Instability in Porous Layers with Depth-Dependent Viscosity and Permeability

Abstract

The paper is concerned with a multicomponent porous horizontal layer L with depth-dependent viscosity and permeability, heated from below and salted partly from below and partly from above. The instability of the thermal conduction solution, due to the growth of the gradient of temperature or of the gradient of chemical species salting L from above, is investigated.

1 Introduction

Many researchers, in the past as nowadays, have paid a great attention to the onset of thermal convection in porous media with depth-dependent viscosity and permeability (cf. Fig. 1 [1–14] and the references therein). This is because the viscosity and permeability depth-dependence is of relevant interest either in geophysical phenomena or in planning artificial porous materials. Denoting by f the ratio between viscosity and permeability, f can be chosen ad hoc (for promoting or inhibiting the onset of convection), in planning artificial porous materials. On the contrary, in the geophysical applications, a big caution is needed: the choice of f has to be motivated by experimental results (increase in viscosity with depth in the earth’s mantle [1], permeability changes due to mineral diagenesis in fractured crust [7], core boring analysis, …). In any case either in planning artificial porous materials or in geophysical applications, the depth-dependence of f has to satisfy the compatibility laws implied by the model chosen for investigating the problem at stake.

Fig. 1

A porous horizontal layer with depth-dependent porosity

We call (strong) compatibility law, the vertical component of the double curl of the Darcy equation in which f appears as coefficient of the seepage velocity. Further, we call weak compatibility law, any integral equation obtained by integrating the strong compatibility law multiplied by f ^β φ, with $\beta\in\mathbb{R}$ and φ suitable assigned weight function. In [12] and [14] the heat and mass transfer has been investigated (in the absence and in the presence of rotation respectively) via weak compatibility laws. In the present paper—in the absence of rotation—our aim is to investigate and put in evidence the:

• (i) strong and weak compatibility laws for f;

• (ii) sets of admissible f by the compatibility laws;

• (iii) influence of f on the thermal and “cold” convection.

Section 2 is devoted to some mathematical preliminaries concerned with the Darcy-Boussinesq scheme. In Sect. 3 the compatibility laws are obtained while in Sect. 4 the existence of admissible f is furnished. In Sect. 5 the influence of the depth-dependence of viscosity and permeability on the onset either of thermal or cold convection, is investigated. The paper ends with some final remarks (Sect. 6).

2 Preliminaries

Let Oxyz be an orthogonal frame of reference with fundamental unit vectors i,j,k (k pointing vertically upwards).

We assume that m different chemical species (“salts”) S _α (α=1,2,…,m), have dissolved in the fluid and have concentrations C _α (α=1,2,…,m), respectively, and that the equation of state is

$$\rho=\rho_0 \Biggl[1-\alpha_* (T-T_0)+\displaystyle \sum_{\alpha=1}^mA_\alpha (C_\alpha- \hat{C}_\alpha) \Biggr], $$

where $\rho_{0}, T_{0}, \hat{C}_{\alpha}$ (α=1,2,…,m), are reference values of the density, temperature and salt concentrations, while the constants α _∗,A _α denote the thermal and solute S _α expansion coefficients respectively. Combining Darcy’s Law with (thermal) energy and mass balance together with the Boussinesq approximation, we obtain the fundamental equations governing the isochoric motions

$$ \left \{ \begin{array}{l} \nabla P=-\displaystyle \frac{\mu}{K}\mathbf{v}-\mathbf{g}\rho_0 \Biggl[1-\alpha_* (T-T_0)+\displaystyle \sum_{\alpha=1}^mA_\alpha(C_\alpha-\hat{C}_\alpha)\Biggr], \\ \nabla\cdot\mathbf{v}=0,\\ T_t+\mathbf{v}\cdot\nabla T=k\varDelta T,\\ C_{\alpha t}+\mathbf{v}\cdot\nabla C_\alpha=k_\alpha \varDelta C_\alpha, \end{array} \right . $$

(1)

P = pressure field, $\mu=\bar{\mu}f_{1}(z)$ = dynamic viscosity, $K=\bar{K} f_{2}(z)$ = permeability, v = (seepage) velocity, $\bar{\mu}$ = reference dimensional viscosity, $\bar{K}$ = reference dimensional permeability, g = gravity, k = thermal diffusivity, k _α  = diffusivity of the solute S _α .

