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 Total Downloads : 823
 Authors : R. Sumithra
 Paper ID : IJERTV1IS5002
 Volume & Issue : Volume 01, Issue 05 (July 2012)
 Published (First Online): 02082012
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Exact Solution Of Triple Diffusive Marangoniconvection In A Composite Layer
R. SUMITHRA Department of Mathematics
Government Science College Bangalore560 001, Karnataka,INDIA.
Abstract
The TripleDiffusive Marangoniconvection problem is investigated in a two layer system comprising an incompressible three component fluid saturated porous layer over which lies a layer of the same fluid. The lower surface of the porous layer is rigid and the upper free surface are considered to be insulating to temperature and solutes concentration perturbations. At the upper free surface, the surface tension effects depending on temperature and both the solute concentrations are considered. At the interface, the normal and tangential components of velocity, heat and solute concentrations and their fluxes are assumed to be continuous. The resulting eigenvalue problem is solved Exactly and an analytical expression for the Thermal Marangoni Number is obtained. The effect of variation of different physical parameters on the same is investigated in detail.

Introduction
Hydrothermal growth is a crystal growth from aqueous solution at high temperature and pressure. Even under hydrothermal conditions most of the materials grown have very low solubilities in pure water. Thus to achieve reasonable solubilities large quantities of other materials called mineralizers are added which do not react with the material being grown but affect the density gradients. The convection involved is multi component convection There are many fluid systems in which more than two components are present. The problem under investigation also has many applications like solidification of alloys, the materials processing, the moisture migration in thermal insulation and stored grain, underground spreading of chemical pollutants, waste and fertilizer migration in saturated soil and petroleum reservoirs.
. For example, Degens et al [3] have reported that the saline waters of geothermally heated Lake Kivu are strongly stratified by temperature and salinity which is the sum of comparable concentrations of many salts, while the oceans contain many salts in concentrations less than a few percent of the sodium chloride concentration i. e. one can expect a multicomponent system. Even in laboratory experiments on double diffusive convection, dyes or small temperature anamolies introduce a third property which affects the density of the fluid. In these cases the study of double diffusive convection becomes very restrictive. Therefore, one has to consider the stability of multi component systems. Turner et al [17] and Griffiths [4] have initiated the work in this direction by conducting laboratory experiments in which the fluxes of several components across diffusive interfaces are measured. Shivakumara [13] has investigated the onset of triple diffusive convection, where the effect of third diffusing component upon the onset of marginal, oscillatory convection and bifurcation from the static solution are discussed.
The problems of triple diffusive convection in clear fluids are also studied by Pearlstein et al [8] and Lopez et al [5]. Rudraiah and Vortmeyer [11] have studied the linear stability of three component system in a porous medium in the presence of a gravitationally stable density gradient. Poulikakos [9] has in his brief communication established the presence of a third diffusing component with small diffusivity can seriously alter the nature of the convective instabilities in the system. Triple diffusive convection in composite layers is not given much importance. Where as Single component convection in composite layers is investigated by Many of the researchers started by Nield [7] , Rudraiah [12], Taslim and Narusawa [16], McKay [6], Chen [2] . Recently I. S. Shivakumara et. al [14] have investigated the onset of surface tension driven convection in a two layer system comprising an incompressible fluid saturated porous layer over which lies a layer of the same fluid. The critical Marangoni number is obtained for insulating
boundaries both by Regular Perturbation technique and also by exact method. They also have compared the results obtained by both the methods and found in
0
q
T 2
q
(1)
P 2 (2)
agreement.
0 t
q q q
Double diffusive convection in composite layers has wide applications in crystal growth and solidification of alloys. Inspite of its wide applications not much work has been done in this area. Chen and Chen [1] have considered the problem of onset of finger convection using BJslip condition at the interface. The problem of double diffusive convection for a thermohaline system consisting of a horizontal fluid layer above a saturated porous bed has been investigated experimentally by Poulikakos and Kazmierczak [10]. Venkatachalappa et al [17] have investigated the
double diffusive convection in composite layer conducive for hydrothermal growth of crystals with the lower boundary rigid and the upper boundary
t q T T
q 1 1 1
C1 C 2C
t
q 2 2 2
C2 C 2C
t
For the porous layer,
m qm
0
1 1
qm
(3)
(4)
(5)
(6)
free with deformation. The double diffusive
0 t
2 qm m qm
magneto convection in a composite layer bounded by
rigid walls is investigated in Sumithra [15] .
m Pm m
2
qm
q
K m
(7)

