International Journal of Engineering Science 48 (2010) 1519–1533
Contents lists available at ScienceDirect
International Journal of Engineering Science
journal homepage: www.elsevier.com/locate/ijengsci
Long-time behaviour of multi-component fluid mixtures in porous media
Salvatore Rionero
University of Naples Federico II, Department of Mathematics and Applications ‘‘R. Caccioppoli”, Complesso Universitario Monte S. Angelo,
Via Cinzia, 80126 Naples, Italy
a r t i c l e
i n f o
Article history:
Received 12 April 2010
Accepted 15 July 2010
Available online 23 August 2010
To K.R. Rajagopal with best wishes.
Keywords:
Multi-component fluid mixtures
Porous media
Absorbing sets
Convection
Stability
a b s t r a c t
The long-time behaviour of a triply convective–diffusive fluid mixture saturating a porous
horizontal layer in the Darcy–Oberbeck–Boussinesq scheme, is investigated. It is shown
that the L2- solutions are bounded, uniquely determined (by the initial and boundary data)
and asymptotically converging toward an absorbing set of the phase-space. The stability
analysis of the conduction solution is performed. The linear stability is reduced to the stability of ternary systems of O.D.Es and hence to algebraic inequalities. The existence of an
instability area between stability areas of the thermal Rayleigh number (‘‘instability
island”), is found analytically when the layer is heated and ‘‘salted” (at least by one ‘‘salt”)
from below. The validity of the ‘‘linearization principle” and the global nonlinear asymptotic stability of the conduction solution when all three effects are either destabilizing or
stabilizing, are obtained via a symmetrization.
Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction
The behaviour of convective–diffusive fluid mixtures in a porous layer can describe several phenomena. In particular
there are numerous applications in geophysical situations like salt movement underground, contaminant transport and
underground water flow, ice melting, etc. {cfr. [1–3] and the references therein}. Although the subject of double-diffusive
convection of a fluid-saturated porous layer is still a very active research area {cfr. for instance, [1–20,29,30] and the references therein}, the same subject with more than two different chemical species dissolved in – although more difficult – has
also attracted, in the past as nowadays, the attention of many authors {cfr. [21–25] and the references therein}. This is because the multi-component diffusive convection presents a picture of theoretical behaviours increasing together with the
number of components and new behaviours, like the existence of ‘‘instability islands” between stability areas of the thermal
Rayleigh number [23–25]. In [23] – via the energy method – the analysis of the linear and nonlinear stability of the conduction solution is performed and new behaviours are envisaged numerically. In particular, for certain values of the Rayleigh and
Prandtl numbers, the existence of an ‘‘instability island” is envisaged numerically. In the present paper we reconsider the
problem studied in [23] aimed to:
(1) study the long-time behaviour of the L2-solutions;
(2) reduce the (linear) stability of the conduction solution to the stability of the null solution of ternary linear systems of
O.D.Es [28];
(3) formulate the stability conditions through algebraic inequalities, for any values of the Rayleigh and Prandtl numbers;
(4) show analytically the existence of an ‘‘instability island ”, when the layer is heated and salted (at least by one salt) from
below.
E-mail address: rionero@unina.it
0020-7225/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijengsci.2010.07.007
1520
S. Rionero / International Journal of Engineering Science 48 (2010) 1519–1533
The plan of the paper is as follows: Section 2 is devoted to preliminaries. Successively, in Section 3, the boundedness and
uniqueness of the solutions is shown. Section 4 is devoted to the existence of absorbing sets. In the Section 5 the linear stability of the conduction solution is reduced to the stability of the null solution of systems of three O.D.Es. Then, applying the
Routh–Hurwitz conditions, the stability conditions are obtained – in closed forms – via algebraic inequalities in terms of Rayleigh and Prandtl numbers. The existence of an ‘‘instability island ” when the layer is heated and salted (by at least one salt)
from below, is obtained in Section 6. Successively (Section 7) it is shown that the most destabilizing case of a layer heated
from below and salted from above, is symmetrizable and hence the coincidence between the conditions of global nonlinear
stability and of linear stability is reached choosing the L2-norm of the perturbations as Liapunov function. The paper ends,
(Section 8), with some final remarks.
2. Preliminaries
Let Oxyz be a cartesian frame of reference with fundamental unit vectors i, j, k, with k pointed vertically upward.
