nach oben

Acta Mechanica

Erschienen in:

Open Access 23.06.2021 | Original Paper

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

verfasst von: Florinda Capone, Roberta De Luca, Giuliana Massa

Erschienen in: Acta Mechanica | Ausgabe 9/2021

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Patentsuche

Aus

Abstract

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 $L^2$-norm is obtained.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

1 Introduction

Thermal convection in single porous layers has attracted—in the past as nowadays—the interest of many authors due to its numerous applications in industrial processes, in geophysics and in astronomy, for instance. Starting from a detailed theoretical analysis [20, 26, 29‐31], several mathematical models have been introduced in order to better describe different aspects of the phenomenon (see [1‐4, 8‐11, 14, 17, 18, 23‐25] and the references therein). In particular, in [2] mixed convection flow has been numerically studied; in [4, 11], the effect of an external magnetic field induced in an electrically conducting fluid has been investigated; in [17] double-diffusive convection is analysed, while [24] deals with double-diffusive convection according to the Brinkman model.

Recently, the attention of researchers has turned to the onset of thermal convection in bi-disperse porous media (BDPM). A BDPM, as defined in [13], 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. The fundamental theory for thermal convection in BDPM can be found in [22, 26]. In particular, as regards the porosity, a bi-disperse porous medium is a double porosity material. Precisely, denoting by $\varphi $ the porosity of the macropores and by $\epsilon $ the porosity of the micropores, $(1- \varphi ) \epsilon $ represents the fraction of volume occupied by the micropores, $\varphi + (1- \varphi ) \epsilon $ is the fraction of volume occupied by the fluid, and $(1- \epsilon )(1-\varphi )$ is 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, 26‐28] and references therein).

The study of fluid flow in rotating porous media is motivated by its theoretical significance and practical applications (see [3, 5‐8, 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.

2 Mathematical model

Let Oxyz be a reference frame with fundamental unit vectors $\mathbf{i},\mathbf{j},\mathbf{k}$ ($\mathbf{k}$ pointing vertically upward). Let L be a bi-disperse porous layer of thickness d uniformly heated from below and rotating about the vertical axis z, with constant angular velocity $\mathbf{\Omega }=\Omega \mathbf{k}$. Let L be saturated by a homogeneous incompressible fluid at rest state, and let us assume the validity of the local thermal equilibrium between the f-phase and the p-phase, i.e. $T^f=T^p=T$. The saturated bi-disperse porous medium is also supposed to be horizontally isotropic. Let the axes (x, y, z) be the principal axes of the permeability, so the macropermeability tensor and the micropermeability tensor may be written as:

$$\begin{aligned} \begin{aligned} \mathbf{K}^f&= \text {diag}(K^f_x,K^f_y,K^f_z)=K^f_z \ \mathbf{K}^{f*}, \\ \mathbf{K}^p&= \text {diag}(K^p_x,K^p_y,K^p_z)=K^p_z \ \mathbf{K}^{p*}, \end{aligned}\\ \mathbf{K}^{f*}=\text {diag}(k,k,1),\quad \mathbf{K}^{p*}=\text {diag}(h,h,1), \end{aligned}$$

where

$$\begin{aligned} k=\frac{K^f_x}{K^f_z}=\frac{K^f_y}{K^f_z}, \quad h=\frac{K^p_x}{K^p_z}=\frac{K^p_y}{K^p_z}. \end{aligned}$$

A Boussinesq approximation is used, whereby the density is constant except in the buoyancy forces, which are linear in temperature. Taking into account the Coriolis terms due to the uniform rotation of the layer about the vertical axis z for the micropores and the macropores [6] and extending the Brinkman model for a simple porous medium to BDPM [21], the relevant equations are:

$$\begin{aligned} \!\!\!\! {\left\{ \begin{array}{ll} \mathbf{v}^f\! =\! \mu ^{-1} \mathbf{K}^f \! \cdot \! \Bigl [ - \zeta (\mathbf{v}^f\! -\! \mathbf{v}^p) \!-\! \nabla p^f \! + \! \varrho _F \alpha g T \mathbf{k}\! -\! \dfrac{2 \varrho _F \Omega }{\varphi } \mathbf{k} \times \mathbf{v}^f \!+\! {\tilde{\mu }}_f \varDelta \mathbf{v}^f \Bigr ], \\ \mathbf{v}^p \!=\! \mu ^{-1} \mathbf{K}^p \! \cdot \! \Bigl [ - \zeta (\mathbf{v}^p \!-\! \mathbf{v}^f) \!-\! \nabla p^p \! + \! \varrho _F \alpha g T \mathbf{k} \!-\! \dfrac{2 \varrho _F \Omega }{\epsilon } \mathbf{k} \times \mathbf{v}^p \!+\! {\tilde{\mu }}_p \varDelta \mathbf{v}^p \Bigr ], \\ \nabla \cdot \mathbf{v}^f = 0, \\ \nabla \cdot \mathbf{v}^p = 0, \\ (\varrho c)_m T_{,t} + (\varrho c)_f (\mathbf{v}^f + \mathbf{v}^p) \cdot \nabla T = k_m \varDelta T, \end{array}\right. } \end{aligned}$$

(1)

where

$$\begin{aligned} p^s=P^s-\frac{\varrho _F}{2} \vert \mathbf{\Omega } \times \mathbf{x} \vert ^2, \quad s=\{f,p\} \end{aligned}$$

is the reduced pressure, with $\mathbf{x}=(x,y,z)$, $\mathbf{v}^s$ = seepage velocity, $P^s$= pressure, $\varrho $= density, $\zeta $ = interaction coefficient between the f-phase and the p-phase, $\mathbf{g}=-g \mathbf{k}$ = gravity, $\mu $ = fluid viscosity, ${\tilde{\mu }}_s$ = effective viscosity, $\varrho _F$ = reference constant density, $\alpha $ = thermal expansion coefficient, c = specific heat, $c_p$ = specific heat at a constant pressure, $(\varrho c)_m=(1-\varphi )(1-\epsilon )(\varrho c)_{sol}+\varphi (\varrho c)_f+\epsilon (1-\varphi )(\varrho c)_p$, $k_m=(1-\varphi )(1-\epsilon )k_{sol}+\varphi k_f+\epsilon (1-\varphi )k_p$ = thermal conductivity (The subscript sol refers to the solid skeleton).

To (1), the following boundary conditions are appended:

$$\begin{aligned} \begin{array}{l} \mathbf{v}^s \cdot \mathbf{n}=0\,, \quad s=\{f,p\}\qquad \text{ on }\quad z=0,d,\\ T=T_L\,,\quad \text{ on }\quad z=0\,,\qquad T=T_U\,, \quad \text{ on }\quad z=d \end{array} \end{aligned}$$

(2)

where $\mathbf{n}$ 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:

$$\begin{aligned} \mathbf{{\overline{v}}}^f=\mathbf{0},\quad \mathbf{{\overline{v}}}^p=\mathbf{0}, \quad {\overline{T}}=- \beta z + T_L, \end{aligned}$$

where $\beta =\dfrac{T_L-T_U}{d}$ is the temperature gradient. Denoting by $\{ \mathbf{u}^f, \mathbf{u}^p, \theta , \pi ^f, \pi ^p \}$ a perturbation to the steady solution, one recovers that the evolutionary system governing the perturbation fields is given by