To (1) we append the boundary conditions

$$ \left \{ \begin{array}{l} T(0)=T_1,\qquad T(d)=T_2,\\ C_{\alpha}(0)=C_{\alpha_l},\qquad C_{\alpha}(d)=C_{\alpha_u}\quad \alpha=1,2,\ldots,m,\\ \mathbf{v}\cdot\mathbf{k}=0,\quad \mbox{at}\ z=0,d, \end{array} \right . $$

(2)

with T ₁, T ₂(<T ₁), $C_{\alpha_{l}}$, $C_{\alpha_{u}}$ (α=1,2,…,m), positive constants and d = layer depth. The boundary value problem (1)–(2) admits the conduction solution ($\tilde{\mathbf{v}}$, $\tilde{p}$, $\tilde{T}$, $\tilde {C}_{\alpha}$) given by

$$ \left \{ \begin{array}{l} \tilde{\mathbf{v}}=0,\qquad \tilde{T}=T_1-\beta z,\qquad \beta=\displaystyle \frac{T_1-T_2}{d},\\ \tilde{C}_\alpha= C_{\alpha_l}-\displaystyle \frac{z(\delta C_\alpha)}{d}, \qquad C_{\alpha_l}-C_{\alpha_u}=\delta C_\alpha,\\ \tilde{P}=p_0+\rho_0 g z^2 \Biggl[-\displaystyle \frac{\alpha_*\beta}{2}+\displaystyle \sum_{\alpha=1}^mA_\alpha \displaystyle \frac{(\delta C_\alpha)}{2d} \Biggr]\\ \phantom{\tilde{P}=} -\rho_0 g z \Biggl[1-\alpha_*(T_1-T_0)+\displaystyle \sum_{\alpha=1}^mA_\alpha (C_{\alpha l}-\hat{C}_\alpha) \Biggr], \end{array} \right . $$

(3)

where p ₀ is a constant. Setting

$$ \mathbf{v}=\tilde{\mathbf{v}}+\mathbf{u}, \qquad p=\tilde{P}+ \varPi, \qquad T=\tilde{T}+\theta, \qquad C_\alpha= \tilde{C}_\alpha+\varPhi_\alpha, $$

(4)

and introducing the scalings

$$ \left \{ \begin{array} {l} t=t^* \displaystyle \frac{d^2}{k},\quad \mathbf{u}=\mathbf{u}^*\displaystyle \frac{k}{d}, \quad \varPi=\varPi^*\displaystyle \frac{\mu k}{K},\quad \mathbf{x}= \mathbf{x}^*d,\quad \theta=\theta ^* T^{\sharp}, \\ \varPhi_\alpha=(\varPhi_\alpha)^*\varPhi^\sharp_\alpha, \quad T^\sharp= \biggl(\displaystyle \frac{\mu k |\delta T|}{\alpha_* \rho_0 gK d} \biggr)^{\frac{1}{2}},\quad \varPhi^\sharp_\alpha= \biggl( \displaystyle \frac{\mu k P_\alpha|\delta C_\alpha |}{A_\alpha\rho_0 g K d} \biggr)^{\frac{1}{2}}, \\ R= \biggl(\displaystyle \frac{\alpha_* \rho_0 g K d |\delta T|}{\mu_1 k} \biggr)^{\frac{1}{2}},\quad R_\alpha= \biggl(\displaystyle \frac{A_\alpha\rho_0 g K d P_\alpha|\delta C_\alpha |}{\mu_1 k} \biggr)^{\frac{1}{2}}, \\ \delta T=T_1-T_2,\quad H=\mbox{sgn}(\delta T),\quad H_\alpha=\mbox {sgn}(\delta C_\alpha),\quad P_\alpha=\displaystyle \frac{k}{k_\alpha}, \end{array} \right . $$

