and micropores are different, in particular we set {h 1 = 3.3, h 2 = 0.9, k 1 = 0.2, k 2 = 1.1} (see [27]) and we vary h s , k s for s = 1, 2 in turn to see how each parameter affects the Rayleigh number. As in [27], we numerically find out a very complex relationship between the macro and micro permeability parameters and the critical Rayleigh and wave numbers. For increasing h 1 , h 2 , k 1 , k 2 , we can see a similar trend, i.e., R 2 L increases up to a maximum before decreasing. From 4(a) and from 5(a), we can see a first transition from rolls along x-axis to three-dimensional cells and then another transition to rolls along y-axis, while 4(b) and 5(b) displays a mirror behavior with respect to 4(a) and 5(a) , respectively.

Conclusions
The onset of thermal convection in a horizontal layer of anisotropic BDPM, uniformly rotating about a vertical axis and uniformly heated from below, has been analyzed, according to Darcy's law in both micropores and macropores. In particular, it has been proved that: • the strong version of the principle of exchange of stabilities holds, and hence, when the convection arises, 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. Moreover, it has been numerically analyzed the influence of the rotation and the influence of the anisotropy on the onset of convection.