International Journal of Application or Innovation in Engineering & Management (IJAIEM)
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Volume 3, Issue 10, October 2014
ISSN 2319 - 4847
EFFECTS OF PARABOLIC AND INVERTED
PARABOLIC SALINITY GRADIENTS ON
DOUBLE DIFFUSIVE MARANGONI
CONVECTION IN A COMPOSITE LAYER
AN EXACT STUDY
1
R. Sumithra and 2B. Komala
1
Department of Mathematics, Government Science College, Bangalore – 560 001,
Karnataka, INDIA.
2
Department of Mathematics, BTLITM, Bangalore-560 099,
Karnataka, INDIA.
ABSTRACT
The Effects of Parabolic and Inverted parabolic Salinity gradients on the onset of Double Diffusive Marangoni Convection in
a two-layer system, comprising an incompressible two component fluid saturated porous layer over which lies a layer of the
same fluid in the microgravity condition, are investigated. The upper boundary of the fluid layer is free and the lower
boundary of the porous layer is rigid and both the boundaries are insulating to heat and mass. At the interface, the velocity,
shear stress, normal stress, heat, heat flux, mass and mass flux are assumed to be continuous conducive for Darcy-Brinkman
model. The resulting Eigen value problem is solved exactly. The Thermal Marangoni numbers for linear, parabolic and
inverted parabolic salinity profiles are obtained. The effects of different physical parameters on the onset of double diffusive
Marangoni convection are investigated for above profiles in detail.
Keywords: Double diffusive convection, Salinity gradients, Thermal Marangoni numbers, Darcy-Brinkman model
1. INTRODUCTION
In the generation of techno-savvy world, the chips made up of pure crystals are of great demand leading to the
enormous scope for the evolutions and explorations in the industry of crystal growth. There are many methods of
growing crystals and these can be classified on the basis of method of producing super saturation as Isothermal methods
(constant temperature method) ex., Hydrothermal growth and Non isothermal methods (temperature variation
method) In the case of Isothermal methods, any property of the crystal that is temperature dependent will be under
better control. 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 solvents. Thus to achieve
reasonable solubilities, large quantities of other materials are added which do not react with the material being grown.
These materials are called mineralizers. The apparatus consists of an autoclave consisting two layers. Nutrients in the
lower part (nutrient region) of the autoclave dissolve in the fluid (solvent + Mineralizers + crystal material), which is
kept to hotter than the upper part (growth region) of the autoclave. The temperature difference causes convection from
the nutrient region to the growth region and the upper fluid is supercooled which drives the crystallization. Since the
fluid has more than one diffusive component of different molecular diffusivities (heat, concentration of mineralizer) the
convection is multi diffusive and the materials in the nutrient chips can be regarded as a porous medium. This method
of growing crystals exactly simulates the double (if one mineralizer is added), triple (if two mineralizers are added) and
multi component (if more than two mineralizers are added) diffusive convection in a horizontal composite layer( a fluid
layer overlying a fluid saturated porous layer ). In the case of non isothermal methods of growing crystals that is, when
the temperature gradient is imposed on the system, the main advantage is that the diffusion path is usually shorter, so
reasonable rates are achieved without elaborate control or apparatus investment. In these situations, maintaining a
uniform temperature and salinity gradients is a limitation and occurrence of non-uniform temperature and salinity
gradients is a reality. The study of non uniform gradients is not given much attention and the non uniform salinity
gradients are rarely touched. Though some literature is available on the study of non uniform temperature gradients,
but the non uniform salinity gradients is at scarce. Recently Subbarama Pranesh et al (2012) have investigated the
effect of non uniform basic concentration gradient on the onset of double diffusive convection in a micropolar fluid
layer heated and saluted from below and cooled from above. The Eigen values are obtained using Galerkian method for
free-free, rigid-free, rigid-rigid velocity boundary combination with isothermal on spin-vanishing permeable
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Volume 3, Issue 10, October 2014
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boundaries. One linear and five non linear concentration profiles are considered and their comparative influence on
onset is discussed and results are depicted graphically. It is observed that the fluid layer with suspended particles heated
and saluted from below is more stable compared to the classical fluid layer without suspended particles. Here we make
an attempt to study the effects of two non uniform salinity gradients (parabolic and inverted parabolic profile) on the
onset of surface tension driven double diffusive convection in a horizontal composite layer by Exact method [6]. Here
we give some literature on the effects of non uniform temperature gradients on Marangoni convection in single
horizontal fluid and porous layers separately. Nanjundappa Rudraiah and Pradeep G Siddheshwar (2000) have
investigated the effect of non-uniform basic temperature gradients on the onset of Marangoni convection in a horizontal
layer of a Boussinesq fluid with suspended particles. It is observed that the fluid layer with suspended particles heated
from below is more stable compared to the classical fluid layer without suspended particles. The problem has possible
applications in microgravity situations [4]. Shivakumara et al (2002) have investigated the effect of different basic
temperature gradients on the onset of ferro convection driven by combined surface tension and buoyancy forces are
studied. The results indicate that the stability of Rayleigh-Bernard-Marangoni Ferro convection is significantly affected
by basic temperature gradients and the mechanism for suppressing or augmenting the same is discussed in detail. It is
shown that the results obtained under the limiting conditions compare well with the existing ones [1]. Melviana
Johnson Fu et al (2009) have studied the effect of six different non-uniform basic state temperature gradients on the
onsets of Marangoni convection in a horizontal micropolar fluid layer bounded below by a rigid plate and above by
non-deformable free surface subjected to a constant heat flux. They used Rayleigh Ritz technique to solve the resulting
Eigen value problem and discussed the influence of the various parameters on the onset of Marangoni convection [3].
Siti Suzillian Putri Mohamed Isa et al (2009) have investigated the effect of six different non-uniform basic
temperature gradients on the onset of Marangoni convection in a horizontal layer with a free-slip bottom heated from
below and cooled from above. They solved the resulting the Eigen value problem using single-term Galerkian
expansion procedure and have discussed the effect of the various parameters on the onset of Marangoni convection [5].
Coming to the single porous layers, Shivakumara et al (2012) have investigated the effect of different forms of basic
temperature gradients on the criterion for the onset of convection in a layer of an incompressible couple stress fluid
saturated porous medium is investigated. It is shown that the principle of exchange of stability is valid, and the Eigen
value problem is solved numerically using the Galerkian technique. The parabolic and inverted parabolic basic
temperature profiles have the same effect on the onset of convection [2]. Sumithra and Manjunath (2014) have
investigated the an exact study of Magneto-Marangoni-convection in a two layer system comprising an
incompressible electrically conducting fluid saturated porous layer over which lies a layer of the same fluid in the
presence of a vertical magnetic field in the microgravity condition. The lower rigid surface of the porous layer and the
upper free surface are considered to be insulating to temperature perturbations. At the upper free surface, the surface
tension effects depending on temperature are considered. At the interface, the normal and tangential components of
velocity, heat and heat flux are assumed to be continuous. The resulting Eigen value problem is solved exactly for both
parabolic and inverted parabolic temperature profiles and analytical expressions of the Thermal Marangoni Number
are obtained. Effects of variation of different physical parameters on the Thermal Marangoni Number for both profiles
are compared [7].
2. FORMULATION OF THE PROBLEM
We consider a horizontal two – component fluid saturated, isotropic, sparsely packed porous layer of thickness
dm underlying a two component fluid layer of thickness d , in the microgravity condition. The lower surface of the
porous layer is rigid and the upper surface of the fluid layer is free with the surface tension effects depending on both
temperature and concentration. Both the boundaries are kept at different constant temperatures and concentrations. A
Cartesian coordinate system is chosen with the origin at the interface between porous and fluid layers and the z – axis,
vertically upwards. The continuity, momentum, energy and concentration equations are,