Formulation of the problem
We consider a horizontal three – component fluid
A Tm T 2 T
qm m m m m m
t
(8)
Cm2 2
saturated isotropic sparsely packed porous layer of thickness dm underlying a three component fluid
Cm1 C
qm m m1
t
2
C
m1 m m1
(9)
layer of thickness d. The lower surface of the porous layer is considered to rigid and the upper surface of the fluid layer is free at which the surface tension effects depending on temperature and both the species concentrations. Both the boundaries are kept at different constant temperatures and salinities. A Cartesian coordinate system is chosen with the origin
t qm m Cm2 m2mCm2
(10)
at the interface between porous and fluid layers and
the z axis, vertically upwards as shown in Fig.1.
Where the symbols in the above equations have the
following meaning.
q u, v, w
is the velocity
vector, t is the time, is the fluid viscosity, P is
the pressure, 0
is the fluid density, T is the
temperature, is the thermal diffusivit C1
is the
species concentration1 or the salinity field 1, 1 is
the solute1 diffusivity of the fluid, C2
is the species
concentration2 or the salinity field2, 2
is the
Fig1. Physical Configuration
. The continuity, momentum, energy, species concentration1 and species concentration2 equations
solute1 diffusivity of the fluid, the ratio of heat capacities, Cp
0Cp
p
A m is
C
f
is the specific heat,
are,
K is the permeability of the porous medium. The
subscripts m and f refer to the porous medium and the fluid respectively.
The basic steady state is assumed to the quiescent and we consider the solution of the form,
T dmTu mdTl ,
m m
0 d d
d
C
1dmC1u 1mdC1l
d
10
1 m 1m ,
1 2 d C dC
C 2 m 2u 2m 2l
are the interface
d
d
0, 0, 0, Pb z,Tb z,C1b z ,C2b z (11)
in the fluid layer and in the porous layer
um , vm , wm , Pm ,Tm ,C1m ,C2m
0, 0, 0, Pmb zm ,Tmb zm ,C1mb zm ,C2mb zm
(12)
Where the subscript b denotes the basic state. The temperature and species concentration distributions
20
2 m 2m
temperature and concentrations.
In order to investigate the stability of the basic solution, infinitesimal disturbances are introduced in the form,
q, P,T ,C1,C2 0, Pb z ,Tb z ,C1 z ,C2 z
q 1 2
, P, , S , S (19)
Tb z , Tmb zm , Cb z , Cm b z ,m and
And
C z , C z , respectively are found to be
qm , Pm ,Tm ,Cm1,Cm2
2b 2mb m
T T z
0, Pmb zm ,Tmb zm ,Cm1b zm ,Cm2b zm
b
0
d
q
, P , , S
, S
(20)
T z T 0 u
in 0 z d
(13)
T z
T