We assume that the fluid has dissolved in two different chemical components (or ‘‘salts”) Sa (a = 1, 2), having the concentration Ca (a = 1, 2), respectively and we assume that the equation of state is given by
h
i
q ¼ q0 1 aðT T 0 Þ þ A1 C 1 Cb 1 þ A2 C 2 Cb 2 ;
b a ða ¼ 1; 2Þ, are reference values of density, temperature and salt concentration respectively and the conwhere q0 ; T 0 ; C
stants a, Aa denote respectively the thermal and solute Sa expansion coefficient respectively (a = 1, 2). Combining the Darcy’s
Law
rp ¼
l
K
v þ qg;
together with the equations of conservation of temperature and solute, the equations governing the isochoric motions can be
written as
8
h
i
>
rp ¼ Kl v gq0 1 aðT T 0 Þ þ A1 C 1 Cb 1 þ A2 C 2 Cb 2 ;
>
>
>
>
>
>
< r v ¼ 0;
T t þ v rT ¼ kDT;
>
>
>
>
>
C þ v r C 1 ¼ k 1 DC 1 ;
>
> 1t
:
C 2t þ v rC 2 ¼ k2 DC 2 ;
ð1Þ
where p = pressure field, l = dynamic viscosity, K = porosity, v = velocity, g = gravity, k = thermal diffusivity, Ka = diffusivity of
the solute Sa, To (1) we append the boundary conditions
8
>
< Tð0Þ ¼ T 1 ; TðdÞ ¼ T 2 ;
C a ð0Þ ¼ C al ; C a ðdÞ ¼ C au
>
:
v k ¼ 0; at z ¼ 0; d;
a ¼ 1; 2;
ð2Þ
with T 1 ; T 2 ; C al ; C au ða ¼ 1; 2Þ, positive constants. The boundary value problem (1) and (2) admits the conduction solution
e a Þ given by Tracey [23]
e; C
~; T
~; p
ðv
8
2
v~ ¼ 0; Te ¼ T 1 bz; b ¼ T 1 T
;
>
d
>
>
>
>
zðdC
Þ
e a ¼ C a a ; C a C a ¼ dC a ;
>
<C
u
l
l
d
h
i
ðdC 1 Þ
ðdC 2 Þ
ab
2
~
>
p
¼
p
þ
q
gz
þ
A
1 2d þ A2 2d þ
0
>
0
2
>
>
h
i
>
>
: q gz2 1 aðT T Þ þ A C C
b 1 þ A2 C 2l C
b2 ;
1
0
1
1l
0
ð3Þ
where p0 is a constant. Setting
v ¼ v~ þ u;
~ þ p;
p¼p
and introducing the scalings
T ¼ Te þ h;
e a þ Ua ;
Ca ¼ C
ð4Þ
8
2
>
t ¼ t dk ; u ¼ u dk ; p ¼ p lKk ; x ¼ x d; h ¼ h T ] ;
>
>
>
>
1
1
>
>
lkjdTj 2
>
a jdC a j 2
< Ua ¼ ðUa Þ ua ; T ] ¼ aq
; ua ¼ lAkP
;
a q0 gKd
0 gKd
1
1
>
2
>
a jdC a j 2
>
R ¼ aq0 gKdjdTj
; Ra ¼ Aa q0 gKdP
;
>
>
lk
lk
>
>
>
: dT ¼ T T ; H ¼ sgnðdTÞ; H ¼ sgnðdC Þ; P ¼
1
2
a
a
a
ð5Þ
k
ka
;
S. Rionero / International Journal of Engineering Science 48 (2010) 1519–1533
1521
the dimensionless equations governing the perturbation {u*, p*, h*, (Ua)*}, omitting the stars, are
8
rp ¼ u þ ðRh R1 U1 R2 U2 Þk;
>
>
>
>
>
>
< r u ¼ 0;
ht þ u rh ¼ HRu k þ Dh;
>
>
>
P1 ðU1t þ u rU1 Þ ¼ H1 R1 u k þ DU1 ;
>
>
>
:
P2 ðU2t þ u rU2 Þ ¼ H2 R2 u k þ DU2 ;
ð6Þ
ðu iÞz ¼ ðu jÞz ¼ u k ¼ h ¼ U1 ¼ U2 ¼ 0 on z ¼ 0; 1:
ð7Þ
under the boundary conditions (free boundary conditions)
In (5) and (6) R and Ra are the thermal and salt Rayleigh numbers while Pa are the salt Prandtl numbers.
3. Boundedness and uniqueness
To (6) and (7) we append the initial conditions
uðx; 0Þ ¼ u0 ðxÞ;
hðx; 0Þ ¼ h0 ðxÞ;
Ua ðx; 0Þ ¼ Ua0 ðxÞ;
a ¼ 1; 2:
ð8Þ
The initial-boundary value problem (6)–(8) will be studied under the assumptions that the perturbations (u, h, U1, U2) are
periodic with periods 2apx and 2apy in the x and y directions (ax > 0, ay > 0) and belong to L2(X)"t P 0,
X ¼ 0;
2p
2p
0;
½0; 1;
ax
ay
ð9Þ
being the periodicity cell.
Theorem 1. The solutions of the boundary value problem
@F
þ av rF ¼ bDF;
@t
in X;
r v ¼ 0;
v k ¼ 0;
ð10Þ
z ¼ 0; 1;
½F z¼0 ¼ F 1 ; ½Fz¼l ¼ F 2 ;
with a, b, Fi(i = 1, 2) positive constants, are bounded according to the point-wise estimate
sup jFj < F 1 þ supðF 0 F 1 Þþ þ supðF 0 F 2 Þ ;
XRþ
X
ð11Þ
X
with F0 = [F]t=0.
Proof. We give the proof only in the case F1 i F2, since a completely analogous proof holds in the case F1 6 F2. Setting [26]:
e
F ¼ F fðF F 1 Þþ ðF F 2 Þ g;
ð12Þ
it follows that
8
>
< F > F1 > F2 )
F2 < F < F1 )
>
:
F < F2 < F1 )
Therefore
e
F ¼ F ðF F 1 Þ ¼ F 1 ;
e
F ¼ F;
e
F ¼ F ðF F 2 Þ ¼ F 2 :
ð13Þ
e
F 6 F1;
ð14Þ
and (12) implies
jFj < F 1 þ ðF F 1 Þþ þ ðF F 2 Þ :
ð15Þ
2
Denoting by h , i and k k the scalar product and the norm in L (X) and setting X = X1(t) + X2(t) with
ðF F 1 Þþ
P 0 in X1
;
0 in X2
ðF F 1 Þ
0 in X1
P 0 in X2
it follows that
rðF F 1 Þþ DðF F 1 Þþ 0 in X2 ;
rðF F 1 Þ DðF F 1 Þ 0 in X1 :
1522
S. Rionero / International Journal of Engineering Science 48 (2010) 1519–1533
By virtue of (10) it follows that
@
ðF F 1 Þ þ av rðF F 1 Þ ¼ bDðF F 1 Þ:
@t
ð16Þ
On the other hand (via the Reynolds transport theorem)
Z
ðF F 1 Þþ
Z
ðF F 1 Þþ ðF F 1 Þ dX1 0:
X1
Z
@
ðF F 1 Þþ ðF F 1 Þ dX1
@t
X1
Z
d
¼
ðF F 1 Þþ ðF F 1 Þ dX1 0;
dt X1
@
ðF F 1 Þ dX1 ¼
@t
Z
X1
ðF F 1 Þ
@
ðF F 1 Þþ dX1
@t
since
X1
Hence in view of the previous relations
(
ðF F 1 Þþ z¼0 ¼ 0;
< v ; rðF F 1 Þpþ >¼ 0;
R
p2
2
< ðF F 1 Þp1
þ ; DðF F 1 Þþ >¼ ðp 1Þ X ðF F 1 Þþ jrðF F 1 Þþ j dX;
ð17Þ
on multiplying (16) by (F F1)p1 for p > 1 and integrating over X one easily obtains
Z
X
1p Z
1p
ðF F 1 Þpþ dX 6
ðF 0 F 1 Þpþ dX :
ð18Þ
X
Letting p ? 1, for the dominate convergence theorem, it turns out that
supðF F 1 Þþ 6 supðF 0 F 1 Þþ ;
X
X
8t P 0:
ð19Þ
8t P 0;
ð20Þ
Since, analogously, one easily obtains
supðF F 2 Þ 6 supðF 0 F 2 Þ ;
X
X
Eq. (11) immediately follows.
h
Remark 1. In view of (6)1 one obtains
kuk2 6 ðRkhk þ R1 kU1 k þ R2 kU2 kÞku kk:
ð21Þ
But in view of Theorem 1 exists a constant m > 0 such that
kuk2 6 mku kk 6 mkuk;
ð22Þ
kuk 6 m:
ð23Þ
hence
Theorem 2. System(6)–(8) can admit only one solution for any assigned initial data.