(5)

since in the case at stake the layer is heated from below, salted from below by S _α (with α=1,2,…,r) and from above by S _α with (α=r+1,…,m), it follows that H=H _α =1 (for α=1,…,r) and H _α =−1 (for α=r+1,…,m) and the equations governing the dimensionless perturbations {u ^∗,Π ^∗,θ ^∗,(Φ _α )^∗}, omitting the stars ans setting $f= \frac{f_{1}}{f_{2}}$, are

$$ \left \{ \begin{array} {l} \nabla\varPi=-f(z)\mathbf{u}+ \Biggl(R\theta-\displaystyle \sum_{\alpha=1}^mR_\alpha \varPhi_\alpha \Biggr)\mathbf{k}, \\ \nabla\cdot\mathbf{u}=0, \\ \theta_t+\mathbf{u} \cdot\nabla\theta=R \mathbf{u}\cdot\mathbf{k}+ \varDelta \theta, \\ P_\alpha \biggl(\displaystyle \frac{\partial\varPhi_{\alpha}}{\partial t}+\mathbf{u}\cdot\nabla \varPhi_\alpha \biggr)=R_\alpha\mathbf{u}\cdot\mathbf{k}+\varDelta \varPhi_\alpha,\quad \alpha=1,2,\ldots,r, \\ P_\alpha \biggl(\displaystyle \frac{\partial\varPhi_{\alpha}}{\partial t}+\mathbf{u}\cdot\nabla \varPhi_\alpha \biggr)=-R_\alpha\mathbf{u}\cdot\mathbf {k}+\varDelta \varPhi_\alpha,\quad \alpha=r+1,\ldots,m, \end{array} \right . $$

(6)

under the boundary conditions

$$ \mathbf{u}\cdot\mathbf{k}=\theta=\varPhi_\alpha=0\quad \mbox{on } z=0,1. $$

(7)

In (5)–(6) R is the thermal Rayleigh number and R _α , P _α are the salt S _α Rayleigh and Prandtl numbers respectively.

We assume, as usually done in stability problems in layers, that

• (i) the perturbations (u,v,w,θ,Φ ₁,…,Φ _m ) are periodic in the x and y directions, respectively of periods 2π/a _x , 2π/a _y ;

• (ii) Ω=[0,2π/a _x ]×[0,2π/a _y ]×[0,1] is the periodicity cell;

• (iii) u, Φ _α (α=1,…,m), θ belong to W ^2,2(Ω) and are such that all their first derivatives and second spatial derivatives can be expanded in a Fourier series uniformly convergent in Ω,

and denote by $L^{*}_{2}(\varOmega)$ the set of functions Φ such that

• (1) $\varPhi:(\mathbf{x}, t)\in\varOmega\times\mathbb{R}^{+}\rightarrow\varPhi (\mathbf{x}, t)\in\mathbb{R}$, Φ∈W ^2,2(Ω), $\forall t\in \mathbb{R}^{+}$, Φ is periodic in the x and y directions of period $\frac{2\pi }{a_{x}}$, $\frac{2\pi}{a_{y}}$ respectively and [Φ] _z=0=[Φ] _z=1=0;

• (2) Φ, together with all the first derivatives and second spatial derivatives, can be expanded in a Fourier series absolutely uniformly convergent in Ω, $\forall t\in\mathbb{R}^{+}$.

Remark 1

We require

$$\displaystyle \int_\varOmega u\,d\varOmega= \displaystyle \int _\varOmega v\,d\varOmega=0, $$

in order to eliminate the presence of laminar rigid motions.

3 Compatibility Laws

This section is devoted to obtain via the b.v.p.

$$ \left\{\begin{array}{l} \nabla\varPi=-f(z)\mathbf{u}+ \Biggl(R \theta-\displaystyle \sum_{\alpha=1}^mR_\alpha \varPhi_\alpha \Biggr)\mathbf{k}, \\ \nabla\cdot\mathbf{u}=0, \\ \mathbf{u}\cdot\mathbf{k}=\varPhi_\alpha=0,\quad\mbox{on } z=0,1, \ \alpha=1,2,\ldots,m, \end{array} \right. $$

(8)

the compatibility laws for f.