q  0
(1)




 q
 0    q    q   P   2 q
 t

T

  q   T   2T
t
C

  q   C   s  2 C
t
For the porous layer,

m  qm  0
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
 0  qm 
 
2
(6)

    m Pm  m qm  qm
  t 
K
T

A m   qm   m  Tm   m m2 Tm
(7)
t
C

 m   qm .  Cm   sm 2Cm
(8)
t

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, 0 is the fluid density, A 
 C 
 C 
0
p m
p
is the ratio of heat capacities, C p is the
f
specific heat, K is the permeability of the porous medium, T is the temperature,  is the thermal diffusivity of the
fluid, C is the concentration,  s is the solute diffusivity of the fluid,  is the porosity, and the subscripts m and
f refer to the porous medium and the fluid respectively. The basic steady state is assumed to be the quiescent and we
consider the solution of the form,
u , v, w, P, T , C    0, 0, 0, Pb  z  , Tb  z  , Cb  z 
(9)
in the fluid layer
and in the porous layer
(10)
um , vm , wm , Pm , Tm , Cm   0, 0, 0, Pmb  zm  , Tmb  zm  , Cmb ( zm ) 
where the subscript ‘b’ denotes the basic state. The temperature distributions Tb  z  , Tmb  zm  , are found to be
T  T  z in 0  z  d
Tb  z   T0  u 0
(11)
d
T
 T  z
Tmb  zm   T0  l 0 m in 0  zm  dm
(12)
dm
 d T   m dTl
is the interface temperature.
T0  m u
 dm   m d
The concentration distributions Cb  z  , Cmb  z m  , are found to be
Cb C0  Cu