e 1; U
e 2; u
e Þ be two solutions of (6)–(8). Setting
~; P
h; U
Proof. Let (h, U1, U2, u, P) and ð~
W ¼ ~h h;
it follows that
e a Ua ;
Wa ¼ U
~ u;
U¼u
e P;
P ¼ P
8
U ¼ rP þ ðRW R1 W1 R2 W2 Þk;
>
>
>
>
>
>
>
> r U ¼ 0;
>
>
<
~ rW þ U rh ¼ HU k þ DW;
Wt þ u
>
>
>
>
>
~ rW1 þ U rU1 Þ ¼ H1 R1 U k þ DW1 ;
>
P1 ðW1t þ u
>
>
>
>
:
~ rW2 þ U rU2 Þ ¼ H2 R2 U k þ DW2 ;
P2 ðW2t þ u
a ¼ 1; 2;
ð24Þ
ð25Þ
S. Rionero / International Journal of Engineering Science 48 (2010) 1519–1533
1523
with
W ¼ W1 ¼ W2 ¼ 0 for z ¼ 0; 1;
ðU iÞz ¼ ðU jÞz ¼ ðU kÞ ¼ 0;
z ¼ 0; 1:
ð26Þ
Then one easily obtains
81
>
>
2
>
>
<1
2
Wk2 6 jHj kU kk kWkþ < U rW; h > krWk2 ;
d
k
dt
P1 dtd kW1 k2 6 jH1 j kU kk kW1 kþ < U rW1 ; U1 > krW1 k2 ;
>
1
>
P2 d kW2 k2 6 jH2 j kU kk kW2 kþ < U rW2 ; U2 > krW2 k2 ;
>
>
: 2 dt
kUk 6 RkWk þ R1 kW1 k þ R2 kW2 k;
ð27Þ
In view of
8
kU kk kWk 6 kUk kWk 6 RkWk2 þ ðR1 kW1 k þ R2 kW2 kÞkWk
>
>
>
>
>
>
< 6 R þ 12 ðR1 þ R2 Þ kWk2 þ 12 R1 kW1 k2 þ R2 kW2 k2 ;
>
kU kk kW1 k 6 kUkkW1 k 6 R1 þ 12 ðR þ R2 Þ kW1 k2 þ 12 ðRkWk2 þ R2 kW2 k2 Þ;
>
>
>
>
>
: kU kk kW k 6 R þ 1 ðR þ R ÞkW k2 þ 1 RkWk2 þ R kW k2 ;
2
2
1
2
1
1
2
2
ð28Þ
setting
(
E ¼ 12 kWk2 þ P1 kW1 k2 þ P2 kW2 k2 ;
m ¼ maxðjH1 j; jH2 j; jHjÞ;
ð29Þ
it follows that
2 dE
6 ð4R þ R1 þ R2 ÞkWk2 þ ðR þ 4R1 þ R2 ÞkW1 k2 þ ðR þ R1 þ 4R2 ÞkW2 k2 þ j < U rW; h > j þ j
m dt
2
< U rW1 ; U1 > j þ j < U rW2 ; U2 > j þ krWk2 þ krW1 k2 þ krW2 k2 :
m
ð30Þ
But in view of Theorem 1 exists a positive constant m1 such that
sup ðjhj; jU1 j; jU2 jÞ 6 m1 ;
ð31Þ
XRþ
hence one obtains
2
8
2
kUk
1
>
>
> j < U rW; h > j 6 m1 < jUj; jrWj >6 2 m1 e þ ekrWk ;
>
<
2
2
;
þ
e
j < U rW1 ; U1 > j 6 12 m1 kUk
1 krW1 k
e
1
>
>
2
>
>
2
kUk
1
:
j < U rW2 ; U2 > j 6 2 m1 e2 þ e2 krW2 k :
ð32Þ
Choosing
e ¼ e1 ¼ e2 ¼
1
;
m
ð33Þ
in view of (23), it turns out that
dE
6 qE () E 6 E0 eqt ;
dt
ð34Þ
with q = positive constant. Hence
E0 ¼ 0 ) EðtÞ ¼ 0;
8t 2 Rþ :
ð35Þ
4. Absorbing sets
Lemma 1. Let (u, h, U1, U2) 2 L2(X)6 be a solution of (6) and (7). Then (a = 1, 2)
h ¼ ~h þ h;
~a þ U
a;
Ua ¼ U
in X Rþ ;
ð36Þ
with
j~hj 6 1;
~ a j 6 1;
jU
ð37Þ
1524
S. Rionero / International Journal of Engineering Science 48 (2010) 1519–1533
a , decreasing functions of t such that
and
h; U
(
khð; tÞk 6 fkðh 1Þþ k þ kðh þ 1Þ kgt¼0 egt ;
a ð; tÞk 6 fkðU
a 1Þ k þ kðW
a þ 1Þ kg egt ;
kU
t¼0
þ
with
g ¼ p2 inf 1;
1 1
;
:
P1 P2
Proof. See [25] pp. 136–137
ð38Þ
ð39Þ
h
Lemma 2. Let (u, h, U1, U2) 2 L2(X)6 be a solution of (6) and (7). Then
d 2
khk þ P1 kU1 k2 þ P2 kU2 k2 6 2p2 khk2 þ P1 kU1 k2 þ P2 kU2 k2 þ 3RjH þ 1jkhk2 þ j3R1 jH1 1j P1 j kU1 k2
dt
þ j3R2 jH2 1j P2 jkU2 k2 :
ð40Þ
Proof. In view of (6) and (7), one obtains
with
81 d
>
< 2 dt e ¼< Dh; h > þ < DU1 ; U1 > þ < DU2 ; U2 >
þ < HRh þ H1 R1 U1 þ H2 R2 U2 ; x > kuk2
>
:
þ < Rh R1 U1 R2 U2 ; x >¼ 0;
1
2
e ¼ ðkhk2 þ P1 kU1 k2 þ P2 kU2 k2 Þ;
ð41Þ
ð42Þ
and hence
1 de
¼ krhk2 þ krU1 k þ krU2 k2 kuk2 þ < RðH þ 1Þh þ R1 ðH1 1ÞU1 þ R2 ðH2 1ÞU2 ; x > :
2 dt
ð43Þ
1 de
6 p2 ðkhk2 þ kU1 k2 þ kU2 k2 Þ kuk2 þ < RðH þ 1Þh þ R1 ðH1 1ÞU1 þ R2 ðH2 1ÞU2 ; x > :
2 dt
ð44Þ
8
>
< RðH þ 1Þh; x >6 32 RjH þ 1j khk2 þ 13 kuk2 ;
>
<
< R1 ðH1 1ÞU1 ; x >6 32 R1 jH1 1j kU1 k2 þ 13 kuk2 ;
>
>
:
< R2 ðH2 1ÞU2 ; x >6 32 R2 jH2 1j kU2 k2 þ 13 kuk2 ;
ð45Þ
By virtue of the Poincarè inequality one obtains
But
hence (40) immediately follows.