We begin by remarking that—since {sinnπz} (n=1,2,…), is a complete orthogonal system for L ²[(0,1)], by virtue of the periodicity, it turns out that $\forall\varPsi\in L^{*}_{2}(\varOmega)$, there exists a sequence $\{\tilde{\varPsi}_{n}(x,y,t)\}$ such that

$$ \left\{\begin{array}{l} \varPsi=\displaystyle \sum _{n=1}^\infty\tilde{\varPsi}_n(x,y,t)\sin n\pi z,\qquad \varPsi_n=\tilde{\varPsi}_n\sin n\pi z, \\ \varDelta _1\varPsi=-a^2\varPsi,\qquad \varDelta \varPsi=- \displaystyle \sum_{n=1}^\infty\xi _n \tilde{\varPsi}_n\sin n\pi z, \\ \xi_n=a^2+n^2\pi^2,\qquad a^2=a^2_x+a^2_y, \\ \varDelta =\varDelta _1+\displaystyle \frac{\partial^2}{\partial z^2},\qquad \varDelta _1=\displaystyle \frac{\partial^2 }{\partial x^2}+\displaystyle \frac{\partial^2}{\partial y^2}, \end{array} \right. $$

(9)

the series appearing in (9) being absolutely uniformly convergent in Ω.

Theorem 1

Let (u,θ,Φ ₁,…,Φ _m ) be solution of (8). Then:

• (i) the (strong) compatibility law is given by

  $$ f\varDelta w+f^\prime w_z=\varDelta _1 \Biggl(R\theta-\displaystyle \sum_{\alpha=1}^m R_\alpha\varPhi_\alpha \Biggr), $$

  (10)

• (ii) the first two components u,v of u are given by

  $$ \begin{array} {l} u=\displaystyle \sum _{n=1}^\infty u_n(x,y,t)\displaystyle \frac{d}{dz}(\sin n\pi z),\qquad u_n=\displaystyle \frac{1}{a^2}\displaystyle \frac{\partial w_n}{\partial x}, \\ \\ v=\displaystyle \sum_{n=1}^\infty v_n(x,y,t)\displaystyle \frac{d}{dz}(\sin n\pi z),\qquad v_n=\displaystyle \frac{1}{a^2}\displaystyle \frac{\partial w_n}{\partial y}. \end{array} $$

  (11)

Proof

The proof—essentially already given in [13]—is sketched here for the sake of completeness.

Setting

$$ \zeta=(\mbox{rot}\mathbf{u})\cdot\mathbf{k}=\displaystyle \frac{\partial v}{\partial x}- \displaystyle \frac{\partial u}{\partial y}, $$

(12)

from (8)₂ one obtains

$$ \varDelta _1 u=-\displaystyle \frac{\partial^2 w}{\partial x\partial z}- \displaystyle \frac {\partial\zeta}{\partial y},\qquad \varDelta _1 v=-\displaystyle \frac{\partial^2 w}{\partial y\partial z}+\displaystyle \frac {\partial\zeta}{\partial x}. $$

(13)

On the other hand, in view of

$$ \left\{\begin{array}{l} \nabla\times(f\mathbf{u})=\bigl[fw_y-(fv)_z\bigr]\mathbf{i}+ \bigl[(fu)_z-fw_x\bigr]\mathbf{j}+f\zeta\mathbf{k},\\ \nabla\times \Biggl(R\theta-\displaystyle \sum_{\alpha=1}^m R_\alpha\varPhi_\alpha \Biggr)\mathbf{k}= \Biggl(R\theta-\displaystyle \sum_{\alpha=1}^m R_\alpha\varPhi_\alpha \Biggr)_y\mathbf{i}- \Biggl(R\theta-\displaystyle \sum_{\alpha=1}^m R_\alpha\varPhi_\alpha \Biggr)_x\mathbf{j}, \end{array} \right.$$

(14)

it follows that the third component of the curl of (8)₁ gives

$$ f\zeta=0\quad\Leftrightarrow\quad\zeta=0. $$

(15)

Therefore it easily follows that

$$ \left\{\begin{array}{l} \mathbf{k}\cdot\nabla\times\bigl[\nabla\times(f\mathbf{u})\bigr]=-f^\prime w_z-f\varDelta w,\\ \mathbf{k}\cdot\nabla\times \Biggl[\nabla\times \Biggl(R\theta-\displaystyle \sum_{\alpha=1}^m R_\alpha\varPhi_\alpha \Biggr) \Biggr]\cdot\mathbf{k}= -\varDelta _1 \Biggl(R\theta-\displaystyle \sum_{\alpha=1}^m R_\alpha\varPhi_\alpha \Biggr) \end{array} \right.$$