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathbf{u}^f\! =\! \mu ^{-1} \mathbf{K}^f \!\cdot \! \Bigl [ -\! \zeta (\mathbf{u}^f \!-\! \mathbf{u}^p) \!-\! \nabla \pi ^f \!+\! \varrho _F \alpha g \theta \mathbf{k} \!-\! \dfrac{2 \varrho _F \Omega }{\varphi } \mathbf{k} \times \mathbf{u}^f \!+\! {\tilde{\mu }}_f \varDelta \mathbf{u}^f \Bigr ], \\ \mathbf{u}^p \!=\! \mu ^{-1} \mathbf{K}^p \!\cdot \! \Bigl [ - \!\zeta (\mathbf{u}^p \!-\! \mathbf{u}^f)\! -\! \nabla \pi ^p \!+\! \varrho _F \alpha g \theta \mathbf{k}\! -\! \dfrac{2 \varrho _F \Omega }{\epsilon } \mathbf{k} \times \mathbf{u}^p \!+\! {\tilde{\mu }}_p \varDelta \mathbf{u}^p \Bigr ], \\ \nabla \cdot \mathbf{u}^f = 0, \\ \nabla \cdot \mathbf{u}^p = 0, \\ (\varrho c)_m \theta _{,t} + (\varrho c)_f (\mathbf{u}^f + \mathbf{u}^p) \cdot \nabla \theta = (\varrho c)_f \beta (w^f+w^p) + k_m \varDelta \theta , \end{array}\right. } \end{aligned}$$

(3)

where $\mathbf{u}^f=(u^f,v^f,w^f),\,\mathbf{u}^p=(u^p,v^p,w^p)$. Introducing the non-dimensional parameters

$$\begin{aligned} \! \mathbf{x}^{*}= & {} \displaystyle \frac{\mathbf{x}}{d}, \ t^{*}=\displaystyle \frac{t}{\tilde{t}}, \ \theta ^{*} = \displaystyle \frac{\theta }{\tilde{T}}, \ \mathbf{u}^{s*} = \displaystyle \frac{\mathbf{u}^s}{\tilde{u}}, \ \pi ^{s*} = \displaystyle \frac{\pi ^s}{\tilde{P}}, \quad \text {for} \ s=\{f,p\},\\ \eta= & {} \displaystyle \frac{\varphi }{\epsilon }, \ \sigma =\displaystyle \frac{{\tilde{\mu }}_p}{{\tilde{\mu }}_f}, \ \gamma _1=\displaystyle \frac{\mu }{K^f_z \zeta }, \ \gamma _2=\displaystyle \frac{\mu }{K^p_z \zeta }, \end{aligned}$$

where the scales are given by

$$\begin{aligned} \tilde{u}=\frac{k_m}{(\varrho c)_f d}, \ \tilde{t} = \frac{d^2 (\varrho c)_m}{k_m}, \ \tilde{P} = \frac{\zeta k_m}{(\varrho c)_f}, \ \tilde{T} = \sqrt{\frac{\beta k_m \zeta }{(\varrho c)_f \varrho _F \alpha g}}, \end{aligned}$$

and setting

$$\begin{aligned} \mathcal {T}= \dfrac{2 \varrho _F \Omega K^f_z}{\varphi \mu }, \qquad Da_f=\dfrac{{\tilde{\mu }}_f K^f_z}{d^2 \mu }, \qquad R = \sqrt{\dfrac{\beta d^2 (\varrho c)_f \varrho _F \alpha g}{k_m \zeta }}, \end{aligned}$$

which are the Taylor number $\mathcal {T}$, the Darcy number $Da_f$, and the thermal Rayleigh number R, respectively, and the resulting non-dimensional perturbation equations, dropping all the asterisks, are

$$\begin{aligned} {\left\{ \begin{array}{ll} \gamma _1 (\mathbf{K}^f)^{-1} \mathbf{u}^f\! +\! (\mathbf{u}^f \!-\! \mathbf{u}^p) \!=\! -\! \nabla \pi ^f \!+\! R \theta \mathbf{k} \!-\! \gamma _1 \mathcal {T} \mathbf{k} \times \mathbf{u}^f \!+\! Da_f \gamma _1 \varDelta \mathbf{u}^f, \\ \gamma _2 (\mathbf{K}^p)^{-1} \mathbf{u}^p \!-\! (\mathbf{u}^f \!-\! \mathbf{u}^p) \!=\! -\! \nabla \pi ^p \!+\! R \theta \mathbf{k} \!-\! \eta \gamma _1 \mathcal {T} \mathbf{k} \times \mathbf{u}^p \!+\! Da_f \gamma _1 \sigma \varDelta \mathbf{u}^p, \\ \nabla \cdot \mathbf{u}^f = 0, \\ \nabla \cdot \mathbf{u}^p = 0, \\ \theta _{,t} + (\mathbf{u}^f + \mathbf{u}^p) \cdot \nabla \theta = R (w^f+w^p) +\varDelta \theta , \end{array}\right. } \end{aligned}$$

(4)

under the initial conditions

$$\begin{aligned} \mathbf {u}^s(\mathbf {x},0)=\mathbf {u}^s_0(\mathbf {x})\,,\qquad \varphi ^s(\mathbf {x},0)=\varphi _0(\mathbf {x})\,,\qquad \theta (\mathbf {x},0)=\theta _0(\mathbf {x}) \end{aligned}$$

with $\nabla \cdot \mathbf {u}_0^s=0,\, s=\{f,p\}$, and the stress-free boundary conditions [12]

$$\begin{aligned} u^f_{,z}=v^f_{,z}=u^p_{,z}=v^p_{,z}=w^f=w^p=\theta =0 \quad \text {on} \ z=0,1. \end{aligned}$$

(5)

Moreover, according to experimental results, let us assume that the perturbation fields are periodic functions in the x, y directions and denote by

$$\begin{aligned} V=\Big [ 0,\frac{2 \pi }{l} \Big ] \times \Big [ 0,\frac{2 \pi }{m} \Big ] \times [0,1] \end{aligned}$$

the periodicity cell.

3 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 $\mathbf{u}^f, \mathbf{u}^p, \theta , \pi ^f, \pi ^p$ have time dependence like $e^{\overline{\sigma }t}$, i.e.

$$\begin{aligned} {\left\{ \begin{array}{ll} \gamma _1 (\mathbf{K}^f)^{-1} \mathbf{u}^f \!+\! (\mathbf{u}^f \!-\! \mathbf{u}^p) \!=\! -\! \nabla \pi ^f \!+\! R \theta \mathbf{k} \!-\! \gamma _1 \mathcal {T} \mathbf{k} \times \mathbf{u}^f \!+\! Da_f \gamma _1 \varDelta \mathbf{u}^f, \\ \gamma _2 (\mathbf{K}^p)^{-1} \mathbf{u}^p \!-\! (\mathbf{u}^f \!-\! \mathbf{u}^p) \!=\! -\! \nabla \pi ^p \!+\! R \theta \mathbf{k} \!-\! \eta \gamma _1 \mathcal {T} \mathbf{k} \times \mathbf{u}^p \!+\! Da_f \gamma _1 \sigma \varDelta \mathbf{u}^p, \\ {\overline{\sigma }} \theta = R (w^f+w^p) +\varDelta \theta . \end{array}\right. } \end{aligned}$$

(6)