h
Lemma 3. Let (u, h, U1, U2) 2 L2(X)6 be a solution of (6)–(8). Then
kuk 6 Rkhk þ R1 kU1 k þ R2 kU2 k:
ð46Þ
Proof. Since
< h; x >6 khk kxk 6 khk kuk;
< Ua ; x >6 kUa k kxk;
a ¼ 1; 2;
by virtue of (6)1 and (46) immediately follows.
h
Theorem 3. For any g > 0, the set
n
S ¼ ðu; h; U1 ; U2 Þ 2 ½L2 ðXÞ6 : kðh 1Þþ k þ kðh þ 1Þ k < g; kðU1 1Þþ k þ kðU1 þ 1Þ k < g; kðU2 1Þþ k
o
1
þ ðU2 þ 1Þ k < g; kuk 6 ðR þ R1 þ R2 ÞðjXj2 þ gÞ ;
where jXj denotes the measure of X, is an absorbing set of the solutions (u, h, U1, U2) 2 L2(X)6 of (6) and (7).
S. Rionero / International Journal of Engineering Science 48 (2010) 1519–1533
1525
Proof. Let B a bounded set of the phase-space and let (a = 1, 2)
8
< sup½kðh 1Þþ k þ kðh þ 1Þ k < M;
B
: sup½kðUa 1Þþ k þ kðUa þ 1Þ k < M:
B
From
M expðgtÞ < g;
with g given by (39), it follows that for
1
t > t1 ¼
g
ln
M
g
;
all the solutions starting from B satisfy the inequality
sup kðh 1Þþ k þ kðh þ 1Þ k; kðU1 1Þþ k þ kðU1 þ 1Þ k; kðU2 1Þþ k þ kðU2 þ 1Þ k < g:
ð47Þ
Further from (36)–(38) and (47) we deduce that
(
1
khk 6 k~hk þ khk 6 jXj2 þ g;
e a k þ kUa k 6 jXj12 þ g;
kUa k 6 k U
a ¼ 1; 2 8t > t1 ;
ð48Þ
and hence, by means of (46)
1
kuk 6 ðR þ R1 þ R2 Þ jXj2 þ g :
ð49Þ
Therefore, all the solutions starting from B belong to S* at time t > t1. Further, since k
hk and kUa kða ¼ 1; 2Þ, are decreasing
functions of t, S* is invariant. h
5. Linear instability of the conduction solution via a new approach
Our aim here is to reduce the linear stability of the conduction solution to the stability of the null solution of ternary systems of O.D.Es and hence to the application of the Routh–Hurwitz stability conditions.
The set of functions fsinðnpzÞgn2N is a complete orthogonal set for L2[0, 1]. Then, for any function L 2 fx; h; U1 ; U2 g it
follows that
L¼
1
X
n¼1
e n ðx; y; tÞ sinðnpzÞ;
L
ð50Þ
with (by virtue of the periodicity in the x and y directions)
fn ¼ a2 L
e n;
D1 L
D1 ¼
@2
@2
þ 2;
2
@x
@y
a2 ¼ a2x þ a2y :
ð51Þ
Setting
f ¼ ðr uÞ k;
ð52Þ
by virtue of r u = 0, u = (u, v, x), one obtains
D1 u ¼
@ 2 x @f
;
@x@z @z
D1 v ¼
@ 2 x @f
þ :
@y@z @x
ð53Þ
On the other hand, (6)1 implies f 0, hence
D1 u ¼
@2x
;
@x@z
D1 v ¼
@2x
:
@y@z
ð54Þ
Therefore by virtue of the periodicity - one obtains
8
1
P
>
>
~ n ðx; y; tÞ dzd sinðnpzÞ;
u
>u ¼
>
>
n¼1
>
>
>
1
<
P
v ¼ v~ n ðx; y; tÞ dzd sinðnpzÞ;
n¼1
>
>
>
>
~ n ¼ a2 u
~ n ; D1 v~ n ¼ a2 v~ n ;
>
D
1u
>
>
>
~n
~n
:u
@
x
x
1
~ n ¼ a2 @x ; v~ n ¼ a12 @@y
:
ð55Þ
1526
S. Rionero / International Journal of Engineering Science 48 (2010) 1519–1533
Then x, h, U1, U2 are the effective perturbation fields. These fields are not independent. In fact in view of (a = 1, 2)
8
~ n ðx; y; tÞ sinðnpzÞ;
xn ¼ x
Dxn ¼ ða2 þ n2 p2 Þxn ;
>
>
>
<
Dhn ¼ ða2 þ n2 p2 Þhn ;
hn ¼ ~hn ðx; y; tÞ sinðnpzÞ;
>
>
>
:
e an ðx; y; tÞ sinðnpzÞ; DUan ¼ ða2 þ n2 p2 ÞUan ;
Uan ¼ U
ð56Þ
and (6) it turns out that
8
< xn ¼ gn ðRhn R1 U1n R2 U2n Þ;
:u ¼ 1
n
a2
@ 2 xn
@x@z
i þ a12
@ 2 xn
@y@z
j þ xn k;
nn ¼ a2 þ n2 p2 ;
gn ¼ ann2 ;
ð57Þ
verify (6)1 and (6)2, and
xn ¼ hn ¼ Wan ¼ 0 on z ¼ 0; 1;
ð58Þ
Therefore, the independent fields- in view of (57)1 - are reduced to three only, precisely to h, U1, U2.