h  z  in 0  z  d
z
d
C
C  C0
 mb  L
hm  z m  in 0  zm  dm
zm
dm

Here
h( z ), hm ( zm )
salinity
gradients
in
fluid
and
(14)
porous
layers
respectively
the
basic
state.
such
dm
d
that
are
(13)
 h  z  dz  d and  h  z  dz
m
0
m
m
 d m .The
b denotes
subscript
At
the
0
interface h  z   hm  zm  and note that C0 
 d mCu   m dCl
 dm  md
is concentration at the interface.
In order to investigate the stability of the basic solution, infinitesimal disturbances are introduced in the form,


 q, P, T , C   0, Pb  z  , Tb  z  , Cb  z     q , P,  , S 
(15)
And


 qm , Pm , Tm , Cm    0, Pmb  zm  , Tmb  zm  , Cmb  z m    qm , Pm ,  m , S m 
(16)
where the primed quantities are the perturbed ones over their equilibrium counterparts. Now Equations (15) and (16)
are substituted into the Equations (1) to (8) and are linearized in the usual manner. Next, the pressure term is
eliminated from (2) and (6) by taking curl twice on these two equations and only the vertical component is retained.
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d2 
The variables are then non-dimensionalized using d ,
,
, T0  Tu and C0  Cu as the units of length, time
 d
d 2 m
velocity, temperature, and the concentration in the fluid layer and dm , m ,
, Tl  T0 , Cl  C0 as the
 m dm
corresponding 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 manner the detailed flow fields in both the fluid and porous layers can be clearly obtained for all the depth
ratios dˆ 
dm
.
d
The dimensionless equations for the perturbed variables are given by,
1  2 w
 4w
(17)
Pr t

 w   2
(18)
t
S
 w h( z )    2 S
(19)
t
 2  2m wm
ˆ 2 4m wm   2m wm
 
(20)
Prm t

A m  wm   m2  m
(21)
t
S
 m  wm hm ( zm )   m m2 Sm
(22)
t


For the fluid layer Pr 
is the Prandtl number,   s is the diffusivity ratio in the fluid layer. For the porous


 m
K

layer, Prm 
is the Prandtl number,  2  2  Da is the Darcy number, ˆ  m is the viscosity

m
dm

ratio,  m  sm is the diffusivity ratio in the porous layer. We make the normal mode expansion and seek solutions for
m
the dependent variables in the fluid and porous layers according to
 w  W  z  
      z  f x , y e nt

     
 S   S  z  
(23)
And
 wm  Wm  zm  
      z  f x , y e nmt
(24)
 m   m  m   m m 
 S m   S m  zm  
2
2
2
2
With  2 f  a f  0 and  2 m f m  am f m  0 , where a and am are the non-dimensional horizontal wave numbers,
n and nm are the frequencies. Since the dimensional horizontal wave numbers must be the same for the fluid and
a am
ˆ . Substituting Equations (23) and (24) into the Equations
porous layers, we must have
and hence am  da

d dm


(17) to (22) and denoting the differential operator
and
by D and Dm respectively, an Eigen value
z
z m
problem consisting of the following ordinary differential equations, is obtained,
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In 0 
ISSN 2319 - 4847
z 1,
n 2
 2
2
2
 D  a    D  a W  0
Pr


2
 D  a2  n   W  0
(25)
(26)
  D 2  a 2   n  S  W h( z )  0 (27)


In
0  zm  1
 2
n 2 
ˆ 2  m  1  Dm2  am2 Wm  0
 Dm  am2  
Prm


(28)
2
2
(29)
 Dm  am  nm A m  Wm  0
 m  Dm2  am2   nm  Sm  Wm hm ( zm )  0


(30)
It is known that the principle of exchange of instabilities holds for Double Diffusive Marangoni convection in both fluid
and porous layers separately for certain choice of parameters. Therefore, we assume that the principle of exchange of
instabilities holds even for the composite layers. In other words, it is assumed that the onset of convection is in the
form of steady convection and accordingly, we take n  nm  0 .
We get,
In
0  z 1
2 2
 D  a  W (z)  0
 D  a  ( z )  W ( z )  0
  D  a  S ( z)  W ( z ) h  z   0
ˆ  1  D  a W
In 0  z  1  D  a  