Let $(\cdot , \cdot )$ and $\Vert \cdot \Vert $ 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 $\mathbf{u}^{f*}$, (6)$_{2}$ by $\mathbf{u}^{p*}$ and (6)$_{3}$ by $\theta ^{*}$, the integration over V leads to:

$$\begin{aligned} {\overline{\sigma }} \Vert \theta \Vert ^2= & {} - \gamma _1 (\mathbf{K}^f)^{-1} \Vert \mathbf{u}^f \Vert ^2 - \gamma _2 (\mathbf{K}^p)^{-1} \Vert \mathbf{u}^p \Vert ^2 - \Vert \mathbf{u}^f - \mathbf{u}^p \Vert ^2 - \Vert \nabla \theta \Vert ^2 \nonumber \\&+ R[(\theta ,w^{f*}+w^{p*}) + (w^f+w^p,\theta ^{*})] \nonumber \\&- \gamma _1 \mathcal {T}(\mathbf{k} \times \mathbf{u}^f, \mathbf{u}^{f*}) - \eta \gamma _1 \mathcal {T}(\mathbf{k} \times \mathbf{u}^p,\mathbf{u}^{p*}) \nonumber \\&- Da_f \gamma _1 \Vert \nabla \mathbf{u}_f \Vert ^2 - Da_f \sigma \gamma _1 \Vert \nabla \mathbf{u}^p \Vert ^2. \end{aligned}$$

(7)

If ${\overline{\sigma }}=\overline{\sigma }_r+i \overline{\sigma }_i$, the imaginary part of (7) is

$$\begin{aligned} \overline{\sigma }_i \Vert \theta \Vert ^2 = - \gamma _1 \mathcal {T}(\mathbf{k} \times \mathbf{u}^f, \mathbf{u}^{f*}) - \eta \gamma _1 \mathcal {T}(\mathbf{k} \times \mathbf{u}^p, \mathbf{u}^{p*})\,. \end{aligned}$$

(8)

On the other hand, applying the same procedure to the complex conjugate of (6) and multiplying by $\mathbf{u}^{f}$, $\mathbf{u}^{p}$, $\theta $, one gets:

$$\begin{aligned} \overline{\sigma }_i \Vert \theta \Vert ^2 = - \gamma _1 \mathcal {T}(\mathbf{k} \times \mathbf{u}^{f*}, \mathbf{u}^{f}) - \eta \gamma _1 \mathcal {T}(\mathbf{k} \times \mathbf{u}^{p*}, \mathbf{u}^{p})\,. \end{aligned}$$

(9)

Adding (8) and (9), one obtains:

$$\begin{aligned} 2\overline{\sigma }_i\Vert \theta \Vert ^2=0\end{aligned}$$

(10)

and hence necessarily $\overline{\sigma }_i=0$, i.e., ${\overline{\sigma }} \in {\mathbb {R}}$ and the strong form of the principle of exchange of stability holds, i.e., the oscillatory convection cannot arise.

Denoting by

$$\begin{aligned} \begin{array}{l} \varDelta _1 f=f_{,xx}+f_{,yy}\,, \quad \varDelta ^m \equiv \underbrace{ \varDelta \varDelta \cdots \varDelta }_{m}\,, \qquad \omega ^s_3=(\nabla \times \mathbf{u}^s) \cdot \mathbf{k},\,\, s=\{f,p\}, \\ \qquad \qquad \qquad \qquad {\overline{a}}=\displaystyle \frac{\gamma _1}{k}+1\,,\quad {\overline{b}}=\displaystyle \frac{\gamma _2}{h}+1 \end{array} \end{aligned}$$

and defining the operators

$$\begin{aligned} A\equiv \overline{a}-Da_f \gamma _1 \varDelta \,, \qquad B\equiv \overline{b}-Da_f \sigma \gamma _1 \varDelta \,, \qquad \varPsi \equiv (AB-1), \end{aligned}$$

(11)

the third components of curl and of double curl of (4)$_{1,2}$ are, respectively, given by

$$\begin{aligned} {\left\{ \begin{array}{ll} A\omega _3^f -\omega _3^p=\gamma _1\mathcal {T}w_{,z}^f,\\ -\omega _3^f+B\omega _3^p=\eta \gamma _1\mathcal {T}w_{,z}^p \end{array}\right. }\end{aligned}$$

(12)

and

$$\begin{aligned} {\left\{ \begin{array}{ll} \! - \dfrac{\gamma _1}{k} w^f_{,zz} \! - \! \gamma _1 \varDelta _1 w^f \! - \! \varDelta w^f \! + \! \varDelta w^p \! \! = \! \! - \! R \varDelta _1 \theta \! + \! \gamma _1 \mathcal {T}\omega ^f_{3,z} \! - \! Da_f \gamma _1 \varDelta ^2 w^f \! , \\ \! - \dfrac{\gamma _2}{h} w^p_{,zz} \! - \! \gamma _2 \varDelta _1 w^p \! + \! \varDelta w^f \! - \! \varDelta w^p \! \! = \! \! - \! R \varDelta _1 \theta \! + \! \eta \gamma _1 \mathcal {T}\omega ^p_{3,z} \! - \! Da_f \gamma _1 \sigma \varDelta ^2 w^p \! . \end{array}\right. } \end{aligned}$$

(13)

Applying the operator B to (12)$_1$, by virtue of (12)$_2$, one has

$$\begin{aligned} \varPsi \omega _3^f=\gamma _1\mathcal {T}Bw_{,z}^f+\eta \gamma _1\mathcal {T}w_{,z}^p. \end{aligned}$$

This equation, together with that one obtained by applying the operator $\varPsi $ to (12)$_2$, leads to

$$\begin{aligned} {\left\{ \begin{array}{ll} \varPsi \omega _3^f=\gamma _1\mathcal {T}B w_{,z}^f+\eta \gamma _1\mathcal {T}w_{,z}^p,\\ \varPsi B \omega _3^p=\gamma _1 \mathcal {T}B w_{,z}^f+\eta \gamma _1 \mathcal {T}AB w_{,z}^p. \end{array}\right. } \end{aligned}$$

(14)

Applying the operator $\varPsi $ to (13)$_1$ and $\varPsi B$ to (13)$_2$, one gets

$$\begin{aligned} {\left\{ \begin{array}{ll} -\overline{a} \varPsi w^f_{,zz} - {\hat{\gamma }}_1 \varPsi \varDelta _1 w^f + \varPsi \varDelta _1 w^p + \varPsi w^p_{,zz} \\ \qquad = - R \varPsi \varDelta _1 \theta + \gamma _1 \mathcal {T}\varPsi \omega ^f_{3,z} -Da_f \gamma _1 \varPsi \varDelta ^2 w^f, \\ -\overline{b} \varPsi B w^p_{,zz} -\hat{\gamma _2} \varPsi B \varDelta _1 w^p + \varPsi B \varDelta _1 w^f +\varPsi B w^f_{,zz} \\ \qquad = -R \varPsi B \varDelta _1 \theta + \eta \gamma _1 \mathcal {T}\varPsi B \omega ^p_{3,z} -Da_f \sigma \gamma _1 \varPsi B \varDelta ^2 w^p, \ \end{array}\right. } \end{aligned}$$

(15)

with ${\hat{\gamma }}_r=\gamma _r+1$, for $r=1,2$.

