“Cold convection” in porous layers salted from above

A multicomponent fluid mixture saturating a porous rotating horizontal layer, heated from below and salted partly from below and partly from above, in the Darcy–Boussinesq scheme, is investigated. Conditions guaranteeing the “cold convection” i.e. the instability of the thermal conduction solution irrespective of the temperature gradient, are furnished.


Introduction
The heat and mass transfer by convection in horizontal layers, either because of its great geophysical relevance (engineering geology, subsurface fluid motions,…) or the frequent applications (industrial processes, crystal growth, thermal engineering,…), in the past as nowadays, has attracted the attention of many scientists (cfr.  and the references therein). The essential feature of the phenomenon is as follows: a horizontal layer L of fluid at rest state is heated from below in such a way that an adverse temperature gradient b is maintained. Because of thermal expansion the fluid at bottom expands as it becomes hotter. When b reaches a critical value b C , the buoyancy overcomes gravity, the fluid raises and a pattern of cellular motion can be seen. This is the onset of the thermal convection [6]. The phenomenon is called Bènard convection because of his experiments (1900). Although the exact parameter for determining the onset of the thermal convection, is not b C but the non-dimensional Rayleigh number depending on b and other parameters (cf (2.5) 9 ), the onset of instability of the thermal conduction solution is normally associated to the growing of b. On the other hand, when L is heated from below and salted from above and below then the thermal conduction solution is stabilized by the chemicals (''salts'') salting L from below and destabilized by the chemicals salting L from above. Therefore one expects that the growth of the gradient of a chemical species salting L from above (by alone or together with the growth of the gradients of other species salting L from above) can produce the instability of the thermal conduction solution.
The present paper is devoted to this type of instability of the thermal conduction solution, but under a further relevant request. Precisely, we look for conditions guaranteeing the instability of the thermal conduction solution irrespective of the temperature gradient i.e. V b [ 0. As far as we know this instability (analogous but different from the Marangoni effect)-named by us ''cold convection''-is neither mentioned nor investigated in the literature. 1 We consider an m -component (m = 1, 2,…) fluid mixture saturating a porous horizontal rotating layer L-heated from below and salted from below by the salts S 1 , S 2 ,…S r (0 B r \ m) and from above by the salts S r?1 ,…, S m ; in the Darcy-Boussinesq scheme. Denoting by R the thermal Rayleigh number, R a the salt S a Rayleigh number, T the Taylor-Darcy number (cfr. Sect. 2) and by the critical thermal Rayleigh number in the absence of salts in L (m = 0), our aim is to obtain-among other results-the following general one: or (in particular) 9" a 2 fr þ 1; . . .; mg : guarantees the onset of the ''cold convection''.
Section 2 is devoted to the introduction of the Darcy-Boussinesq equations governing the perturbations to the thermal conduction solution. In Sect. 3 the Routh-Hurwitz instability conditions are applied. In the subsequent Sect. 4 , conditions guaranteeing the onset of the ''cold convection'' irrespective of the temperature gradient, are furnished, Sect. 5 is devoted to the discussion. The paper ends with an Appendix in which some useful estimates are obtained.