(16)

and (10)₁ is given by the third component of the double curl of (8)₁. Finally, since ζ=0 implies

$$ \varDelta _1 u=-\displaystyle \frac{\partial^2 w}{\partial x\partial z},\qquad \varDelta _1 v=-\displaystyle \frac{\partial^2 w}{\partial y\partial z}, $$

(17)

(11) immediately follow. □

Theorem 2

Let f>0 be two times derivable a.e. in (0,1). Then either

$$ \displaystyle \int_0^1 \Biggl[ \displaystyle \frac{1}{1+\beta} \bigl(f^{1+\beta} \bigr)_{zz}w-f^{1+\beta } \varDelta w+f^\beta \varDelta _1 \Biggl(R\theta-\displaystyle \sum _{\alpha=1}^m R_\alpha \varPhi_\alpha \Biggr) \Biggr]\,dz=0, $$

(18)

with

$$ \beta\neq-1,\quad \beta\in\mathbb{R}, $$

(19)

or

$$ \displaystyle \int_0^1 \Biggl[(\log f)_{zz}w-\varDelta w+f^{-1} \varDelta _1 \Biggl(R\theta - \displaystyle \sum_{\alpha=1}^m R_\alpha \varPhi_\alpha \Biggr) \Biggr]\,dz=0, $$

(20)

are weak compatibility laws.

Proof

In view of (10), $\{\forall\beta\in\mathbb{R},\beta\neq-1\}$, one obtains

$$ f^{1+\beta} \varDelta w+\displaystyle \frac{1}{1+\beta} \bigl(f^{1+\beta} \bigr)_zw_z-f^\beta \varDelta _1 \Biggl(R\theta-\displaystyle \sum _{\alpha=1}^m R_\alpha\varPhi_\alpha \Biggr)=0. $$

(21)

Therefore integrating (21), (18) immediately follows. Analogously, since (10) implies

$$ \varDelta w+(\log f)_zw_z-f^{-1} \varDelta _1 \Biggl(R\theta-\displaystyle \sum _{\alpha =1}^m R_\alpha\varPhi_\alpha \Biggr)=0, $$

(22)

(20) immediately follows. □

Remark 2

By virtue of (11) it is enough to characterize the influence of f on w for characterizing the influence of u. The simplest way for evaluating the influence on w is to consider the equation governing only the vertical component of ∇×∇×(f u).

4 Sets of Admissible f

In the absence of depth-dependence (f=1), (10) implies the basic relation [15, 16]

$$ w_n=\eta_n \Biggl(R \theta_n- \sum_{\alpha=1}^{m}R_{\alpha} \phi_{\alpha } \Biggr),\quad \eta_n=\displaystyle \frac{a^2}{\xi_n}, $$

(23)

which plays a relevant role since:

• (i) reduces the unknown fields to θ,ϕ ₁,…,ϕ _m ;

• (ii) the procedures and the stability-instability results obtained in [15, 16] are based on its validity.

Our aim in the present section is to furnish general sets of f—admitted by the strong compatibility law (10) or by the weak compatibility law (18) in the case β=0—guaranteeing the existence of (23) with η _n suitably modified.

Theorem 3

The weak compatibility equation

$$ \displaystyle \int_0^1 \Biggl[f''w-f\varDelta w+\varDelta _1 \Biggl(R \theta-\sum_{\alpha =1}^mR_{\alpha} \phi_{\alpha} \Biggr) \Biggr]\,dz=0 $$

(24)

is verified by

$$ f=a_0+\sum_{r=1}^{\infty}a_r \sin r\pi z $$

(25)

with a ₀, a _r (r=1,…,∞) real constants such that

$$ \left\{\begin{array}{l} f>0 \quad\mathit{a.e.\ in}\ (0,1)\\ \displaystyle \sum_{r=1}^{\infty}a_r\tilde{w}_r=0 \end{array} \right.$$

(26)

and $\tilde{w}_{n}$ given by (23) with

$$ \eta_n=\displaystyle \frac{a^2}{a_0\xi_n}. $$

(27)