Tl T0 zm
in 0 z d
m m m m1 m2
d
mb m 0
m
m m
(14)
Where the primed quantities are the perturbed ones over their equilibrium counterparts. Now Eqs. (19)
C1b
z C10
C10 C1u z
d
in 0 z d
(15)
and (20) are substituted into the Eqs. (1) to (10) and are linearised in the usual manner. Next, the pressure term is eliminated from (2) and (7) by taking curl twice on these two equations and only the vertical
component is retained. The variables are then
C z
C
C1l C10 zm
in 0 z d
d 2
1mb m 10
m
m m nondimensionalised using d , ,
d , T0 Tu ,
d
(16)
C10 C1u and
C20 C2u
as the units of length,
C z C
C20 C2u z
in 0 z d
time, velocity, temperature, species concentrations in
2b 20 d
d 2
(17)
the fluid layer and
dm , m , m ,
Tl T0 ,
m dm
C z
C
C2l C20 zm in
C1l C10
and
C2l C20
as the corresponding
d
2mb m 20
m
0 zm dm
(18)
characteristic quantities in the porous layer. Note that the separate length scales are chosen for the two layers so that each layer is of unit depth.
In this way the detailed flow fields in both the fluid and porous layers can be clearly obtained for
all the depth ratios
d dm
. The dimensionless
Where
equations for the perturbed variables are given by, in
0 z 1
1 2 w
Pr t
4 w
(21)
w
W z
z
f x, y ent
(29)
w 2
(22)
S1
1 z
t
S 2
S
2
2 z
1 w 1 S1 (23)
t
S
wm
And
Wm zm
2 w 2 S (24)
2 2
z
t
m
m m f
x , y
enmt
Sm1
m1
zm
m m m
In 1 zm 0
2 2 w
Sm2
m2 zm
(30)
m m 24 w
2 w
R 2
With
2 f a2 f 0
and 2 f

a2 f
0 ,
Prm t
m m m m m
2m m
(25)
2 2m m m m
where a and am are the nondimensional horizontal
A m w 2
(26)
wavenumbers, n and nm are the frequencies. Since
t m m m
S 2
the dimensional horizontal wavenumbers must be the same for the fluid and porous layers, we must have
m1 w S
(27) a a
t m m1
S
m m1
2
m
d dm
and hence am da .
m2 w S
t m m2 m m2
(28)
Substituting Eqs. (29) and (30) into the Eqs.(21) to (28) and denoting the differential
For the fluid layer Pr
is the Prandtl number,
operator
z
and
zm
by D and Dm respectively,
1
1
is the ratio salinity1 diffusivity to thermal
an eigenvalue problem consisting of the following ordinary differential equations is obtained,
diffusivity, 2
2
is the ratio salinity2 diffusivity
In 0 z 1,
to thermal diffusivity. For the porous layer,
m
D2 a2
n D2 a2 W 0
(31)
Prm
is the Prandtl number,
Pr
m
2 K Da is the Darcy number, m is
D2 a2 n W 0
(32)
d
2
m
the viscosity ratio,
m1
1 1
is the ratio salinity1
D2 a2 n W 0
(33)
m1
D2 a2 n W 0
(34)
diffusivity to thermal diffusivity, m2
m2
2 2
is the
ratio salinity2 diffusivity to thermal diffusivity.
We make the normal mode expansion and
seek solutions for the dependent variables in the fluid
In 1 zm 0
n 2
and porous layers according to
2 D2 a2 m 1 D2 a2 W 0
m m
Prm
m m m
(35)
m m m m m
D2 a2 An W 0
m1 m m m m1 m
D2 a2 n W 0
(36)
(37)
The upper boundary is assumed to be free insulating both temperature and species concentrations so, the appropriate boundary
conditions at z d ,
D2 a2 n W 0
m2 m m m m2 m
(38)
T C C
w 0,
0, 1 0, 2 0
(48)
z z z
t T t C
It is known that the principle of exchange of instabilities holds for triple diffusive convection in both fluid and porous layers separately for certain
One more velocity condition at the free surface is the continuity of the tangential stress given by
choice of parameters. Therefore, we assume that the
principle of exchange of instabilities holds even for
2 w 2 2
t 2C
the composite layers. In otherwords, it is assumed that the onset of convection is in the form of steady
z2
T 2
C 2 1 C 2 2
1 2
(49)
convection and accordingly we take n nm 0 .
Where t is the surface tension and is
In 0 z 1,
given by
t 0
TT C C1
C2 C2
D2 a2 2 W 0
(39)
t
1
, t ,
T T C1 C
0
1 10
D2 a2 W 0
(40)
T T
1 C C
D2 a2 W 0
(41)
1 1
2 2
D2 a2 W 0
(42)
C2
t
C2
C2 C20
In 1 zm 0
At the interface (i.e., at z 0, zm 0 ), the
m m m m m
2 D2 a2 1 D2 a2 W
0
(43)
normal component of velocity, tangential velocity, temperature, heat flux, species concentration and
mass flux are continuous and respectively yield following Nield (1977),
m m m m
D2 a2 W 0
(44)
w w ,
w wm ,
2 2
m z z
m
m1 Dm am nm m1 Wm 0
(45)
T T
D2 a2 n W 0
m2
m m m m2
m
m
z
mz
T T ,
m ,
(46)
m
m
C C , C1 Cm1 ,
Thus we note that, in total we have a
1 m1 1 z
m1 z
twentyth order ordinary differential equation and we
C C ,
C2 Cm2
(50)
need twenty boundary conditions to solve them.
2 m2 2 z
m2 z
m