 D  a   (z )  W (z )  0
  D  a  S (z )  W ( z ) h  z   0
2
2
2
2
2
m
m
2
m
2
m
m
(32)
2
m
2
m
(31)
m
2
m
m
2
2
m
m
2
m
(33)
m ( z m )  0 (34)
(35)
2
m
m
m
m
m
m
m
(36)
Thus to solve the above ordinary differential equation we need 16 boundary conditions.
3. BOUNDARY CONDITIONS
The bottom boundary is assumed to be rigid and insulating to temperature and concentration so the boundary
conditions at
zm  0 are
wm  0,
wm
Tm
S m
 0,
 0,
 0
zm
zm
zm
(37)
The upper boundary is assumed to be free, insulating to temperature and concentration so the appropriate boundary


2w
T
S
  t 22T   t  22 S  ,
 0,
 0 where
conditions
at
z  d are
2
z
T
C
z
z
  
  
 t   0   T T   S S is the surface tension, here  T    t 
, S   t 
At the interface
 T T  T0
 S C  C0
w  0,
(i.e., at
ˆ
z  0, zm  dm ), the normal component of velocity, tangential velocity, temperature, heat flux, mass and mass
flux are continuous and respectively yield (following Nield (1977)),
T
T
S
S
(38)
  m m , S  Sm , 
  sm m
z
z m
z
z m
We note that two more velocity conditions are required at z  0. Since we have used the Darcy-Brinkman equations of
w  wm ,
w wm

, T  Tm ,
z zm

motion for the flow through the porous medium, the physically feasible boundary conditions on velocity are the
following, at the interface z  0 and zm  dm
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Pm  2 m
wm
w
 P  2
z m
z
ISSN 2319 - 4847
(39)
which will reduce to


 2  w
 w
 2  w
  3 22  2 
  m m   m  2  3 22 m  2  m
z  z
K z m
zm  zm


The other appropriate velocity boundary condition at the interface z  0, zm  d m can be,
  2 wm

 2w
2 
   2   2 w   m   2   22 m wm 
 z

 zm

(40)
All the Sixteen boundary conditions (35) to (40) are non-dimensionalised and are subjected to Normal mode expansion
and are given by
D 2W (1)  M a 2  (1)  M s a 2 S (1)  0,
W (1)  0,
D  (1)  0,
DS (1)  0
ˆ (0)  W (1), TdDW
ˆˆ
ˆ ˆ 2  D 2  a 2 W (0)  ˆ  D 2  a 2 W (0)
TW
(0)  DmWm (1), Td
m
m
m
m
ˆ ˆ 3 2  D3W (0)  3a 2 DW (0)    D W 1  
ˆ 2  Dm3Wm 1  3am2 DmWm 1 
Td
m m
(0)  Tˆ  m (1),
wm  0   0,
ˆ (1),
D(0)  Dm  m (1), S (0)  SS
m
Dm wm (0)  0,
DS (0)  Dm S m (1),
Dm m (0)  0, Dm Sm (0)  0
(41)
 t T0  Tu  d
 t  C0  Cu  d
is the thermal Marangoni number, M s  
is the solute
T
v
S
v
Marangoni number, while Tˆ   TL  T0  /  T0  TU  , Sˆ   CL  C0  /  C0  CU  , and dˆ  d m / d is the depth
ratio. We see that ˆ   /   dˆ / Tˆ and ˆ   /   dˆ / Sˆ because the steady state heat and mass fluxes are
Where
M 
m
s
sm
s
continuous across the interface. The Equations (31) to (36) are to be solved with respect to the above boundary
conditions (41).
4. EXACT SOLUTION
The solutions of the Equations (31) and (34) are independent of   z  ,
can be solved and expressions for
S  z  ,  m  zm  , S m  zm  and thus
W and Wm can be obtained as,
W  z   C1Cosh  az   C2 zCosh  az   C3 Sinh(az )  C4 zSinh(az )
Wm  z m   C5Cosh  am zm   C6 Sinh  am zm   C7Cosh  zm   C8 Sinh  zm 
where

(42)
(43)
1
 am2 , and the expressions for W ( z ) and Wm ( z ) are
2
ˆ am
W  z   C1 Cosh  az   A1 zCosh  az   A2 Sinh(az )  A3 zSinh(az ) 
(44)
Wm  zm   C1  A4Cosh  am z m   A5 Sinh  am zm   A6Cosh  zm   A7 Sinh  zm  
(45)
where
A1 , A2 , A3 , A4 , A5 , A6 , A7 are constants which are determined using corresponding velocity the boundary
conditions
(41)1 , (41)5 , (41)6 , (41)7 , (41)8 , (41)13 , (41)14 as
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Volume 3, Issue 10, October 2014
A1  A4 3  A5 4 ,
A4 
A5 
A2  A4 1  A5  2 ,
1 ˆ
10
T 
9 
8 9  10 7
A3   a  A4  5  A5 6
 aSinh(a)  Cosh(a)  
 aSinh(a)  Cosh(a)  
9
 Tˆ 7
8 9  10  7
,
ISSN 2319 - 4847
A6   A4 ,
9