In view of (14), (15) can be written as

$$\begin{aligned} {\left\{ \begin{array}{ll} {[} -\overline{a}\varPsi -(\gamma _1 \mathcal {T})^2 B ] w^f_{,zz} - {\hat{\gamma }}_1 \varPsi \varDelta _1 w^f + \varPsi \varDelta _1 w^p \\ \quad + {[} \varPsi -\eta (\gamma _1 \mathcal {T})^2] w^p_{,zz} +Da_f \gamma _1 \varPsi \varDelta ^2 w^f= -R \varPsi \varDelta _1 \theta , \\ {[} -\overline{b} \varPsi B -(\eta \gamma _1 \mathcal {T})^2 AB ] w^p_{,zz} - {\hat{\gamma }}_2 \varPsi B \varDelta _1 w^p + \varPsi B \varDelta _1 w^f \\ \quad + {[} \varPsi B-\eta (\gamma _1 \mathcal {T})^2 B] w^f_{,zz} +Da_f \sigma \gamma _1 \varPsi B \varDelta ^2 w^p= -R \varPsi B \varDelta _1 \theta . \end{array}\right. } \end{aligned}$$

(16)

Therefore, to find the linear instability threshold, we consider (6)$_3$, (16)$_1$ and (16)$_2$ with $\overline{\sigma }=0$, i.e.

$$\begin{aligned} {\left\{ \begin{array}{ll} {[} -\overline{a}\varPsi -(\gamma _1 \mathcal {T})^2 B ] w^f_{,zz} - {\hat{\gamma }}_1 \varPsi \varDelta _1 w^f + \varPsi \varDelta _1 w^p \\ \quad + {[} \varPsi - \eta (\gamma _1 \mathcal {T})^2 ] w^p_{,zz} +Da_f \gamma _1 \varPsi \varDelta ^2 w^f= -R \varPsi \varDelta _1 \theta , \\ {[} -\overline{b} \varPsi B -(\eta \gamma _1 \mathcal {T})^2 AB ] w^p_{,zz} - {\hat{\gamma }}_2 \varPsi B \varDelta _1 w^p + \varPsi B \varDelta _1 w^f \\ \quad + {[} \varPsi B-\eta (\gamma _1 \mathcal {T})^2 B] w^f_{,zz} +Da_f \sigma \gamma _1 \varPsi B \varDelta ^2 w^p= -R \varPsi B \varDelta _1 \theta , \\ R (w^f+w^p) +\varDelta \theta =0. \end{array}\right. } \end{aligned}$$

(17)

Employing normal modes in (17), i.e. assuming the following representation [12]

$$\begin{aligned} \begin{aligned} w^f&= W^f_0 \sin (n \pi z) e^{i(lx+my)}, \\ w^p&= W^p_0 \sin (n \pi z) e^{i(lx+my)}, \\ \theta&= \Theta _0 \sin (n \pi z) e^{i(lx+my)}, \end{aligned} \end{aligned}$$

(18)

with $W^f_0,W^p_0,\Theta _0$ real constants, setting $a^2=l^2+m^2$ and $\Lambda _n=a^2+n^2 \pi ^2$, from (17), it turns out that

$$\begin{aligned} {\left\{ \begin{array}{ll} \Big [\! \Lambda _n e (A_1 M \!+\! \sigma f n^2 \pi ^2) \!+\! \Lambda ^2_n e (M e \sigma \!+\! B_1) \!+\! f \overline{b} n^2 \pi ^2 \!+\! B_1 M \\ \ + \ e^2 \Lambda _n^3 A_1 \!+\! e^3 \sigma \Lambda _n^4 \Big ] \! W^f_0 \!+\! \Big [ \!-\! B_1 \Lambda _n \!-\! e A_1 \Lambda _n^2 \!-\! e^2 \sigma \Lambda _n^3 \!+\! \eta f n^2 \pi ^2 \Big ] \! W^p_0 \\ \ \ -R a^2 \Big [ B_1 + e \Lambda _n A_1 +e^2 \sigma \Lambda _n^2 \Big ] \Theta _0 =0, \\ \Big [ \Lambda _n (e \sigma n^2 \pi ^2 \eta f - \overline{b} B_1)+\eta f n^2 \pi ^2 \overline{b} -\Lambda _n^2 e C \\ \ \ -\Lambda _n^3 e^2 \sigma (A_1+\overline{b}) -\Lambda _n^4 \sigma ^2 e^3 \Big ] W^f_0 \\ \ \ +\Big \{ \Lambda _n e (CN+\eta ^2 f A_1 n^2 \pi ^2) + \Lambda _n^2 e \sigma [e(A_1+\overline{b})N+\overline{b}B_1 + e \eta ^2 f n^2 \pi ^2]\\ \ \ + B_1 \overline{b} N +e^2 \Lambda _n^3 \sigma (C+e \sigma N)+\Lambda _n^4 e^3 \sigma ^2 (A_1+\overline{b}) +\Lambda _n^5 e^4 \sigma ^3 \\ \ \ +\eta ^2 \! f n^2 \pi ^2 \overline{a} \overline{b} \Big \} W^p_0 \!-\! R a^2 \Big [ \overline{b} B_1 \!+\! e \Lambda _n C \!+\! \Lambda _n^2 e^2 \sigma (A_1 \!+\! \overline{b}) \!+\! e^3 \sigma ^2 \Lambda _n^3 \Big ]\! \Theta _0 \!=\! 0, \\ \\ R W^f_0+R W^p_0 - \Lambda _n \Theta _0=0, \end{array}\right. } \end{aligned}$$

(19)

where

$$\begin{aligned} \begin{array}{l} A_1 = \sigma {\overline{a}} +{\overline{b}}, \,\,\, B_1 = \dfrac{\gamma _1}{k} \dfrac{\gamma _2}{h} + \dfrac{\gamma _1}{k} +\dfrac{\gamma _2}{h}, \,\,\, C = \sigma (2 B_1 + 1) + {{\overline{b}}}^2, \\ M = \dfrac{\gamma _1}{k} n^2 \pi ^2 + \gamma _1 a^2 + \Lambda _n, \,\,\, N = \dfrac{\gamma _2}{h} n^2 \pi ^2 + \gamma _2 a^2 + \Lambda _n, \\ e = Da_f \gamma _1, \,\,\,\, f = (\gamma _1 \mathcal {T})^2. \end{array} \end{aligned}$$

(20)

Setting

$$\begin{aligned} \begin{aligned} h_{11} =&\Lambda _n e (A_1 M+\sigma f n^2 \pi ^2) + \Lambda ^2_n e (M e \sigma +B_1) + f \overline{b} n^2 \pi ^2 + B_1 M +e^2 \Lambda _n^3 A_1 +e^3 \sigma \Lambda _n^4, \\ h_{12} =&-B_1 \Lambda _n -e A_1 \Lambda _n^2 -e^2 \sigma \Lambda _n^3 +\eta f n^2 \pi ^2, \\ h_{13} =&B_1 + e \Lambda _n A_1 +e^2 \sigma \Lambda _n^2, \\ h_{21} =&\Lambda _n (e \sigma n^2 \pi ^2 \eta f - \overline{b} B_1)+\eta f n^2 \pi ^2 \overline{b} -\Lambda _n^2 e C -\Lambda _n^3 e^2 \sigma (A_1+\overline{b}) -\Lambda _n^4 \sigma ^2 e^3, \\ h_{22} =&\Lambda _n e (CN+\eta ^2 f A_1 n^2 \pi ^2) + \Lambda _n^2 e \sigma [e(A_1+\overline{b})N+\overline{b}B_1 + e \eta ^2 f n^2 \pi ^2] \\&+ B_1 \overline{b} N+ e^2 \Lambda _n^3 \sigma (C+e \sigma N)+\Lambda _n^4 e^3 \sigma ^2 (A_1+\overline{b}) +\Lambda _n^5 e^4 \sigma ^3 +\eta ^2 f n^2 \pi ^2 \overline{a} \overline{b}, \\ h_{23} =&\, \overline{b} B_1 + e \Lambda _n C + \Lambda _n^2 e^2 \sigma (A_1+\overline{b}) + e^3 \sigma ^2 \Lambda _n^3, \end{aligned} \end{aligned}$$