Boundary conditions
The bottom boundary is assumed to be rigid and insulating to both temperature and species concentrations, so that at zm dm ,
We take two more boundary conditions at the interface. Since we have used the Darcy Brinkman equations of motion for the flow through the porous medium, the physically feasible boundary conditions on velocity are the following, at z 0
wm 0,
wm 0,
z
Tm 0,
z
Cm1 0,
z
Cm2 0
z
and zm 0
m m m m
(47)
P 2 wm P 2 w
m m
T T d
zm z
Where
M t 0 u
T
is the thermal
which will reduce to
C10 C1u d
t
2 w
Marangoni number,
M s1
32
2
C1
z2 z
is the solute1 Marangoni number,
w
2
w
t C20 C2u d
m m 2 32
m
m
m
M s 2
is the solute2
K zm
2m z2
zm
(51)
C2
Marangoni number,
d
is the depth ratio,
The other appropriate velocity boundary condition at dm
the interface
z 0, zm 0 can be ,
2 w
2 w
t is the ratio of thermal diffusivities of
2 w m 2 w m
2
m
m
z2
z2
2m m
fluid to porous layer ,
s1
is the ratio of
(52)
s1
s1m
All the twenty boundary conditions (47) to (52) are
solute1 diffusivities of fluid to porous layer ,
nondimenstionalised by using the same scale factors that of equations and are subjected to normal mode
analysis and they are given .
s 2
s 2
s 2m
is the ratio of solute2 diffusivities of
W (1) 0, D2W (1) a2M 1
fluid to porous layer.
s1 1
a2 M
1 a2M
s2 2
1 0,
The Eqs.(41) to (46) are to be solved with respect to the boundary conditions (53).
D(1) 0,
D1(1) 0,
D2 (1) 0
2

Exact Solution
W (0) Wm (0), DW (0) DmWm (0),
t t The equations (39) and (43) are independent of
D2 a2
W (0)
3
D2 a2 W (0)
, 1, 2 and
m , m1
, m2 respectively and
m m m
t
D3W (0) 3a2 DW (0)
2
t Da
DmWm
0
they can be solved independently to get the general solutions in the form,
W z A1Cosh az A2 zCosh az
A Sinhaz A zSinhaz
4 3 2
3 4 (54)
DmWm 0 3am DmWm 0
W z A Cosh a z A Sinh a z
t m 5
m m 6 m m
(0) t
m (0),
D(0) Dmm (0),
A7Cosh zm A8Sinh zm
(55)
(0) s1 (0), D (0) D
(0),
Where A1 to A4 and A5 to A8 constants to be
1 m1 1
m m1
determined using the velocity boundary conditions of
531 ,536 , 537 , 538 , 539 , 5310 , 5311
(0) s 2 (0), D (0) D
(0),
2 m2 2
m m2
and obtain
W 1 0, D W 1 0, D 1 0,
W z A1[Cosh az a1zCosh az
m m m m m
D 1 0, D 1 0
(53)
a2Sinh az a3 zSinh az
m m1
m m2
(56)
W z A [a Cosh a z
a Sinh a z
a2 1
m 1 4
m m 5 m m
m 2
a6Cosh zm a7 Sinh zm ]
(57)
1
2
Da am
3a2 4 a3 4