 Tˆ  7 


A7 
 A5 am

's
And  i are
1 
1
 1  3am2 
ˆ 2  am Sinh(am )   Sinh( )   
ˆ 2 am3 Sinh(am )   3 Sinh( ) 
3 ˆ ˆ3 2 

2a Td 
2 
1
 1  3am2 
ˆ 2  Cosh(am )  Cosh( )  am  
ˆ 2 am2 Cosh(am )   2Cosh( ) am 
3 ˆ ˆ3 2 

2a Td 
3 
1
ˆ ˆ  ,   1 a (Cosh(a )  Cosh( ))  Tda
ˆˆ 
am Sinh(am )   Sinh( )  Tda
1
4
m
m
2
ˆ
ˆ
ˆ
ˆ
Td
Td








1
ˆ ˆ2
2aTd
1
6 
ˆ ˆ2
2aTd
5 








 ˆ 2am2 Cosh(am )   2  am2 Cosh( ) 


  2
am 2

2
 ˆ  2am Sinh(am )     am Sinh( )  

 
7  3Cosh(a )  1Sinh(a)   5 Sinh(a ),  8   4Cosh(a)   2 Sinh(a)   6 Sinh(a )


 9  Cosh  am   Cosh   , 10  Sinh  am  
am
Sinh  

The Temperature distributions are obtained from the Equations (32) and (35) by substituting expressions for
W and
Wm , are as below
1
Az
Az
[ Sinh (az )(2 z  A1 z 2  3 )  Cosh (az )(2 A2 z  1  A3 z 2 )]}
4a
a
a
Az
 m  z m   C1{ A10Cosh  am zm   A11Sinh  am z m   4 m Sinh(am zm ) 
2 am
  z   C1{ A8Cosh  az   A9 Sinh  az  
A5 zm
A
A
Cosh(am zm )  2 6 2 Cosh  zm   2 7 2 Sinh  z m }
2am
  am
  am
The constants
A8 , A9 , A10 , A11 are determined using temperature boundary conditions (41)3 , (41)9 , (41)10 ,
(41)15 and are obtained as below.
ˆ
A8  A10TCosh
 am    4 , aA9  A10 am Sinh  am    5
A10 
Where
1  a 4 Sinh  a    5Cosh  a 
ˆ
TaCosh
 am  Sinh  a   amCosh  a  Sinh  am 
,
A11 
A 
1  A5
 27 2

am  2am   am 
 i's are
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Volume 3, Issue 10, October 2014
1 

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A 
A 
1 


aCosh  a   2  A1  3   Sinh  a   2  2 A1  3 

4a 
a
a 


A
A
1 




aSinh  a   2 A2  1  A3   Cosh  a   2 A2  1  2 A3 

4a 
a
a




2 
A4
A
A
A
Sinh  am   5 Cosh  am   2 6 2 Cosh    2 7 2 Sinh  
2am
2am
  am
  am
3 
A
A
A
A4
Sinh  am   amCosh  am    5  Cosh  am   am Sinh  am    2 6 2 Sinh    2 7 2 Cosh  

2am
2am
  am
  am
4 
1  A5
A  ˆ
 2 7 2  TSinh
 am   Tˆ2

am  2am   am 
5 
1 
A1   A5
A 
 2 7 2  Cosh  am   3
 2 A2    
4a 
a   2am   am 
4.1 Linear Salinity Profile
We consider linear salinity profile of the form h
 z   hm  zm   1 ,
substituting this in eq.(33) and (36) the
expressions for S  z  and S m  z m  are obtained as
S  z   C1  A12Cosh  az   A13Sinh  az  
1 
Az
Az


 
Sinh  az   2z  A1z2  3   Cosh  az   2A2 z  1  A3 z2 

4a 
a 
a


 
S m  zm   C1  A14Cosh  am zm   A15 Sinh  am zm  
 
1  A4 zm
Az
A
A
Sinh  am zm   5 m Cosh  am zm   2 6 2 Cosh  z m   2 7 2 Sinh  z m   

 pm  2am
2am
  am
  am
 
The constants
A12 , A13 , A14 , A15 are determined using salinity boundary conditions (41)4 , (41)11 , (41)12 , (41)16 and
are obtained as below.
 