(19) can be written as

$$\begin{aligned} {\left\{ \begin{array}{ll} h_{11} W^f_0 +h_{12} W^p_0 - R a^2 h_{13} \Theta _0 = 0, \\ \\ h_{21} W^f_0 +h_{22} W^p_0 - R a^2 h_{23} \Theta _0 = 0, \\ \\ R W^f_0 + R W^p_0 - \Lambda _n \Theta _0 = 0. \end{array}\right. } \end{aligned}$$

(21)

The smallest value of $R^2$ which vanishes the determinant of (21) is the critical Rayleigh number for the onset of instability, i.e.

$$\begin{aligned} R^2_L=\min _{(n,a^2) \in \mathbb {N}\times \mathbb {R}^{+}} f_L^2(n,a^2) \end{aligned}$$

(22)

with

$$\begin{aligned} f_L^2(n,a^2)=\dfrac{\Lambda _n}{a^2} \dfrac{(h_{11} h_{22} - h_{12} h_{21})}{h_{13} h_{22} - h_{12} h_{23} + h_{11} h_{23} - h_{21} h_{13}}\,. \end{aligned}$$

(23)

We have proved that: (i) both numerator and denominator of (23) are strictly positive; (ii) the minimum of $f_L^2$ with respect to n, by using numerical computations, is attained at $n=1$. Hence,

$$\begin{aligned} R^2_L=\displaystyle \min _{a^2\in \mathbb {R}^+} f^2_L(1,a^2). \end{aligned}$$

(24)

The minimum of $f_L^2(1,a^2)$ 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:

$$\begin{aligned} \!\!\!\!\!\!\!\!R_L\!\!^2 \! \! =\!\min _{n,a^2}\! \dfrac{\Lambda _n}{a^2} \dfrac{B_1 MN \! - \! B_1 \Lambda _n^2 \!+\! \eta ^2 f^2 n^4 \pi ^4 + f n^2 \pi ^2 (\overline{b}N+\eta ^2 \overline{a}M+2 \eta \Lambda _n)}{B_1(M+N+2 \Lambda _n) +f n^2 \pi ^2 (\overline{a} \eta ^2 -2 \eta +\overline{b})} \end{aligned}$$

(25)

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.

$$\begin{aligned} R_L^2 \! =\! \min _{n,a^2} \dfrac{\Lambda _n}{a^2} \dfrac{\varGamma ^2 \Lambda _n^2 \! + \! \gamma _1^2 \mathcal {T}^2 n^2 \pi ^2 \Lambda _n [\eta ^2(\gamma _1+1)^2 \! + \! (\gamma _2+1)^2 \! + \! 2 \eta ] \! + \! \gamma _1^4 \mathcal {T}^4 n^4 \pi ^4 \eta ^2}{\varGamma \Lambda _n (\gamma _1+\gamma _2+4) + \gamma _1^2 \mathcal {T}^2 n^2 \pi ^2 [\eta ^2 \gamma _1 +\gamma _2 + (\eta -1)^2]} \end{aligned}$$

where $\varGamma =\gamma _1 \gamma _2 +\gamma _1 + \gamma _2$;

(iii)

if $h\!=\!k\!=\!1$, ${\tilde{\mu }}_p=0$, in the limit as $\mathcal {T}\rightarrow 0$, i.e. assuming the validity of the Brinkman’s law only in the momentum equation for the macropores and in the absence of rotation, from (22) one obtains

$$\begin{aligned} \! R^2 \! \! =\min _{n,a^2} \dfrac{\Lambda _n^2}{a^2} \dfrac{\gamma _1 \gamma _2 + \gamma _1 + \gamma _2 + e (\gamma _2+1) \Lambda _n}{e \Lambda _n+\gamma _1+\gamma _2+4} \end{aligned}$$

(26)

and (26) coincides with the critical threshold found in [16].

4 Nonlinear stability

In order to study the nonlinear stability of the conduction solution, by virtue of (11), since

$$\begin{aligned} \varPsi \equiv \overline{a} \overline{b} -e A_1 \varDelta +e^2 \sigma \varDelta ^2 -1\,, \quad \varPsi B \equiv \overline{b} B_1 -e C \varDelta +e^2 \sigma (A_1 +\overline{b}) \varDelta ^2 -e^3 \sigma ^2 \varDelta ^3 \end{aligned}$$

(16) and (4)$_5$ can be written as:

$$\begin{aligned} {\left\{ \begin{array}{ll} L_1 w^f + L_2 w^p +R L_3 \theta =0, \\ M_1 w^f + M_2 w^p +R M_3 \theta =0, \\ \theta _{,t} + (\mathbf{u}^f + \mathbf{u}^p) \cdot \nabla \theta = R (w^f+w^p) +\varDelta \theta , \end{array}\right. } \end{aligned}$$

(27)

where the following differential operators have been defined:

$$\begin{aligned} \begin{aligned} L_1\equiv&- \overline{a}B_1 \partial _{zz} + e (\overline{a}A_1+\sigma f) \partial _{zz} \varDelta -f \overline{b} \partial _{zz} -\hat{\gamma _1} B_1 \varDelta _1 - \overline{a} \sigma e^2 \varDelta ^2 \partial _{zz} \\&+\hat{\gamma _1} e A_1 \varDelta \varDelta _1 -\hat{\gamma _1} e^2 \sigma \varDelta ^2 \varDelta _1 +e B_1 \varDelta ^2 -e^2 A_1 \varDelta ^3 +e^3 \sigma \varDelta ^4, \\ L_2\equiv&B_1 \varDelta _1 +B_1 \partial _{zz} -e A_1 \varDelta ^2 +e^2 \sigma \varDelta ^3 - \eta f \partial _{zz}, \\ L_3\equiv&\, B_1 \varDelta _1 -e A_1 \varDelta \varDelta _1 + e^2 \sigma \varDelta ^2 \varDelta _1 \\ M_1\equiv&\, \overline{b}B_1 \varDelta - e C \varDelta ^2 + e^2 \sigma (A_1+\overline{b}) \varDelta ^3 -e^3 \sigma ^2 \varDelta ^4 -\overline{b} \eta f \partial _{zz} + e \eta f \sigma \varDelta \partial _{zz}, \\ M_2\equiv&\, -\overline{b} (\overline{b}B_1 +\overline{a} \eta ^2 f) \partial _{zz}+\sigma e \overline{b} B_1 \varDelta ^2 -\hat{\gamma _2} \overline{b} B_1 \varDelta _1\\&+e( \overline{b}C+\eta ^2 f A_1 )\varDelta \partial _{zz} -e^2 \sigma [\overline{b}(A_1+\overline{b})+\eta ^2 f] \varDelta ^2 \partial _{zz}+e^3 \overline{b} \sigma \varDelta ^3 \partial _{zz}\\&+\hat{\gamma _2} e C \varDelta \varDelta _1 -\hat{\gamma _2} e^2 \sigma (A_1+\overline{b}) \varDelta ^2 \varDelta _1 \\&+\hat{\gamma _2} e^3 \sigma ^2 \varDelta ^3 \varDelta _1 -\sigma e^2 C \varDelta ^3 +\sigma ^2 e^3 (A_1+\overline{b}) \varDelta ^4 -e^4 \sigma ^3 \varDelta ^5, \\ M_3\equiv&\, \overline{b} B_1 \varDelta _1 -e C \varDelta \varDelta _1 +e^2 \sigma (A_1+\overline{b}) \varDelta ^2 \varDelta _1 -e^3 \sigma ^2 \varDelta ^3 \varDelta _1. \end{aligned} \end{aligned}$$