Preliminaries
Let Oxyz be an orthogonal frame of reference with fundamental unit vectors i; j; kðk pointing vertically upwards).
We assume that m different chemical species (''salts'') S a (a = 1, 2,…, m), have dissolved in the fluid and have concentrations C a (a = 1, 2,…, m), respectively, and that the equation of state is where q 0 ; T 0 ;Ĉ a ða ¼ 1; 2; . . .; mÞ; are reference values of the density, temperature and salt concentrations, while the constants a * , A a denote the thermal and solute S a 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, when L rotates about the vertical axis with constant velocity x = xk p 1 is the pressure field, l is the dynamic viscosity, K is the porosity, v is the velocity, P ¼ p 1 À 1 2 q 0 ½x Â x 2 ; g is the gravity, k is the thermal diffusivity, K a is the diffusivity of the solute S a , To (2.1) we append the boundary conditions with T 1 , T 2 (\T 1 ), C a_l , C a_u (a = 1, 2,…, m), positive constants and d = layer depth. The boundary value problem (2.1), (2.2) admits the conduction solution ðṽ;p;T;C a Þ given bỹ and introducing the scalings ð2:5Þ since in the case at stake the layer is heated from below, salted from below by S a (with a = 1, 2,…, r) and from above by S a with (a = r ? 1,…, m), it follows that H = H a = 1, (for a = 1,…, r) and H a = -1 (for a = r ? 1,…, m) and the equations governing the dimensionless perturbations fu Ã ; P Ã ; h Ã ; ðU a Þ Ã g; omitting the stars, are under the boundary conditions In (2.5), (2.6) R and R a are the thermal and salt S a Rayleigh numbers respectively while P a and T are the salt S a Prandtl number and the Taylor-Darcy number.
We assume, as usually done in stability problems in layers, that the horizontal laminar flows are avoided and that (i) the perturbations ðu; v; w; h; U 1 ; . . .; U m Þ are periodic in the x and y directions, respectively of periods 2p/a x , 2p/a y ; (ii) X ¼ ½0; 2p=a x Â ½0; 2p=a y Â ½0; 1 is the periodicity cell; (iii) u; U a ; ða ¼ 1; . . .; mÞ; h belong to W 2;2 ðXÞ and are such that all their first derivatives and second spatial derivatives can be expanded in a Fourier series uniformly convergent in X; and denote by L Ã 2 ðXÞ the set of functions U such that (1) U : ðx; tÞ 2 X Â R þ ! Uðx; tÞ 2 R; U 2 W 2;2 ðXÞ; 8t 2 R þ ; U is periodic in the x and y directions of period 2p a x ; 2p a y respectively and ½U z¼0 ¼ ½U z¼1 ¼ 0; (2) U; together with all the first derivatives and second spatial derivatives, can be expanded in a Fourier series absolutely uniformly convergent in X; 8t 2 R þ : Let u = (u, v, w) and U 2 ðw; U 1 ; . . .; U m Þ: Since the sequence fsinðnpzÞg is a complete orthogonal system for L 2 (0,1) (under (2.7)), it follows that, 8U 2 L Ã 2 ðXÞ; there exists a sequence fŨ n ðx; y; tÞg n2N such that On the other hand setting it follows that Appendix (Proof of (2.10)) and in view of (2.8)-(2.10) and DU n ¼ Àn n U n ; U n 2 ðw n ; h n ; U an Þ; ð2:11Þ it follows that (2.6), (2.7) are equivalent to o ot a 0;0 ¼ R 2 g n À n n ; a 0;a ¼ ÀRR a g n ; a ¼ 1; 2; . . .; m; a 1;0 ¼ RR 1 P 1 g n ; a 1;1 ¼ À r;1 ¼ RR r P r g n ; a r;a ¼ À R r R a P r g n ; a r;r ¼ À R 2 r g n þn n P r ; a 6 ¼ r m;0 ¼ À RR m P m g n ; a m;2 ¼ R m R a P m g n ; a m;m ¼ R 2 m g n Àn n P m ; a ¼ 1; . . .; m ð2:15Þ

Routh-Hurwitz instability conditions
The equation governing the eigenvalues k a n of (2.14) and hence the stability-instability of the thermal conduction solution, can be written  are necessary and sufficient in order for all the roots of (3.1) to have negative real parts.
Theorem 31 implies the following ones. (i.e. one of the coefficients of (3.2) is negative for at least one n 2 N) it is sufficient for guaranteeing the instability of the thermal conduction solution.

Remark 3.1
The characteristic values I sn are given in terms of the entries of (2.14). Precisely I sn is obtained by adding the principal minors of orders s of 2.14.
By virtue of the previous remark, the following theorem holds.