Sinh am   A5
A7  
A12  Sˆ  6 10 Cosh am  
  2 2  7 
am pm  2am  am  
 9
 
Cosh am   A5
A7  
6 10
1  A5
A7 
aA13   6 10 amSinh am  
, A15 
  2 2  3, A14 
  2 2
pm  2am  am  
9
am pm  2am  am 
 9
Where
 i' s , for i = 6 to 10 are
6 
1
4a

A3 
A3 


 Sinh  a   2  2 A1  a   aCosh  a   2  A1  a  





A
A 


Cosh  a   2 A2  2 A3  1   aSinh  a   2 A2  A3  1 
a
a 


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Volume 3, Issue 10, October 2014
ISSN 2319 - 4847
7 

1  A4
A
A
A
Sinh  am   5 Cosh  am   2 6 2 Cosh    2 7 2 Sinh   

 pm  2am
2am
  am
  am

8 
1
4a
A1  1

 2 A2   
a   pm

 A4
A
A
A
Sinh  am   4 Cosh  am   5 Cosh  am   5 Sinh  am 

2
2am
2
 2am

A6
A
Sinh    2 7 2 Cosh   
2
  am
  am

2
ˆ
 9  aSSinh
 a  Cosh  am   amCosh  a  Sinh  am 
10 
ˆ
aSSinh
 a  Sinh  am   A5
A  ˆ
 2 7 2   SaSinh
 a  7 

am pm
 2a m   a m 
Cosh  a  Cosh  am   A5
A 
 2 7 2    8Cosh  a 

 pm
 2am   am 
The Thermal Marangoni number for Linear Salinity Profile is obtained by the boundary condition
Mt 

 11  M s a 2 12
(41) 2 as

2
a 13
Where




11  C1  a 2Cosh  a   A1 2aSinh  a   a 2Cosh  a   A2 a 2 Sinh  a   A3 2aCosh  a   a 2 Sinh  a  

A3 
A1


 
 Sinh  a   2  A1  a   Cosh  a   2 A2  a  A3   



 

1
A
A
13  C1{ A8Cosh  a   A9 Sinh  a   [ Sinh (a )(2  A1  3 )  Cosh (a )(2 A2  1  A3 )]}
4a
a
a
12  C1  A12Cosh  a   A13 Sinh  a  
1
4a
4.2 Parabolic Salinity Profile
We consider parabolic salinity profile of the form h
 z   2z, hm  zm   2zm , substituting this in eq.(33) and (36), the
expressions for S  z  and S m  z m  are obtained as
 z 2 A1 z 3 A1 z A2 z A3 z 2 
2
S  z   C1  A16Cosh  az   A17 Sinh  az    Sinh  az  

 3 2


4a 2 
 4a 12a 4a 4a
 z
A z 2 A z 2 A z 3 A z   
Cosh  az    2  1 2  2  3  3 3   
4a
4a
12a 4a   
 4a
 A4 zm2 A5zm 
 A5zm2 A4zm 
2 
Sm  zm   C1  A18Cosh amzm   A19Sinh amzm    Sinh amzm  
 2   Cosh amzm  
 2 
 pm 
 4am 4am 
 4am 4am 

Az
2 A7
Cosh  zm   2 6 m 2 
   am  2  a 2
m


The constants



  Sinh  z   A7 z m  2 A6
m
2

  2  am2
 2  am2






2 


A16 , A17 , A18 , A19 are determined using the salinity boundary conditions (41) 4 , (41)11 , (41)12 ,
(41)16 and are obtained as below.
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Volume 3, Issue 10, October 2014
ISSN 2319 - 4847
p

p
p
A16  Sˆ  7 Cosh  am   7 Sinh  am   p3  , aA17  7 am Sinh  am   p5Cosh  am   p6 ,
am
p8
 p8

p
A18  7 , am A19  p5
p8
's
Where pi , for i = 1 to 8 are
p1 
p2 
A3 
2  1
 2 3
  4 a 4a 
2
A
A
A 
A
A
A
A 
 1 A1
 1
Sinh  a  

 13  22  32   aCosh  a  
 1  13  22  32 


 2a 4a 4a 4a 2a 
 4a 12a 4a 4a 4a 
A
A
A
A 
A
A
A
A 
 1
 1
Cosh  a   2  12  2  3  33   aSinh  a   2  12  2  3  33 
2a
2a 4a 4a 
4a
4a 12a 4a 
 4a
 4a
p3 
 A
 A5
2 
A5 
A 
 42 
 Sinh  am   4  2   Cosh  am  
 pm 
 4am 4am 
 4am 4am 