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

$$\begin{aligned} E(t)= & {} \frac{1}{2} \Vert \theta \Vert ^2,\nonumber \\ I(t)= & {} (w^f+w^p,\theta ), \quad D(t) = \Vert \nabla \theta \Vert ^2. \end{aligned}$$

(28)

Retaining (27)$ _{1,2}$ as constraints, multiplying (27)$_3$ by $\theta $ and integrating over the periodicity cell, one gets

$$\begin{aligned} \frac{dE}{dt} = I-D \le -D \left( 1-\frac{R}{R_E} \right) , \end{aligned}$$

(29)

where

$$\begin{aligned} \frac{1}{R_E} = \max _{\mathcal {H}^{*}} \frac{I}{D} \end{aligned}$$

(30)

and

$$\begin{aligned} \begin{aligned} \mathcal {H}^{*} = \{&(w^f,w^p,\theta ) \in (H^1)^3 | w^f=w^p=\theta =0 \ \text {on} \ z=0,1; \text {periodic in} \ x,y \\&\text {with periods} \ 2 \pi /l, 2 \pi /m;D<\infty ; \text {verifying} \ (27)_{1,2} \} \end{aligned} \end{aligned}$$

is the space of kinematically admissible solutions.

The variational problem associated with the maximum problem (30) is equivalent to

$$\begin{aligned} \frac{1}{R_E}= \max _{\mathcal {H}} \frac{I + \int _V \lambda ^{'} g_1 \ dV + \int _V \lambda ^{''} g_2 \ dV }{D}, \end{aligned}$$

(31)

where $\lambda ^{'}(\mathbf{x})$ and $\lambda ^{''}(\mathbf{x})$ are Lagrange multipliers and

$$\begin{aligned} g_1=L_1 w^f + L_2 w^p +R L_3 \theta , \quad g_2=M_1 w^f + M_2 w^p +R M_3 \theta ,\\ \begin{aligned} \mathcal {H} = \{&(w^f,w^p,\theta ) \in (H^1)^3 | w^f=w^p=\theta =0 \ \text {on} \ z=0,1; \text {periodic in} \ x,y \\&\text {with periods} \ 2 \pi /l, 2 \pi /m, \ \text {respectively};D< \infty \}. \end{aligned} \end{aligned}$$

Applying the Poincaré inequality, one obtains that $D \ge \pi ^2 \Vert \theta \Vert ^2$, and hence, from (29) it follows that the condition $R<R_E$ implies $E(t) \rightarrow 0$ at least exponentially.

Remark 2

Multiplying (4)$_1$ by $\mathbf{u}^f$, (4)$_2$ by $\mathbf{u}^p$, integrating over the period cell V and adding the resulting equations, one finds

$$\begin{aligned} \begin{aligned} \gamma _1&\! \int _V\! \Big \{ \!k^{-1} [ (u^{f})^2 \! + \! (v^{f})^2 ] \! + \! (w^{f})^2 \Big \}\! dV \! + \gamma _2 \!\! \int _V \Big \{\! h^{-1} [ (u^{p})^2 \! + \! (v^{p})^2 ] \! + \! (w^{p})^2 \Big \}\! dV \\&+\!\Vert \mathbf{u}^f - \mathbf{u}^p \Vert ^2 = R (\theta , w^f+w^p) - Da_f \gamma _1 \Vert \nabla \mathbf{u}^f \Vert ^2 - \sigma Da_f \gamma _1 \Vert \nabla \mathbf{u}^p \Vert ^2 . \end{aligned} \end{aligned}$$

(32)

Setting ${\hat{k}}=\max (k,1)$ and ${\hat{h}}=\max (h,1)$, by virtue of generalized Cauchy inequality and Poincaré-like inequality, from (32) one obtains

$$\begin{aligned} \Big ( \frac{\gamma _1}{{\hat{k}}}+2 Da_f \gamma _1 c_1 \Big ) \Vert \mathbf{u}^f \Vert ^2 + \Big ( \frac{\gamma _2}{{\hat{h}}}+2 Da_f \gamma _1 \sigma c_2 \Big ) \Vert \mathbf{u}^p \Vert ^2 \le R^2 \Big ( \frac{{\hat{k}}}{\gamma _1} + \frac{{\hat{h}}}{\gamma _2} \Big ) \Vert \theta \Vert ^2, \end{aligned}$$

(33)

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 $\mathbf{u}^f$ and $\mathbf{u}^p$.

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

$$\begin{aligned} {\left\{ \begin{array}{ll} R_E(w^f+w^p)+R_E L_3 \lambda ^{'} +R_E M_3 \lambda ^{''} +2 \varDelta \theta =0, \\ R_E \theta +L_1 \lambda ^{'} +M_1 \lambda ^{''}=0, \\ R_E \theta +L_2 \lambda ^{'} +M_2 \lambda ^{''}=0, \\ L_1 w^f + L_2 w^p +R L_3 \theta =0, \\ M_1 w^f + M_2 w^p +R M_3 \theta =0. \end{array}\right. } \end{aligned}$$

(34)

Eliminating the variable $\theta $ in (34), it turns out that

$$\begin{aligned} {\left\{ \begin{array}{ll} -R_E^2 w^f \! - \! R_E^2 w^p \!+ \! (2 \varDelta L_1 \! - \! R_E^2 L_3) \lambda ^{'} \! + \! (2 \varDelta M_1 -R_E^2 M_3) \lambda ^{''} \! \! = \! 0, \\ \! -R_E^2 w^f \! - \! R_E^2 w^p \!+ \! (2 \varDelta L_2 \! - \! R_E^2 L_3) \lambda ^{'} \!+ \! (2 \varDelta M_2 \! - \! R_E^2 M_3) \lambda ^{''} \! \! = \! 0, \\ \! (2 \varDelta L_1 \! - \! R_E^2 L_3) w^f \!+ \! (2 \varDelta L_2 \! - \! R_E^2 L_3) w^p -R_E^2 L_3^2 \lambda ^{'} -R_E^2 L_3 M_3 \lambda ^{''} \! \! = \! 0, \\ \! (2 \varDelta M_1 \! - \! R_E^2 M_3) w^f \!+ \! (2 \varDelta M_2 \! - \! R_E^2 M_3) w^p \! - \! R_E^2 L_3 M_3 \lambda ^{'} \! - \! R_E^2 M_3^2 \lambda ^{''} \! \!= \! 0. \end{array}\right. } \end{aligned}$$

(35)

By using (18)$_1$ and (18)$_2$ and choosing [17]

$$\begin{aligned} \begin{aligned} \lambda ^{'}&= \lambda _0^{'} \sin (n \pi z) e^{i(lx+my)}, \\ \lambda ^{''}&= \lambda _0^{''} \sin (n \pi z) e^{i(lx+my)}, \end{aligned} \end{aligned}$$