Setting
8
>
a1n ¼ HR2 gn nn ; a2n ¼ HRR1 gn ; a3n ¼ HRR2 gn ;
>
>
>
>
<
H R2 g n
1 gn
; b2n ¼ 1 P11 n n ; b3n ¼ H1 RP11R2 gn ;
b1n ¼ H1 RR
P1
>
>
>
>
>
: c ¼ H2 RR2 gn ; c ¼ H2 R1 R2 gn ; c ¼ H2 R22 gn nn ;
1n
2n
3n
P2
P2
P2
ð59Þ
8
1
P
>
>
ht ¼
ða1n hn þ a2n U1n þ a3n U2n Þ u rh;
>
>
>
n¼1
>
>
>
>
<
1
P
U1t ¼
ðb1n hn þ b2n U1n þ b3n U2n Þ u rU1 ;
>
n¼1
>
>
>
>
>
1
>
P
>
>
ðc1n hn þ c2n U1n þ c3n U2n Þ u rU2 :
: U2t ¼
ð60Þ
by virtue of (6) one obtains
n¼1
Disregarding the nonlinear terms and setting
(
hn ðtÞ ¼ nn ðtÞFðx; yÞ sinðnpzÞ;
Uin ¼ nin ðtÞFðx; yÞ sinðnpzÞ;
ði ¼ 1; 2Þ;
it follows that the conduction solution (3) is linearly stable iff 8n 2 N, the null solution of
8 dnn
¼ a1n nn þ a2n n1n þ a3n n2n ;
>
dt
>
>
<
dn1n
¼ b1n nn þ b2n n1n þ b3n n2n ;
dt
>
>
>
: dn2n
¼ c1n nn þ c2n n1n þ c3n n2n ;
dt
ð61Þ
is stable. Setting
8
I 1n ¼ a1n þ b2n þ c3n ;
>
>
>
>
>
>
>
a1n
a1n a2n
>
>
>
I2n ¼
þ
>
>
>
<
b
b
c
1n
2n
1n
>
>
a1n
>
>
>
>
>
>
>
I3n ¼ b1n
>
>
>
>
:
c1n
a2n
a3n
b2n
b3n ;
c2n
c3n
a3n
c3n
þ
b2n
b3n
c2n
c3n
;
ð62Þ
the Routh–Hurwitz conditions for all the eigenvalues of (61) have negative real part can be written {cfr. [27], pp. 112–114}
I1n < 0;
I2n > 0;
I3n < 0;
I1n I2n I3n < 0:
ð63Þ
We remark that if and only if the algebraic inequalities (63) hold, 8ða2 ; nÞ 2 Rþ N, the conduction solution is linearly stable.
S. Rionero / International Journal of Engineering Science 48 (2010) 1519–1533
1527
Setting
8
2
>
>
e2 ¼ 1 þ A
e 1; A
e3 ¼ 1 þ A
e1;
e1 ¼ P 1 ; A
>A
>
Pi
P 1 P2
>
>
i¼1
>
>
>
2
>
P
>
Hi R2i
2
e 2 n2n2 ;
>
>
A
< H1n ¼ HR
Pi
a
i¼1
2
>
P
>
2
1
1
e 3 n2n2 1 HR2 ;
> H2n ¼
þ
R
þ
A
H
>
i
i
Pi
P 1 P2
a
>
eA 1
>
i¼1
>
>
>
>
2
2
P
>
>
> H3n ¼ HR2 Hi R2i ann2 ;
:
ð64Þ
i¼1
it easily follows that
I1n ¼ gn H1n ;
e 1 H2n ;
I2n ¼ gn nn A
I3n ¼ gn n2n H3n ;
ð65Þ
and hence (63) become
H1n < 0;
H2n > 0;
H3n < 0;
2
e 1 H1n H2n < nn H3n ;
A
a2
8ða2 ; nÞ 2 Rþ N:
ð66Þ
Remark 2. Eq. (66) shows the inequalities governing, in the most general case, the linear stability of the conduction solution.
For the sake of simplicity and concreteness, we confine ourselves to the case of a layer heated from below and, precisely,
assume H = 1.
Theorem 4. Let H = 1. Then the conduction solution can be linearly stable only if
R2 < Rc
Rc > 0;
ð67Þ
with
8
Rc ¼ infðRc1 ; Rc2 ; Rc3 Þ;
>
>
>
>
2
P
>
Hi R2i
>
e 2;
>
Rc 1 ¼
þ 4p2 A
>
Pi
>
>
i¼1
<
2
P 1
2
1
2e
1
>
A
¼
þ
R
þ
4
p
;
R
H
c
3
i
> 2
i
Pi
P 1 P2
eA 1
>
>
i¼1
>
>
>
2
>
P
>
2
>
: Rc3 ¼ Hi Ri þ 4p2 :
ð68Þ
i¼1
Proof. In view of
inf
ða2 ;nÞ2Rþ N
n2n
¼ 4p2 ;
a2
ð69Þ
and (64)–(66) and (68) immediately follow. By virtue of
with
8
e2 A
e 2 Un ¼ Q A
e 2 Un ;
>
H1n ¼ R2 Rc1 þ 4p2 A
>
1
>
<
e e
e
H2n ¼ Rc2 R2 4p2 A 3 þ A 3 U n ¼ Q 2 þ A 3 U n ;
eA 1 eA 1
eA 1
>
>
>
:
H3n ¼ R2 Rc3 þ 4p2 U n ¼ Q 3 U n ;
8
2
>
< U n ðaÞ ¼ ann2 ;
e 2;
Q 1 ¼ R2 Rc1 þ 4p2 A
e
2
2
A
>
: Q 2 ¼ Rc2 R 4p2 e3 ; Q 3 ¼ R Rc3 þ 4p2 ;
A1
ð70Þ
ð71Þ
Eq. (66)4 becomes
e
e 2 Un Q þ A 3 Un
e1 Q A
A
1
2
e1
A
!
U n ðQ 3 U n Þ < 0;
ð72Þ
and hence
a U 2n þ 2bU n þ c > 0;
U n P 4p 2
ð73Þ
1528
S. Rionero / International Journal of Engineering Science 48 (2010) 1519–1533
with
e 3;
e2 A
a ¼ 1 þ A
e2Q þ Q ;
e3Q þ A
e1 A
2b ¼ A
1
2
3
e 1Q Q :
c ¼ A
1 2
ð74Þ
Setting
r1;2 ¼
b
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
b a c
;
a
2
for b > a c;
ð75Þ
then immediately- in view of a* > 0 and (69) – the following theorem holds:
Theorem 5. Let H = 1 and (67) hold. Then the conduction solution is stable iff either
2
b a c < 0;
ð76Þ
or
inf
ðR1 ;R2 Þ2ðRþ Þ2
ðb; cÞ P 0:
ð77Þ
Either when (76) or (77) does not hold, the instability begin at Un = r2.