Proof

Setting

$$ f_r=a_r\sin r\pi z, $$

(28)

(24) reduces to

$$ \displaystyle \int_0^1 \Biggl[-a_0\varDelta w+\varDelta _1 \Biggl(R\theta-\sum _{\alpha =1}^{m}R_{\alpha}\phi_{\alpha} \Biggr) \Biggr]\,dz + \sum_{r=1}^{\infty}\displaystyle \int_0^1 \bigl(f''_rw-f_r \varDelta w \bigr)\,dz=0. $$

(29)

On the other hand, in view of (26)₂ and

$$ \displaystyle \int_0^1\sin(r \pi z)\sin(n \pi z)\,dz= \left\{ \begin{array}{l@{\quad }l} 0 & r\neq n\\ \frac{1}{2} & r=n \end{array} \right. $$

(30)

it follows that

$$\begin{aligned} &\displaystyle \sum_{r=1}^{\infty}\displaystyle \int_0^1 \bigl(f''_rw-f_r\varDelta w \bigr)\, dz\\ &\quad {}=\displaystyle \sum_{r=1}^{\infty}a_r\sum_{n=1}^{\infty}\tilde{w}_n \bigl(\xi _n-r^2\pi^2 \bigr)\displaystyle \int_0^1\sin(r\pi z)\sin(n\pi z)\,dz \\ &\quad {}=\displaystyle \frac{a^2}{2}\sum_{n=1}^{\infty}a_n\tilde{w}_n=0. \end{aligned}$$

Hence (29) reduces to

$$ \displaystyle \sum_{n=1}^{\infty} \displaystyle \int_0^1 \Biggl[a_0 \xi_n \tilde{w}_n-a^2 \Biggl(R\tilde{\theta}_n-\displaystyle \sum_{\alpha=1}^{m}R_{\alpha} \tilde{\phi}_{\alpha n} \Biggr) \Biggr]\sin(n\pi z)\,dz=0 $$

(31)

and is verified by

$$ \tilde{w}_n=\displaystyle \frac{a^2}{a_0\xi_n} \Biggl(R\tilde{\theta}_n-\displaystyle \sum _{\alpha=1}^{m}R_{\alpha} \tilde{\phi}_{\alpha n} \Biggr)\quad\forall n\in \{1,2,\ldots \}. $$

(32)

 □

Theorem 4

Let

$$0<h_1<h_2<\cdots <h_{s-1}<h_s=1, $$

$$ f(z)= \left \{ \begin{array}{l@{\quad }l} a_1,& z\in[0,h_1[,\\ a_2, & z\in[h_1,h_2[,\\ \ldots & \\ a_s, &z\in[h_{s-1},h_s=1[. \end{array} \right . $$

(33)

Then the strong compatibility equation is a.e. verified in [0,1] and

$$ w_n=\left \{ \begin{array}{l@{\quad }l} \displaystyle \frac{a^2}{a_1\xi_n} \Biggl(R\theta_n-\displaystyle \sum_{\alpha=1}^m R_\alpha \varPhi_{\alpha n} \Biggr), & z\in[0,h_1[,\\ \ldots & \\ \displaystyle \frac{a^2}{a_s\xi_n} \Biggl(R\theta_n-\displaystyle \sum_{\alpha=1}^m R_\alpha \varPhi_{\alpha n} \Biggr), &z\in[h_{s-1},h_s[. \end{array} \right . $$

(34)

Proof

In fact in any sublayer [h _p−1,h _p [, p∈{1,…,s}, (10) reduces to

$$ a_p\displaystyle \sum_{n=1}^\infty \xi_nw_n=a^2\displaystyle \sum _{n=1}^\infty \Biggl(R\theta _n- \displaystyle \sum_{\alpha=1}^m R_\alpha \varPhi_{\alpha n} \Biggr) $$

(35)

and hence (34) immediately follows. □

Remark 3

We remark that

• (i) since

  $$\displaystyle \int_0^1 \sin n\pi z\,dz= \displaystyle \frac{1-\cos(n\pi)}{n\pi}=0,\quad\mbox{for even}\ n, $$

  (32) could be requested only for odd n;

• (ii) (26)₁ requires

  $$\left\{ \begin{array}{l} a_0=f(0)=f(1)>0,\\ a_0+a_1-a_3+a_5-\cdots =f(1/2)>0, \end{array} \right.$$

• (iii) (26)₂ can be distinguished in the stability problems;

• (iv) (33) appear to be of relevant interest either in geophysical phenomena when core boring analysis suggests the porosity depth-dependence or in the construction of artificial porous materials for devices in which a suitable transfer of heat is needed.