Theorem 3.3
In terms of the entries of 2.14 it follows that (i) for m = 1, r = 0 (double diffusive-convection in a rotating porous horizontal layer heated from below and salted from above by S 1 ) I 2n ¼ 1 P 1 ÀR 2 À R 2 1 þ n n g n n n g n ; (ii) for m = 2, r = 1 (ternary diffusive-convection in rotating porous horizontal layers heated from below and salted from below by S 1 and from above by S 2 ): ð3:10Þ (iii) for m = 2, r = 0 (ternary diffusive-convection in rotating porous horizontal layers heated from below and salted from above by S 1 and S 2 ): ð3:11Þ (iv) in the general case (m 2 N; r\mÞI 1n and I ðmþ1Þn are given by 1 P a ! n n ; # Á g n n m n ; for m even; Á g n n m n ; for m odd: ð3:12Þ Proof In view of (2.15), (3.9)-(3.12) are easily obtained.

Salts critical Rayleigh numbers for the onset of ''cold convection''
Let S a be a chemical specie salting L from above. We will call critical Rayleigh number of S a for the onset of the ''cold convection'' the lowest value R a (c) of R a such that R a ! R ac ) the onset ofthe ''cold convection'': ð4:1Þ Remark 4.1 For the sake of simplicity we confine ourselves to determine R a C only by the applications of theorem 3.3.
Theorem 4.1 Let L be heated from below and salted from above by S 1 . Then the critical Rayleigh number of S 1 for the onset of the ''cold convection'' is given by hence it is unstable if and only if, at least for one couple ða 2 ; nÞ 2 R þ Â N; one of the following relations holds Since (cfr. Appendix, Proof of (4.5)) min ða 2 ;nÞ2R þ ÂN n n g n ¼ p 2 ð1 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ T 2 p Þ 2 ; ð4:5Þ (4.4) become respectively and (4.2) immediately follows.
Theorem 4.2 Let L be heated from below and salted from below by S 1 and from above by S 2 . Then the critical Rayleigh number R 2c of S 2 for the onset of the ''cold convection'' is given by Proof In the case (m = 2, r = 1)-in view of theorem 33 and (3.10)-it follows that the instability is guaranteed by one of the following conditions Then (4.7) immediately follows.

Remark 4.2 In view of
it follows that On the other hand ð4:14Þ Therefore ð4:16Þ As concerns (4.11)-(4.16), we confine ourselves to remark that P 2 C P 1 implies (4.11) while (4.12) requires P 2 \ P 1 . Theorem 4.3 Let L be heated from below and salted from above by two salts S a (a = 1,2). Then the Rayleigh critical numbers of S 1 and S 2 are given respectively either by ; ð4:17Þ or by the smallest coupled value of R 1 and R 2 verifying one of the equations

ð4:19Þ
Proof In view of theorem 33 and (3.11), the proof is easily reached following the procedure used in the proof of theorem 42.

Discussion
(i) The paper is concerned with an m-component fluid mixture saturating a porous rotating horizontal layer L, heated from below and salted partly from below and partly from above. (ii) The instability of the thermal conduction solutionirrespective of the temperature gradient (analogous but different from the Marangoni instability)named by us ''cold convection''-is studied via the Routh-Hurwitz instability conditions. (iii) It is shown that the ''cold convection'' is admissible by the Darcy-Boussinesq porous media and arises when the Rayleigh numbers of the chemicals salting L from above reach certain critical values. These values, in some prototype cases, are furnished.
(iv) The application of the Routh-Hurwitz instability conditions appear to be very appropriate for investigating the ''cold convection''. (v) The onset of the ''cold convection'' eliminates the thermal conduction observability. (vi) In the applications, the ''cold convection'' could be useful in the devices in which is needed that The results obtained, as far as we know, appear to be new in the existing literature.