An alternative way of satisfying (66)4 is to require that
e1
A
inf
H2n
ða2 ;nÞ2Rþ N
n2n =a2
>
sup
ða2 ;nÞ2Rþ N
jH3n j
;
jH1n j
which allows to obtain the stability conditions directly in terms of R2 ; R21 ; R22 . In view of
8
e 2 n2n2 4p2 ;
>
jH1n j ¼ Rc1 R2 þ A
>
a
>
>
>
2
<
jH3n j ¼ Rc3 R2 þ ann2 4p2 ;
>
>
>
e 2
>
>
: H2n ¼ Rc2 R2 þ A 3 ann2 4p2 ;
eA 1
ð78Þ
ð79Þ
setting
Zða; nÞ ¼
n2n
4p2 ;
a2
ð80Þ
one obtains
inf
ða2 ;nÞ2Rþ N
sup
ða2 ;nÞ2Rþ N
8
Rc R2
e
>
ðRc2 R2 Þ þ A 3 Z >
< 42p2
eA 1
H2n
¼
¼ inf
>
e
4p2 þ Z
n2n =a2 Z2Rþ
>
: A3
eA 1
8
1
>
< eA
ðRc3 R2 Þ þ Z
H3n
2
¼
¼ sup
e2Z :
> Rc3 R2
jH1n j Z2Rþ ðRc R2 Þ þ A
1
2
Rc1 R
if
Rc2 R2
if
Rc2 R2
4p2
4p2
eA 3
;
eA 1
e
> A3 ;
eA 1
6
e 2 ðRc R2 Þ;
if ðRc1 R2 Þ > A
3
e 2 ðRc R2 Þ:
if ðRc1 R2 Þ 6 A
3
ð81Þ
ð82Þ
As consequence of (78)–(82) the following theorems hold.
Theorem 6. Let H = 1 and (67) hold either together with
Rc2 R2 6 4p2
or together with
Rc2 R2 6 4p2
e3
A
;
e1
A
e 2 ðRc R2 Þ;
Rc1 R2 6 A
3
ð83Þ
e3
A
;
e1
A
e 2 ðRc R2 Þ:
Rc1 R2 > A
3
ð84Þ
Then in the case (83), setting
p¼
i
1h 2 e
4p A 1 ðRc1 þ Rc2 Þ ;
2
e 1 Rc Rc 4p2 Rc ;
q¼A
1
2
3
r1;2 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
e 1q ;
p p2 A
2
ð85Þ
S. Rionero / International Journal of Engineering Science 48 (2010) 1519–1533
1529
it turns out that
8
2
e
>
>
< either p A 1 q 6 0 or infðp; qÞ >(0 ) stability;
n
o
stability
for R2 R ½r1 ; r 2 ;
2
e
>
>
: p A 1 q > 0; infðp; qÞ < 0 )
instability for R2 2r1 ; r 2 ½
ð86Þ
while in the case (84) the stability is guaranteed only if
R2 þ
4p2
< Rc 2 :
e1 A
e2
A
ð87Þ
Proof. In the case (83), in view of (78)–(82), the stability is implied by
e 1 ðRc R2 Þ Rc R2
A
2
> 3
;
4p2
Rc1 R2
ð88Þ
and hence by
e 1 R4 þ 2pR2 þ q > 0:
A
ð89Þ
Then (85) immediately follows. In the case (84) one arrives to
e 1 ðRc R2 Þ
A
1
2
>
;
e2
4p2
A
ð90Þ
which is equivalent to (87).
h
Theorem 7. Let H = 1 and (67) hold either together with
Rc2 R2 > 4p2
or together with
Rc2 R2 > 4p2
e3
A
;
e1
A
e 2 ðRc R2 Þ;
Rc 1 R2 6 A
3
ð91Þ
e3
A
;
e1
A
e 2 ðRc R2 Þ:
Rc 1 R2 > A
3
ð92Þ
Then, in the case (91), the stability is guaranteed only if
2
e 3 > Rc 3 R ;
A
Rc1 R2
ð93Þ
while in the case (92) further conditions are not required to R2.
Proof. The proof follows very easily. h
Remark 3. We remark that – for any values of H, Hi, Pi, (i = 1, 2)- in the plane R21 ; R22 the stability- instability area is bounded
by the system
Rci ¼ 0;
infþ
ða2 ;nÞ2R N
n2n
jH3n j
¼ sup
;
a2 ða2 ;nÞ2Rþ N jH1n j
i ¼ 1; 2;
ð94Þ
constituted by algebraic equations of first degree in R2i ; ði ¼ 1; 2Þ.
6. Instability islands
The subsequent Section is devoted to the stability of the conduction solution when the layer is heated from below and
salted from above by both the salts. Since in that case the ‘‘instability islands” are not allowed, we concentrated here on
the remaining two cases.
Theorem 8. Let H = H1 = H2 = 1 together with
Rc2 4p2
e3
A
e 2;
– Rc1 4p2 A
e1
A
R2 < Rc :
ð95Þ
1530
S. Rionero / International Journal of Engineering Science 48 (2010) 1519–1533
Then an ‘‘instability island” exists and it is given by
or by
e 2 < R2 < inf
Rc1 4p2 A
Rc2 4p2
Rc2 4p2
!
e3
A
; Rc ;
e1
A
e3
A
e 2 ; Rc Þ:
< R2 < infðRc1 4p2 A
e1
A
ð96Þ
ð97Þ
Proof. Eq. (96) imply (Q1 > 0, Q2 > 0) and hence c = aQ1Q2 < 0. In view of b2 ac > 0, Theorem 5 guarantees that (96) is an
instability zone. h
e
Since R2 < Rc3 ) Q 3 > 0 and R2 < Rc1 4p2 < Rc2 4p2 A 3 ) fQ 1 < 0; Q 2 > 0g it follows that c > 0, b > 0 and hence by
eA 1
virtue of (77)
R2 < Rc1 4p2 ;
ð98Þ
e
is a stability zone. On the other hand, R ¼ Rc2 4p2 A 3 implies {Q2 = 0, c = 0, b > 0} i.e. stability.
eA
In the case (97) it follows that {Q1 < 0, Q2 < 0} hence1 c < 0 and (97) is an instability area. It is easily verified that the area
e2
below (97) is stable. In fact in that area it is {Q1 < 0, Q2 > 0, Q3 > 0} and hence c > 0, b > 0. On the other hand, R2 ¼ Rc1 4p2 A
implies {Q1 = 0, c = 0, b > 0} i.e. stability.
The following theorem is immediately obtained:
2
Theorem 9. Let {H = H1 = 1, H2 = 1} hold together with (95) and
e2;
Rc1 > 4p2 A
Rc2 > 4p2
e3
A
:
e1
A
ð99Þ
Then (96) or (97) is an instability island.
Remark 4. We remark that:
(i) The existence of ‘‘instability islands” is also envisaged by (85). In fact for r1 > 0, R22]r1, r2[]0, Rc[ denotes an ‘‘instability island”;
(ii) in the case (H = 1, H1 = H2 = 1) neither (96) nor (97) hold. In fact in that case either (96) or (97) require R2 < 0.
