Exact Solution Of Triple Diffusive Marangoniconvection In A Composite Layer

Exact Solution Of Triple Diffusive Marangoniconvection In A Composite Layer

R. SUMITHRA Department of Mathematics

Government Science College Bangalore-560 001, Karnataka,INDIA.

Abstract

The Triple-Diffusive Marangoni-convection 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.

1. 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 BJ-slip 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)

2. 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

3. 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

1. 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

2. 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

3. 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

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

2. 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

3. 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.

4. 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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