m m
The heat equations (40) and (44) are then solved
t t t
2
using thermal boundary conditions of (53), the
2
3a2 4 3 4
m
expressions for , m are obtained as,
z A1 a8Cosh az a9Sinh az f (z)
(58)
Da t t t
2a
2
t
3 3
m z A1[a10Cosh am zm
a11Sinham zm fm (zm )]
(59)
4
2 a2
t m
1
3 2 2
a
The Species concentration1equations (41) and (45) are then solved using species1 boundary conditions
of (53), the expressions for , are obtained
m
5
4
a2 2
as,
1 m1
m
6 5 Cosham Cosh
z f (z)
A a Cosh az a Sinh az
1 1
12 13
1
Cosha t Cosh
7 4 m 4
m1 zm A1[a14Cosh am zm
(60)
8 5 Sinh am Sinham
a Sinh a z
fm (zm )]
a Sinha t Sinh
15 m m
m1
9 4 m m 4
(61)
The Species concentration2 equations (42) and (46) are then solved using species2 boundary conditions
10
1 ,
2a3
11
2
2a3
of (53), the expressions for
, are obtained
2a
2
2 m2
a m ,
a
as,
12 10
13 11
t t
z a Sinh az f (z)
Sinha, Cosha Sinha
2 A1 a16Cosh az 17
2
(62)
14 15 12 10
17 Cosha, 16 13Cosha 11Sinha
z
A [a Cosh a z
18
15 ,
19
16 ,
20
17
m2 m
1 18 m m
14 14 14
a Sinh a z
fm (zm )]
21 618 Sinham ,
22 619 Sinh ,
19 m m
,
m2 23 6 20 7
26 8 20 9
(63)
24 818 amCosham ,
25 819 Cosh ,
where
27
22 ,
28
23 ,
21 21
a 28 26 ,
a
a ,
a
a
a 1 {a4 amCosham Sinham
7
5 27 7 28 3 18 5 19 736
20 2a
24 27 25
a1 12a5 13a7 , a2 10a5 11a7 ,
m1

a5
m
a Sinha Cosha }
a a , a t a ,
2am
m m m
4 4 5 3 6 4
1 a Sinh a Sinh
aCosha Sinha
6 7
2 2
2 2
29 2a
m1
am
am

a1 aCosha Sinha aSinha Cosha
36s1aSinha 30aSinha
Cosha 29
4a
a 37
a Sinha 35

a2 aSinha Cosha
2a
m m m1 1
a Cosha s1aCosham Sinha

a3 aSinha 2Cosha aCosha Sinha
38 m
Sinham
4a a
t
a6 t a7
1 a1
a2
1 a5
30 2 a2
2 a2
39
4a2
2a
2a
m m 2 m2 m
31
a1 a2
2
a5
a a Cosha Sinha
4a 2a
2am
40
1 {
4 m m m
2a
a4 amCosham Sinham
m2 m
32
2am

a5
a Sinha


Cosha }

a5

2am
am Sinham Cosham
2a m m m
m
1 a Sinh a Sinh

a6 Sinh a7 Sinh
6 7
2 2
2 2
m m
2 a2
2 a2
m2
am
am
40s 2aSinha
30aSinha
29
32
t aSinha
aSinha
Cosha
41 39Cosha
33
am Sinham
30 31 29
am Sinham
m2 2
t amCosha t aCosham Sinha
a Cosha s 2aCosham Sinha
42 m
34
a11
33 ,
34
a10
Sinham
a11amCosham 32
am Sinham
a19
41 ,
42
a18
Sinham
a19amCosham 40
am Sinham
a am a , a s 2 a
30 ,
a am a
,
a t a
,
17 a 19 39 16
18
m2
9 a 11 31 8
10 30
a a Cosha
1 a a
1 a
a15
37 ,
a14
15 m m 36
a Sinha
1 2 5
38 m m
35
4a2
2a
2a
am s1
30
1
m1 m
a a , a a ,
13 a 15 35 12
14
m1