7. Global nonlinear stability when all three effects are destabilizing
In view of (66), one immediately obtains that all the three effects are destabilizing when the layers is heated from below
and salted from above with both salt fields i.e.
H > 0;
Hi < 0;
i ¼ 1; 2:
ð100Þ
Analogously all the three effects are stabilizing when the layer is heated from above and salted from below with both the salt
fields, i.e. {H < 0, H1 > 0, H2 > 0}. In both these cases can be shown that the problem is symmetrizable. For the sake of concreteness and simplicity, we refer to (100) since with the same procedure, as expected, in the case {H < 0, H1 > 0, H2 > 0} the conduction solution is easily found to be nonlinearly globally asymptotically stable for any value of the Rayleigh and Prandtl
numbers. Introducing the scalings l2 and l3 and setting
8
< hn ¼ X n ; U1n ¼ l2 Y n ; U2n ¼ l3 Z n ;
1
P
X n ; Y ¼ l1 U1 ; Z ¼ l1 U2 ;
:X ¼
2
3
ð101Þ
n¼1
Eq. (60) becomes
8
1
P
>
>
Xt ¼
a1n Y n þ a2n l2 Y n þ a3n l3 Z n u rX;
>
>
>
n¼1
>
>
>
>
<
1
P
l3
b1n
X
þ
b
Y
þ
b
Z
Yt ¼
u rY;
n
2n
n
3n
n
l2
l2
>
n¼1
>
>
>
>
>
1
>
P
c2n l2
>
c1n
>
: Zt ¼
l3 X n þ l3 Y n þ c3n Z n u rZ:
n¼1
ð102Þ
S. Rionero / International Journal of Engineering Science 48 (2010) 1519–1533
1531
Choosing
1
1
H1 2
;
HP 1
H2 2
;
HP 2
ð103Þ
8
1
1
2
2
>
1
2
>
a2n l2 ¼ HH
RR1 gn ; a3n l3 ¼ HH
RR2 gn ;
>
P
P
>
1
2
>
>
<
1
1
2
b1n
HH1 2
b3n ll3 ¼ HP11 HP22 R1 R2 gn ;
> l2 ¼ P1 RR1 gn ;
2
>
>
>
1
1
>
2
>
: c1n ¼ HH2 2 RR2 g ;
c2n ll2 ¼ HP11 HP22 R1 R2 gn ;
n
l
P2
ð104Þ
l2 ¼
l3 ¼
it follows that
3
3
and for any n 2 N the matrix of the coefficients is symmetric. In view of
< sin npz; sin mpz >¼ 0;
n – m;
< u; rF 2 >¼ 0;
F 2 ðX; Y; ZÞ;
ð105Þ
Eq. (102) implies
1 d
2 dt
Z
X
ðX 2 þ Y 2 þ Z 2 ÞdX ¼
1 Z
X
Q n dX;
ð106Þ
X
n¼1
with
1
Qn ¼
c3n Z 2n
b2n Y 2n
þ
þ
a1n X 2n
HH1 2
2
RR1 X n Y n gn þ;
P1
sffiffiffiffiffiffiffiffiffiffiffiffi
HH2
H1 H2
RR2 gn X n Z n þ 2
2
g Y n Zn :
P2
P1 P2 n
ð107Þ
Therefore, the global asymptotic stability is guaranteed by
8
>
>
> c3n < 0;
>
>
>
>
>
>
>
<
>
>
I3n ¼
>
>
>
>
>
>
>
>
:
R2 R2
c3n b2n H1 H2 P11 P22 g2n > 0;
a1n
HH1
P1
HH2
P2
1
2
1
2
HH1
P1
RR1 gn
RR2 gn
1
2
RR1 gn
b2n
H1 H2
P1 P2
1
2
R1 R2 gn
HH2
P2
H1 H2
P1 P2
1
2
1
2
RR2 gn
R1 R2 gn 0:
ð108Þ
c3n
But in view of (59) c3n < 0; 8n 2 N is equivalent to
jH2 jR22 < 4p2 ;
ð109Þ
and
H1 R21 gn nn H2 R22 gn nn
H1 R21 þ H2 R22 gn nn þ n2n
R1 R2 2
H1 H2 R21 R22 g2n
c3n b2n pffiffiffiffiffiffiffiffiffiffi gn ¼
¼
> 0;
P1 P2
P1 P2
P1 P2
P1 P2
requires
n
H1 R21 þ H2 R22 < n ;
gn
8n;
and hence
jH1 jR21 þ jH2 jR22 < 4p2 :
ð110Þ
Since
I3n ¼ a1n b2n c3n þ HH1 H2
R2 R21 R22 3
R2 R1 R2 3
R2 R2
R2 R2
R2 R2
gn þ HH1 H2
gn þ HjH2 j 2 b2n g2n H1 H2 1 2 a1n g2n HjH1 j 1 c3n g2n < 0;
P1 P2
P1 P2
P2
P1 P2
P1
i.e.
8
R2 R2 R2
1
>
ðHR2 gn nn Þ jH1 jR21 gn nn jH2 jR22 gn nn þ 2HH1 H2 P1 P1 2 2 g3n þ
>
P1 P2
>
>
<
R2 R2
H H R2 R2
HjH2 j P1 P22 g2n jH1 jR21 gn nn 1 P12P2 2 g2n ðHR2 gn nn Þ
>
>
>
>
: HjH1 j R2 R2 g2 jH jR2 g n < 0:
2 2 n
n
1 n
P1 P 2
1532
S. Rionero / International Journal of Engineering Science 48 (2010) 1519–1533
one easily obtains
I3n < 0 () HR2 þ jH1 jR21 þ jH2 jR22 <
nn
gn
;
8ða; nÞ 2 Rþ N;
and hence one arrives to
HR2 þ jH1 jR21 þ jH2 jR22 < 4p2 :
ð111Þ
In view of (109)–(111) it follows that (111) is necessary and sufficient for the stability. Further the stability is global and
‘‘instability islands” are not allowed. Eq. (111) in the case {H = 1, H1 = H2 = 1} has already been found – via the energy variational method- in [23].
8. Concluding remarks
This paper is concerned with:
(1) The long-time behavior of a triply convective–diffusive fluid mixture saturating a porous horizontal layer;
(2) the stability of the conduction solution We remark that:
(i) It is shown that the L2-solutions are bounded and asymptotically converging toward an absorbing set of the phasespace. The procedure used for obtaining these results (and the symmetrization when all the effects are either
destabilizing or stabilizing), can be used for a fluid mixture, with n components, 8n 2 N.