5 Influence of Depth-Dependent Viscosity and Permeability on the Instability of Thermal Conduction

For the sake of concreteness, we confine ourselves to consider the prototype case of a horizontal layer L, heated from below and salted from below by S ₁ and from above by S ₂.

In the absence of depth-dependence (f=1), (23) hold and setting

$$ \left\{\begin{array}{l} \mathtt{I}_{1n}= \biggl\{ R^2- \biggl[\displaystyle \frac{R^2_1}{P_1}-\displaystyle \frac {R^2_2}{P_2}+\displaystyle \frac{\xi_n}{\eta_n} \biggl(1+\displaystyle \frac{1}{P_1}+\displaystyle \frac {1}{P_2} \biggr) \biggr] \biggr\} \eta_n,\\ \mathtt{I}_{2n}=\displaystyle \frac{P_1+P_2}{P_1P_2} \biggl[\displaystyle \frac {1+P_1+P_2}{P_1+P_2}\displaystyle \frac{\xi_n}{\eta_n}+\displaystyle \frac {1+P_2}{P_1+P_2}R^2_1-\displaystyle \frac{1+P_1}{P_1+P_2}R^2_2-R^2 \biggr]\xi_n\eta _n,\\ \mathtt{I}_{3n}=\displaystyle \frac{1}{P_1P_2} \biggl[R^2- \biggl(R^2_1-R^2_2+\displaystyle \frac{\xi_n}{\eta_n} \biggr) \biggr]\eta_n\xi^2_n, \end{array}\right. $$

(36)

the following results hold [15].

• (1) the conditions

  $$ \mathtt{I}_{1n}<0,\quad \mathtt{I}_{2n}>0,\quad \mathtt{I}_{3n}<0,\quad \forall \bigl(a^2,n \bigr)\in\mathbb{R}^+\times\mathbb{N}, $$

  (37)

  are necessary for guaranteeing the linear stability of the thermal conduction solution;

• (2) the conditions

  $$ \mathtt{I}_{1n}<0,\quad \mathtt{I}_{3n}<0,\quad \mathtt{I}_{1n} \mathtt{I}_{2n}-\mathtt{I}_{3n}<0, \quad\forall \bigl(a^2,n \bigr)\in\mathbb{R}^+\times \mathbb{N}, $$

  (38)

  are necessary and sufficient for guaranteeing the asymptotic linear stability of the thermal conduction solution;

• (3) when ( 38 ) hold, $\forall(a^{2},n)\in\mathbb{R}^{+}\times\mathbb{N}$, do not exist subcritical instabilities and the linear stability implies nonlinear asymptotic stability;

• (4) if

  $$ R^2_2\geq R^2_{2c}, $$

  (39)

  with R _2C (critical value of the Rayleigh number of the chemical specie salting L from above) given by

  $$\begin{aligned} &R^2_{2c}= \displaystyle \inf \biggl\{ \displaystyle \frac{P_2}{P_1}R^2_1+ \biggl(1+P_2+\displaystyle \frac {P_2}{P_1} \biggr)\displaystyle \frac{\xi_n}{\eta_n},\, \biggl(1+\displaystyle \frac{P_2}{1+P_1} \biggr) \displaystyle \frac{\xi_n}{\eta_n}+\displaystyle \frac {1+P_2}{1+P_1}R^2_1, R^2_1+\displaystyle \frac{\xi_n}{\eta_n} \biggr\} , \\ &\quad\forall \bigl(a^2,n\bigr)\in\mathbb{R}^+\times\mathbb{N}, \end{aligned}$$

  (40)

  then the “cold convection” (i.e. the thermal conduction instability irrespective of the temperature gradient [17]), arises.