A
2 A7
Cosh    2 6 2 
   am  2  a 2
m


p4 
 A4
2 
A
 52
 am Cosh  am  
 pm 
 4am 4 am



  Sinh    A7  2 A6
2

  2  am2
 2  am2




 A4
A
 52
  Sinh  am  

 2 am 4 am

 A5
A
2 A6
A4 
Cosh  am  
 2    Cosh    2 7 2 
   am  2  a 2
 2 am 4 am 
m



A
2 A7
 Sinh    2 6 2 
   am  2  a 2
m


p5 





2 



 A5
A 
 42 
  am Sinh  am  

 4am 4am 

  Sinh    A7 
 2
2
2 

   am 


  Cosh    A6 
 2
2
2 

   am 



2  A4
2 A6
 2  

2
2

 pm  4am
   am





  A6  ,
2
  2  am2 


p6  p4  p1
p ˆ
ˆ
p7  p2  p6Cosh  a   p5Cosh  am  Cosh  a   p3 SaSinh
 a   5 SaSinh
 a  Sinh  am 
am
ˆ
p  SaSinh
 a  Cosh  a   aCosh  a  Sinh  a 
8
m
m
The Thermal Marangoni number for Parabolic Salinity Profile is obtained by the boundary condition
Mt 

 11  M s a 2 12
(41) 2 as

2
a 13
Where




11  C1  a 2Cosh  a   A1 2aSinh  a   a 2Cosh  a   A2 a 2 Sinh  a   A3 2aCosh  a   a 2 Sinh  a  
12  C1  A16Cosh  a   A17 Sinh  a  
Volume 3, Issue 10, October 2014
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Volume 3, Issue 10, October 2014
ISSN 2319 - 4847
2
A
A
A
A 
A
A
A
A  
 1
 1
Sinh  a  
 1  13  22  32   Cosh  a    2  12  2  3  33   


4a 12a 4a   
 4a 12a 4a 4a 4a 
 4a 4a
1
A
A
13  C1{ A8Cosh  a   A9 Sinh  a   [ Sinh (a )(2  A1  3 )  Cosh (a )(2 A2  1  A3 )]}
4a
a
a
4.3 Inverted Parabolic Salinity Profile
We consider parabolic salinity profile of the form h
 z   2 1  z  , hm  zm   2 1  zm  , substituting this in eq.(33)
and (36) the expressions for S  z  and S m  z m  are obtained as
 2 z  A1 z 2  z 2 A1 z 3 A1 z A3 z  A2 z  A3 z 2 
2
S  z   C1  A20Cosh  az   A21Sinh  az    Sinh  az  





4a
12a 4a 3
4a 2


 2 A2 z  A3 z 2  A2 z 2 A3 z 2 A3 z A1 z  z  A1 z 2   
Cosh  az  



 

4a
12a 4a 3
4a 2

  
S m  zm   C1  A22Cosh  am zm   A23 Sinh  am zm  
  A4 zm A4 zm2 A5 zm 
 A z
2 
A z2 A z 

 2   Cosh  am zm   5 m  5 m  4 2m 
 Sinh  am zm  
 pm 
4am 4am 
4am 4am 
 2am
 2am

A
Az
2 A7
Cosh  zm   2 6 2  2 6 m 2 
   am   am  2  a2
m


The constants



  Sinh  z    A7  A7 zm  2 A6
m
2

  2  am2  2  am2  2  a2
m






2 


A20 , A21 , A22 , A23 are determined using salinity boundary conditions (41)4 , (41)11 , (41)12 , (41)16 and
are obtained as below.


I
A20  Sˆ  A22Cosh  am   5 Sinh  am   I3  , aA21  A22 am Sinh  am   I 5Cosh  am   I1  I 4
am


I
A22  6 ,
am A23   I5
I7
's
Where Ii , for
i  1to 7 are
I1 
I2 
A3 2 A2
A 
2  1
 12 
 2 3
  4a 4a
4a 4a 
2
A
A A A 
A
A 
 2A
 1  A1 A1
Sinh  a    1  13  2 2 3  1   aCosh  a  

 13  22 


4a
4a 
12a 4a 4a 
 4a 4a
 4a
A 
A
A 
1
 A 1 2 A 2 A
 A  A3
Cosh  a    1 2  3  3  33   aSinh  a    2
 2  3  33 
4a 12a 4a 
4a
4a 12a 4a 
 4a