The Thermal Marangoni number
The effects of the parameters
a, Da,
s1,
Now the thermal Marangoni number is obtained by
Ms1, Ms2 , 2 ,
m2
and
on the thermal
the boundary condition 532 as
Marangoni number are obtained and portrayed in the Figures 2 to 9 respectively.
D2W (1) a2 M
M
s1 1
1 a2 M
s 2 2
1
Simplifying we get
a2 M
a21
(64)
300
250
s1
f (1)
200
a12Cosh a a13Sinh a
1
a2 M
150
s 2
f (1)
100
a16Cosh a a17 Sinh a
M 2
2 50
a a Cosh a a Sinh a f (1)
8 9
(65) 0
Where
f 1 Sinha a1 Sinha Cosha
2 4 6 8 10
Fig.2. The effects of a on Thermal Marangoni number M
2a 4a a
The effects of the horizontal wave number
a2 a3
Sinha
a , on the thermal Marangoni number M are shown

2a Cosha 4a Cosha a
in Fig.2. The graph has three diverging curves.
The line curve is for
a 3.0 , the big dotted curve
And
1
a2Cosha a a2Cosha 2aSinha
is for 3.1 and the small dotted line curve is for 3.2. Since the curves are diverging, it indicates that the
2 3
increasing values of will have effect only for

a a2 Sinha a
a2 Sinha 2aCosha
larger values of the depth ratio
d
dm
, that is for