  Since

  $$ \displaystyle \inf_{(a^2,n)\in\mathbb{R}^+\times\mathbb{N}} \displaystyle \frac{\xi_n}{\eta _n}= \displaystyle \inf_{(a^2,n)\in\mathbb{R}^+\times\mathbb{N}}\displaystyle \frac{\xi ^2_n}{a^2}= \biggl(\displaystyle \frac{\xi^2_n}{a^2} \biggr)_{(a^2=\pi^2,\,n=1)}=4\pi^2, $$

  (41)

  it follows that [15–17], in the absence of depth-dependence (f=1).

(1′):

the condition

$$ R^2<\displaystyle \min(R_{C1}, R_{C2}, R_{C3}), $$

(42)

with

$$ \left\{ \begin{array}{l} R_{C1}= \displaystyle \frac{R^2_1}{P_1}-\displaystyle \frac{R^2_2}{P_2}+4\pi^2 \biggl(1+\displaystyle \frac{1}{P_1}+\displaystyle \frac{1}{P_2} \biggr),\qquad R_{C3}=R^2_1-R^2_2+4\pi^2,\\ R_{C2}=\displaystyle \frac{1+P_2}{P_1+P_2}R^2_1-\displaystyle \frac {1+P_1}{P_1+P_2}R^2_2+ \biggl(1+\displaystyle \frac{1}{P_1+P_2} \biggr)4\pi^2, \end{array}\right. $$

(43)

is necessary for guaranteeing the linear stability of the thermal conduction solution;

(2′):

the conditions

$$ \left\{\begin{array}{l} R^2<\displaystyle \min(R_{C1}, R_{C2}),\\ \bigl(R_{C1}-R^2\bigr)\bigl(R_{C2}-R^2\bigr)> \displaystyle \frac{4\pi^2}{P_1+P_2} \bigl(R_{C3}-R^2\bigr), \end{array}\right. $$

(44)

are necessary and sufficient for guaranteeing the absence of subcritical instability and the nonlinear asymptotic stability of the thermal conduction solution;

(3′):

the condition

$$\begin{aligned} R^2_2\geq R^2_{2C} =& \displaystyle \min \biggl\{ \displaystyle \frac{P_2}{P_1}R^2_1+ \biggl(1+P_2+\displaystyle \frac{P_2}{P_1} \biggr)4\pi^2,\displaystyle \frac {1+P_2}{1+P_1}R^2_1 \\ &{}+ \biggl(1+\displaystyle \frac{P_2}{1+P_1} \biggr)4\pi^2, R^2_1+4\pi^2 \biggr\} , \end{aligned}$$

(45)

guarantees the “cold convection”.

By virtue of (27) and (34), the previous results can be applied either in the case (25) or in the case (33). For the sake of concreteness, we refer to (33), admitted by the strong compatibility equation (10). Precisely, by virtue of (41)–(45) and (33), one obtains

Theorem 5

Let (33) holds and let

$$ \bar{a} = \min(a_1,a_2,\ldots,a_s)=a_q $$

(46)

Then the instability of the thermal conduction solution begins in the layer

$$ z \in[h_{q-1}, h_q[ $$

(47)

and is

• (i) irrespective of temperature gradient when

  $$ R_2^2\geq\bar{R}_{2C}^2 $$

  (48)

  with $\bar{R}_{2C}^{2}$ given by the right hand side of (45) with $4 \bar{a} \pi^{2}$ at the place of 4π ²;

• (ii) temperature gradient depending (thermal convection) when (48) does not hold and at least one of (45) does not hold.

6 Final Remarks

We remark that

• (i) in the case (25), properties (1′)–(3′) continue to hold with 4a ₀ π ² at the place of 4π ²;

• (ii) when the condition guaranteeing the cold convection hold, then the thermal conduction (being unstable ∀(T ₁−T ₂>0) is not observable;

• (iii) by virtue of Theorem 5, in the general case of f>0, the plane

  $$ \displaystyle z=\bar{z}: \displaystyle \min_{z\in[0,1]} f(z)= f(\bar{z}) $$

  (49)

  appears to be the best candidate for the onset of the thermal convection;

• (iv) the previous point (iii) will be investigated in a forthcoming paper.