(36)

from (35), one obtains

$$\begin{aligned} {\left\{ \begin{array}{ll} -R_E^2 W^f_0 -R_E^2 W^p_0 +(-2 \Lambda _n h_{11} +R_E^2 a^2 h_{13}) \lambda ^{'}_0 \\ \quad + (-2 \Lambda _n h_{21} +R_E^2 a^2 h_{23}) \lambda ^{''}_0 =0, \\ -R_E^2 W^f_0 -R_E^2 W_0^p +(-2 \Lambda _n h_{12} +R_E^2 a^2 h_{13}) \lambda ^{'}_0 \\ \quad + (-2 \Lambda _n h_{22} + R_E^2 a^2 h_{23}) \lambda ^{''}_0 =0, \\ (-2 \Lambda _n h_{11} +R_E^2 a^2 h_{13}) W^f_0 +(-2 \Lambda _n h_{12} +R_E^2 a^2 h_{13}) W^p_0 \\ \quad -R_E^2 a^4 h_{13}^2 \lambda ^{'}_0 -R_E^2 a^4 h_{13} h_{23} \lambda ^{''}_0=0,\\ (-2 \Lambda _n h_{21} +R_E^2 a^2 h_{23}) W^f_0 +(-2 \Lambda _n h_{22} + R_E^2 a^2 h_{23}) W^p_0 \\ \quad -R_E^2 a^4 h_{13} h_{23} \lambda ^{'}_0 -R_E^2 a^4 h_{23}^2 \lambda ^{''}_0=0. \end{array}\right. } \end{aligned}$$

(37)

Requiring a zero determinant for (37), we get

$$\begin{aligned} R_E^2=R_L^2; \end{aligned}$$

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).

5 Numerical results

In this section, we numerically analyse the asymptotic behaviour of $R_L^2$ with respect to the parameters h, k, $\mathcal {T}^2$, $Da_f$ 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 $n=1$.

Table 1

Critical Rayleigh and wave numbers for $k=1,\eta =0.2,\sigma =0.3,\gamma _1=0.9,\gamma _2=1.8,\mathcal {T}^2=10,Da_f=1$ at different h

$R^2_L$	$a^2_c$	h
326.5116	7.6377	0.1
211.6567	6.1968	0.5
190.6873	5.7021	1
171.6278	5.1756	5
169.0585	5.0986	10

Table 2

Critical Rayleigh and wave numbers for $h=1,\eta =0.2,\sigma =0.3,\gamma _1=0.9,\gamma _2=1.8,\mathcal {T}^2=10,Da_f=1$ at different k

$R^2_L$	$a^2_c$	k
208.1330	6.2007	0.1
192.9895	5.7772	0.5
190.6873	5.7021	1
188.7616	5.6372	5
188.5154	5.6287	10

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

Critical Rayleigh and wave numbers for $h=10,k=1,\eta =0.2,\sigma =0.3,\gamma _1=0.9,\gamma _2=1.8,\mathcal {T}^2=10$ at different Darcy numbers

$R^2_L$	$a^2_c$	$Da_f$
56.3828	13.7917	0.001
100.9334	5.4036	0.5
169.0585	5.0986	1
716.2302	4.9566	5

A typical small Darcy number is $Da_f=0.001$, while a typical Darcy number is $Da_f=1$ [22]

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 $R_L^2$ 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.

Table 4

Critical Rayleigh and wave numbers for increasing Taylor numbers

$R^2_L$	$a^2_c$	$\mathcal {T}^2$
(a) $Da_f=0.001$
52.2521	12.8790	0
52.7216	13.0444	0.1
54.5534	13.6882	0.5
56.7498	14.4563	1
71.7551	19.5421	5
86.7523	24.7523	10
211.5075	39.0022	100
(b) $Da_f=1$
323.9093	7.4974	0
323.9359	7.4988	0.1
324.0419	7.5046	0.5
324.1741	7.5118	1
325.2230	7.5685	5
326.5116	7.6377	10
346.4686	8.6407	100

(a) $h=0.1,k=1,\eta =0.2,\sigma =0.3,\gamma _1=0.9,\gamma _2=1.8, Da_f=0.001$. (b) $h=0.1,k=1,\eta =0.2,\sigma =0.3,\gamma _1=0.9,\gamma _2=1.8,Da_f=1$

Table 5

Critical Rayleigh and wave numbers for increasing Taylor numbers

$R^2_L$	$a^2_c$	$\mathcal {T}^2$
(a) $h=1,k=0.1$
206.8022	6.1510	0
206.8155	6.1515	0.1
206.8690	6.1535	0.5
206.9358	6.1560	1
207.4692	6.1760	5
208.1330	6.2007	10
(b) $h=0.1,k=1$
323.9093	7.4974	0
323.9359	7.4988	0.1
324.0419	7.5046	0.5
324.1741	7.5118	1
325.2230	7.5685	5
326.5116	7.6377	10

(a) $h=1,k=0.1,\eta =0.2,\sigma =0.3,\gamma _1=0.9,\gamma _2=1.8,Da_f=1$. (b) $h=0.1,k=1,\eta =0.2,\sigma =0.3,\gamma _1=0.9,\gamma _2=1.8,Da_f=1$

Tables 4 and 5 and Fig. 3 display a similar trend, as the Taylor number $\mathcal {T}^2$ 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 $\mathcal {T}^2$ increases, the wavenumber $a^2$ also increases and this means that the convection cells become narrower.

6 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.

Acknowledgements

This paper has been performed under the auspices of the GNFM of INdAM. R. De Luca and G. Massa 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 referees for suggestions which have led to improvements in the manuscript.

Declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Availability of data and material

This article does not contain any additional data.

Code availability

Not Applicable.

Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Vorheriger Artikel Generalized Emden–Fowler equations related to constant curvature surfaces and noncentral curl forces

Nächster Artikel Rayleigh-type surface wave in nonlocal isotropic diffusive materials

Barletta, A., Magyari, E., Pop, I., et al.: Mixed convection with viscous dissipation in a vertical channel filled with a porous medium. Acta Mech. 194, 123–140 (2007). https://doi.org/10.1007/s00707-007-0459-3CrossRefMATH

Barletta, A., Rees, D.A.S.: Unstable mixed convection flow in a horizontal porous channel with uniform wall heat flux. Transp. Porous Media 129, 385–402 (2019). https://doi.org/10.1007/s11242-019-01294-yMathSciNetCrossRef

Capone, F., De Luca, R.: Ultimately boundedness and stability of triply diffusive mixtures in rotating porous layers under the action of Brinkman law. Int. J. Non-Linear Mech. 47(7), 799–805 (2012)CrossRef

Capone, F., De Luca, R.: Porous MHD convection: effect of Vadasz inertia term. Transp. Porous Media 118(3), 519–536 (2017)MathSciNetCrossRef

Capone, F., De Luca, R.: The effect of the Vadasz number on the onset of thermal convection in rotating bidispersive porous media. Fluids 5(4), 173 (2020). https://doi.org/10.3390/fluids5040173CrossRef

Capone, F., De Luca, R., Gentile, M.: Coriolis effect on thermal convection in a rotating bidisperive porous layer. Proc. R. Soc. A. (2020). https://doi.org/10.1098/rspa.2019.0875CrossRef

Capone, F., De Luca, R., Gentile, M.: Thermal convection in rotating anisotropic bidispersive porous layers. Mech. Res. Commun. (2020). https://doi.org/10.1016/j.mechrescom.2020.103601CrossRef

Capone, F., Gentile, M.: Sharp stability results in LTNE rotating anisotropic porous layer. Int. J. Therm. Sci. 134, 661–664 (2018)CrossRef

Capone, F., Gentile, M., Hill, A.A.: Anisotropy and symmetry in porous media convection. Acta Mech. 208, 205 (2009). https://doi.org/10.1007/s00707-008-0135-2CrossRefMATH

10.