(ii) The linear stability of the conduction solution is expressed in closed forms via algebraic inequalities involving the
Rayleigh and the Prandtl numbers, and allow to bound the instability area of the space of the Rayleigh and Prandtl
numbers.
(iii) The existence of an ‘‘instability island” when the layer is heated and salted (at least by one salt) from below has
been shown analytically.
(iv) The linear stability of the conduction solution has been reduced to the stability of the null solution of ternary linear
systems of O.D.Es.
Acknowledgments
This work has been performed under the auspices of the G.N.F.M. of I.N.D.A.M. This work was supported in part by a grant
from the Leverhulm Trust, ‘‘Tipping points: mathematics, metafhors and meanings”.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
B. Straugha, Stability and wave motion in porous media, Appl. Math. Sci. 165 (2008).
D.A. Nield, A. Bejan, Convection in Porous Media, Springer-Verlag, Berlin, Heidelberg, New York, 1999.
J.N. Flavin, S. Rionero, Qualitative Estimates for Partial Differential Equations: An Introduction, CRC Press, Boca Raton (FL), 1996.
S. Lombardo, G. Mulone, S. Rionero, Global Stability of the Bénard problem for a mixture with superimposed plane parallel shear flows, Math. Methods
Appl. Sci. 23 (2000) 1447.
S. Lombardo, G. Mulone, S. Rionero, Global nonlinear exponential stability of the conduction–diffusion solution for Schmidt numbers greater than
Prandtl numbers, J. Math. Anal. Appl. 262 (2001) 1229.
G. Mulone, On the nonlinear stability of a fluid layer of a mixture heated and salted from below, Continuum. Mech. Thermodyn. 6 (1994) 161.
S. Lombardo, G. Mulone, B. Straughan, Nonlinear stability in the Bénard problem for a double-diffusive mixture in a porous medium, Math. Methods
Appl. Sci. 24 (2001) 1229.
G. Mulone, S. Rionero, Unconditional nonlinear exponential stability in the Bénard problem for a mixture: Necessary and sufficient conditions, Atti
Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei 9 (1998) 221.
F. Capone, S. Rionero, Nonlinear stability of a convective motion in a porous layer driven by horizontally periodic temperature gradient, Continuum.
Mech. Thermodyn. 15 (2003) 529.
S. Lombardo, G. Mulone, Nonlinear stability convection for laminar flows in a porous medium with Brinkman law, Math. Methods Appl. Sci. 26 (2003)
453.
F. Capone, S. Rionero, On the instability of double diffusive convection in porous media under boundary data periodic in space, in: S. Rionero, G.
Romano (Eds.), Trends and Applications of Mathematics to Mechanics, STAMM, 2002, Springer, 2004, pp. 1–8.
F. Capone, M. Gentile, S. Rionero, Influence of linear concentration heat source and parabolic density on penetrative convection onset, in: R. Monaco, G.
Mulone, S. Rionero, T. Ruggeri (Eds.), Proceedings ‘‘Wascom 2005” 13th Conference on Waves and Stability in Continuum Media, World Scientific,
2006, pp. 77–82.
F. Capone, M. Gentile, S. Rionero, On penetrative convection in porous media driven by quadratic sources, in: R. Monaco, G. Mulone, S. Rionero, T.
Ruggeri (Eds.), Proceedings ‘‘Wascom 2005” 13th Conference on Waves and Stability in Continuum Media, World Scientific, 2006, pp. 83–88.
S. Lombardo, G. Mulone, Necessary and sufficient conditions of global nonlinear stability for rotating double-diffusive convection in a porous medium,
Continuum Mech. Thermodyn. 14 (2002) 527–540.
S. Rionero, A new approach to nonlinear L2 – Stability of double diffusive convection in porous media: Necessary and sufficient conditions for global
stability via a linearization principle, J. Math. Anal. Appl. 333 (2007) 1036–1057.
S. Rionero, L. Vergori, Long time behaviour of fluid motions in porous media in the presence of Brinkman law, Acta Mech., Springer Wien 210 (2–3)
(2009) 21–240.
A.A. Hill, S. Rionero, B. Straughan, Global stability for penetrative convection with throughflow in a porous material, IMA J. Appl. Math. 72 (v) (2007)
635–643.
F. Capone, M. Gentile, A. Hill, Anisotropy and symmetry in porous media convection, Acta Mech. 208 (3–4) (2009) 205–214.
S. Rionero / International Journal of Engineering Science 48 (2010) 1519–1533
1533
[19] B. Straughan, Oscillatory convection and the Cattaneo law of heat conduction, Ricerche Mat. 58 (2009) 157–162.
[20] K.R. Rajagopal, G. Saccomandi, L. Vergori, Stability analysis of the Rayleigh–Bénard convection for a fluid with temperature and pressure dependent
viscosity, Z. Angew. Math. Phys. 60 (4) (2009) 739–755.
[21] A.J. Pearlstein, R.M. Harris, G. Terrones, The onset of convective instability in a triply diffusive fluid layer, J. Fluid Mech. 202 (1989) 443–465.
[22] R.A. Noutly, D.G. Leaist, Quaternary diffusion in acqueous KCl–KH2PO4–H3PO4 mixtures, J. Phys. Chem. 91 (1987) 1655–1658.
[23] J. Tracey, Multi-component convection–diffusion in a porous medium, Continuum Mech. Thermodyn. 83 (1996) 61–381.
[24] B. Straughan, D.W. Walker, Multi-component convection–diffusion and penetrative convection, Fluid Dyn. Res. 19 (1997) 77–89.
[25] B. Straughan, J. Tracey, Multi-component convection–diffusion with internal heating or cooling, Acta Mech. 133 (1999) 219–239.
[26] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Series in Applied Sciences, vol. 68, Springer, New York, 1997.
[27] D.R. Merkin, Introduction to the Theory of Stability, Text in Applied Mathematic, vol. 24, Springer, 1997.
[28] S. Rionero, A rigorous reduction of the L2 – stability of the solutions to a nonlinear binary reaction-diffusion system of O.D.Es to the stability of the
solutions to a linear binary system of O.D.Es, J. Math. Anal. Appl. 310 (2006) 372–392.
[29] S.C. Subramanian, K.R. Rajagopal, A note on flow through porous solids at high pressures, Comput, Math. Appl. 53 (2007) 260–275.
[30] K.R. Rajagopal, On a hierarchy of approximate models for flows of incompressible fluids through porous solids, Math. Models Methods Appl. Sci. 17
(2007) 215–252.