Results and discussion
The Thermal Marangoni number M obtained as a function of the parameters is drawn versus the depth ratio and the results are represented graphically
showing the effects of the variation of one physical quantity, fixing the other parameters. The fixed values of the parameters are
0.25, 0.25, 0.25, 0.25, ,
fluid layer dominant composite systems. From the curves one can see that for a fixed value of , increase in the value of a is to increase the value of the thermal Marangoni number i.e., to stabilize the system by delaying the onset of surface tension driven convection.
The effects of the Darcy number Da , on the thermal Marangoni number M are shown in
t s1
s2 1
Fig.3. The graph has three converging curves.
2 0.25, m1 0.25, m2 0.25 ,
The line curve is for Da 10 , the big dotted curve
1, a 3.0, .
Ms1 10, Ms2 100,
Da 10.0,
1.0 .
is for 20 and the small dotted line curve is for 30. Since the curves are converging, it indicates that the increasing values of Da will have effect only for
smaller values of the depth ratio d
, that is for
is for 75. This number has dual effect on the thermal
d Marangoni number. For values of
5 the curves
m
porous layer dominant composite systems. From the curves one can see that for a fixed value of ,
increase in the value of Da is to increase the value of the thermal Marangoni number i.e., to stabilize the system by delaying the onset of surface tension driven convection.
300
250
200
150
100
50
0
2 4 6 8 10
Fig.3 . The effects of Da on the Thermal Marangoni number M
The effects of the ratio of solute1 diffusivity of the fluid in the fluid layer to that of porous layer
s1
are converging and here for a fixed depth ratio the
300
250
200
150
100
50
0
2 4 6 8 10
Fig.4. The effects of s1 on the Thermal Marangoni number M
350
300
250
200
s1
sm1
, on the thermal Marangoni number M
150
are shown in Fig.4. The curves are converging at both the ends. The line curve is for s1 0.25 , the big dotted curve is for 0.5 and the small dotted line curve is for 0.75. It is evident that the effect of s1
100
50
is prominent in the region 2 8
0
and here for a
fixed value of , increase in the value of s1 is to
2 4 6 8 10
increase the value of the thermal Marangoni number M i.e., to stabilize the system by delaying the onset of surface tension driven convection.
Fig.5. The effects of M s1 on the Thermal Marangoni number M
Figure 5 displays the effects of the solute1
increase in value of
M s1
increases the thermal
Marangoni number M , on the thermal Marangoni
marangoni number where as, for the values of depth
s1
number M. The graph has three converging
ratio 5 the curves are diverging, and here for
curves. The line curve is for Ms1 25 , the big
fixed depth ratio the increase in value of
decreases the thermal marangoni number.
M s1
dotted curve is for 50 and the small dotted line curve
Figure 6 displays the effects of the solute1 Marangoni number M s 2 , on the thermal Marangoni number M. The graph has three converging curves. The line curve is for Ms 2 100 , the big dotted curve is for 150 and the small dotted line
d , that is for fluid layer dominant
dm
composite systems. From the curves one can see that for a fixed value of , increase in the value of 2 is to increase the value of the thermal Marangoni number i.e., to stabilize the system by delaying the
curve is for 200. Its effect is similar to that of
M s1 .
onset of surface tension driven convection.
This number has again dual effect on the thermal Marangoni number. For values of 5.5 the
300
curves are converging and here for a fixed depth
ratio the increase in value of
M s 2
increases the
thermal marangoni number where as, for the values of depth ratio 5.5 the curves are diverging, and
here for a fixed depth ratio the increase in value of
M s 2 decreases the thermal marangoni number.
300
250
200
150
250
200
150
100
50
0
2 4 6 8 10
Fig.7. The effects of 2
on the Thermal Marangoni
100
50
0
2 4 6 8 10
Fig.6. The effects of M s 2 on the Thermal Marangoni number M
The effects of the ratio of solute 2 diffusivity to thermal diffusivity in the fluid layer ,
300
250
200
150
100
number M
2
2
on the thermal Marangoni number M are
50
shown in Fig.7. The graph has three diverging curves. The line curve is for 2 0.25, the big dotted cure is for 0.50 and the small dotted line curve is for 0.75. Since the curves are diverging, it
0
2 4 6 8 10
indicates that the increasing values of 2
will have
Fig.8. The effects of m2 on the Thermal Marangoni
effect only for larger values of the depth ratio number M
The effects of the ratio of solute2 diffusivity to thermal diffusivity of the fluid in the porous layer
m2
300
m2
, on the thermal Marangoni number M
250
are shown in Fig.8. The graph has three converging curves. The line curve is for m2 0.25 , the big dotted curve is for 0.50 and the small dotted line curve is for 0.75. Since the curves are converging, it indicates that the increasing values of m2 will have effect only for smaller values of the depth ratio
200
150
100
d dm
, that is for porous layer dominant
50
composite systems. From the curves one can see that for a fixed value of , increase in the value of m2 is to decease the value of the thermal Marangoni number i.e., to destabilize the system so the onset of
0
2 4 6 8 10
surface tension driven convection is faster.
Fig.9. The effects of
on the Thermal Marangoni
The effects of the viscosity ratio m ,
number M
which is the ratio of the effective viscosity of the porous matrix to the fluid viscosity are displayed in Fig.9. The line curve is for 1 , the big dotted
curve is for 2 and the small dotted line curve is for 3. Since the curves are converging, it indicates that the increasing values of will affect the onset of convection only for the values of 10 . From the curves it is evident that for a fixed value of ,
increase in the value of is to increase the value
6. Conclusions

For Fluid layer dominant composite systems, by increasing values of a 2 the surface tension driven triple diffusive convection can be delayed.

For Porous layer dominant composite systems, by increasing the values of Da, and by decreasing
the value of the system can be stabilized.
of the thermal Marangoni number M i.e., to stabilize the system, so the onset of surface tention driven triple diffusive convection is delayed. In other words when the effective viscosity of the porous medium m is made larger than the fluid viscosity
, the onset of the convection in the fluid layer can
be delayed.
m2

Both the solute Marangoni numbers have similar effects on the convection. They exhibit opposite effects for the fluid layer dominant and porous layer dominant systems.

The effect of ratio of solute1 diffusivity of the fluid in fluid layer to porous layer is prominent for a range of values of depth ratio for certain choice of parameters. There is no effect of the ratio of solute2 diffusivity of the fluid in fluid layer to porous layer on the thermal marangoni number.
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