Capone, F., Gentile, M., Hill, A.A.: Double-diffusive penetrative convection simulated via internal heating in an anisotropic porous layer with throughflow. Int. J. Heat Mass Transf. 54(7–8), 1622–1626 (2011)CrossRef

11.

Capone, F., Rionero, S.: Porous MHD convection: stabilizing effect of magnetic field and bifurcation analysis. Ricerche mat. 65(1), 163–186 (2016)MathSciNetCrossRef

12.

Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. Dover Publications, New York (1981)MATH

13.

Chen, Z.Q., Cheng, P., Hsu, C.T.: A theoretical and experimental study on stagnant thermal conductivity of bidispersed porous media. Int. Commun. Heat Mass Transf. 27, 601–610 (2000)CrossRef

14.

De Luca, R., Rionero, S.: Steady and oscillatory convection in rotating fluid layers heated and salted from below. Int. J. Non-Linear Mech. 78(1), 121–130 (2016)CrossRef

15.

Gentile, M., Straughan, B.: Bidispersive thermal convection. Int. J. Heat Mass Transf. 114, 837–840 (2017)CrossRef

16.

Gentile, M., Straughan, B.: Bidispersive thermal convection with relatively large macropores. J. Fluid Mech. 898, A14 (2020). https://doi.org/10.1017/jfm.2020.411MathSciNetCrossRefMATH

17.

Hill, A.A.: A differential constraint approach to obtain global stability for radiation-induced double-diffusive convection in a porous medium. Math. Methods Appl. Sci. 32(8), 914–921 (2009)MathSciNetCrossRef

18.

Kvernvold, O., Tyvand, P.A.: Nonlinear thermal convection in anisotropic porous media. Mech. Appl. Math. 90(4), 609–624 (1977)MATH

19.

Moutsopoulos, K.N., Koch, D.L.: Hydrodynamic and boundary-layer dispersion in bidisperse porous media. J. Fluid Mech. 385, 359–379 (1999)MathSciNetCrossRef

20.

Nield, D.A., Bejan, A.: Convection in Porous Media, 5th edn. Springer, New York (2017)CrossRef

21.

Nield, D.A., Kuznetsov, A.V.: Heat transfer in bidisperse porous media. Transp. Phenom. Porous Med. III (2005). https://doi.org/10.1016/B978-008044490-1/50006-5CrossRef

22.

Nield, D.A., Kuznetsov, A.V.: The onset of convection in a bidisperse porous medium. Int. J. Heat Mass Transf. 49(17–18), 3068–3074 (2006)CrossRef

23.

Nield, D.A., Kuznetsov, A.V.: The effects of combined horizontal and vertical heterogeneity on the onset of convection in a porous medium with vertical throughflow. Transp. Porous Media 90, 465 (2011)MathSciNetCrossRef

24.

Rionero, S., Vergori, L.: Long-time behaviour of fluid motions in porous media according to the Brinkman model. Acta Mech. 210, 221–240 (2010). https://doi.org/10.1007/s00707-009-0205-0CrossRefMATH

25.

Straughan, B.: Global nonlinear stability in porous convection with a thermal non-equilibrium model. Proc. R. Soc. A. 462, 409–418 (2006)MathSciNetCrossRef

26.

Straughan, B.: Convection with Local Thermal Non-equilibrium and Microfluidic Effects. Adv Mechanics and Matematics, vol. 32. Springer, Cham (2015)MATH

27.

Straughan, B.: Anisotropic bidispersive convection. Proc. R. Soc. A. 475, 20190206 (2019)MathSciNetCrossRef

28.

Straughan, B.: Horizontally isotropic double porosity convection. Proc. R. Soc. A 475, 20180672 (2019)CrossRef

29.

Vadasz, P.: Flow and thermal convection in rotating porous media. Handbook of porous media, 395–440 (2000)

30.

Vadasz, P.: Fluid Flow and Heat Transfer in Rotating Porous Media, Springer Brief Thermal Engineering and Applied Sciences (eBook). Springer, Berlin (2016)CrossRef

31.

Vadasz, P.: Instability and convection in rotating porous media: a review. Fluids 4, 1–31 (2019)CrossRef

Titel: Effect of anisotropy on the onset of convection in rotating bi-disperse Brinkman porous media
verfasst von: Florinda Capone
Roberta De Luca
Giuliana Massa
Publikationsdatum: 23.06.2021
Verlag: Springer Vienna
Erschienen in: Acta Mechanica / Ausgabe 9/2021
Print ISSN: 0001-5970
Elektronische ISSN: 1619-6937
DOI: https://doi.org/10.1007/s00707-021-03002-8

Premium Partner

Marktübersichten

Die im Laufe eines Jahres in der „adhäsion“ veröffentlichten Marktübersichten helfen Anwendern verschiedenster Branchen, sich einen gezielten Überblick über Lieferantenangebote zu verschaffen.

Zur Marktübersicht

\(R^2_L\)	\(a^2_c\)	\(\mathcal {T}^2\)
(a) \(Da_f=0.001\)
52.2521	12.8790	0
52.7216	13.0444	0.1
54.5534	13.6882	0.5
56.7498	14.4563	1
71.7551	19.5421	5
86.7523	24.7523	10
211.5075	39.0022	100
(b) \(Da_f=1\)
323.9093	7.4974	0
323.9359	7.4988	0.1
324.0419	7.5046	0.5
324.1741	7.5118	1
325.2230	7.5685	5
326.5116	7.6377	10
346.4686	8.6407	100

\(R^2_L\)	\(a^2_c\)	\(\mathcal {T}^2\)
(a) \(h=1,k=0.1\)
206.8022	6.1510	0
206.8155	6.1515	0.1
206.8690	6.1535	0.5
206.9358	6.1560	1
207.4692	6.1760	5
208.1330	6.2007	10
(b) \(h=0.1,k=1\)
323.9093	7.4974	0
323.9359	7.4988	0.1
324.0419	7.5046	0.5
324.1741	7.5118	1
325.2230	7.5685	5
326.5116	7.6377	10

Springer Professional

Abstract

Publisher's Note

1 Introduction

2 Mathematical model

3 Onset of convection

4 Nonlinear stability

5 Numerical results

6 Conclusions

Acknowledgements

Declarations

Conflict of interest

Funding

Availability of data and material

Code availability

Publisher's Note

Weitere Artikel der Ausgabe 9/2021

Optimal control of the Cattaneo–Hristov heat diffusion model

Stochastic response of a piezoelectric ribbon-substrate structure under Gaussian white noise

A gradient continuous smoothed GFEM for heat transfer and thermoelasticity analyses

Geometrically nonlinear vibrations of FG-GPLRC cylindrical panels with cutout based on HSDT and mixed formulation: a novel variational approach

Melnikov-based criterion to obtain the critical velocity in axially moving viscoelastic strings under a set of non-Gaussian parametric bounded noise

Aeroelastic stability analysis of a flexible panel subjected to an oblique shock based on an analytical model

Premium Partner

Marktübersichten