I3 
 A
2 
A
 Sinh  am    4  52
 pm 
 4am 4am

2 A7
Cosh    
  2  a2
m


Volume 3, Issue 10, October 2014


 A5
A 
 42 
  Cosh  am   

 4am 4am 


  Sinh     2 A6
2

  2  a2
m




2


 

Page 268
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Volume 3, Issue 10, October 2014
I4 
 A
 A 
2 
A 
 amCosh  am    4  52   Sinh  am    52  
 pm 
 4am 4am 
 4am 

2 A6
 Cosh    
  2  a2
m


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


  Cosh    A6  
2

 2  am2  





  A6 
  2  am2 




  Sinh    A7    Sinh     2 A7
2

  2  a2
 2  am2 

m





2  A5
A4
A
2 A6
I5 

 2   2 7 2 
   am  2  a 2
 pm  2am 4am
m



ˆ
I 6   I 2  I 4Cosh  a   I5Cosh  am  Cosh  a   I 3 SaSinh
a 

2

I5 ˆ
SaSinh  a  Sinh  am   I1Cosh  a 
am
ˆ
I 7  SaSinh
 a  Cosh  am   amCosh  a  Sinh  am 
The Thermal Marangoni number for inverted Parabolic Salinity Profile is obtained by the boundary condition
Mt 

 11  M s a 2 12
(41) 2 as

2
a 13
Where




11  C1  a 2Cosh  a   A1 2aSinh  a   a 2Cosh  a   A2 a 2 Sinh  a   A3 2aCosh  a   a 2 Sinh  a  
A
A
A 
2
 1  A1
12  C1  A20Cosh  a   A21 Sinh  a    Sinh  a  
 1  13  22  

12 a 4 a
4a 
 4a
A
1  
  A  A3 A3
Cosh  a   2

 33  2   
12a 4a 4a   
 4a
1
A
A
13  C1{ A8Cosh  a   A9 Sinh  a   [ Sinh (a )(2  A1  3 )  Cosh (a )(2 A2  1  A3 )]}
4a
a
a
5. INTERPRETATIONS
The Thermal Marangoni numbers for the profiles, namely, linear, parabolic and inverted parabolic profiles for different
parameters are presented graphically as a function of depth ratio d̂ by fixing the other parameters. The effects of the
variations of the parameters like
Horizontal Wave number
number M s , Diffusivity ratio  , and the Darcy number
a , Viscosity ratio ˆ 
m
, Solute Marangoni

Da on the thermal Marangoni number is displayed in figures
2,3, 4,5 and 6 .
800
PARABOLIC PROFILE
600
LINEAR PROFILE
400
INVERTED
PARABOLIC PROFILE
200
0
0.0
0.2
0.4
0.6
0.8
1.0
Fig.1: The variation of thermal Marangoni number for Linear, Parabolic and Inverted parabolic profiles with respect to
the depth ratio. Figure1 shows the variation of thermal Marangoni numbers for different profiles with respect to the
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Volume 3, Issue 10, October 2014
depth ratio for fixed values of
ISSN 2319 - 4847
Da  10, Sˆ  1, Tˆ  1,  1, pm  1, a  5, M s  10,   5. Here the thermal
Marangoni numbers for the profiles differ only for smaller values of depth ratios. Graphically it is evident that the
parabolic salinity profile is the most stable one and the inverted salinity profile is the unstable one, so by choosing the
appropriate salinity profile one can control the onset of double diffusive Marangoni convection in a composite layer in
microgravity condition.
500
120
a=5
150
a=5
400
100
a=4
80
300
a=4
a=5
100
60
a=3
a=4
a =3
200
a=3
40
50
100
20
0
0
0
2
4
6
8
10
0
2
4
Fig. (a)
6
8
10
0
0
Fig (b)
2
4
6
8
10
Fig. (c)
a  3, 4,5 on the Thermal Marangoni numbers M t
Fig 2. The effects of horizontal wave number
The effects of ' a ' horizontal wave number on the Thermal Marangoni numbers in linear, parabolic and inverted
parabolic profiles are shown in Figures (a), (b) and (c) respectively for fixed values of
Da  10, Sˆ  1, Tˆ  1,  1, pm  1, M s  10,   5. The line curve is for a  3 , the big dotted curve is for
a  4 and the small dotted line curve is for a  5 . The curves for all the profiles are converging, indicating that for
larger values of depth ratios, the corresponding thermal Marangoni numbers coincide. The effect of horizontal wave
number is same for all the profiles, that is the increase in the value of the horizontal wavenumber a, the value of the
thermal Marangoni number increases, so the onset of double diffusive Marangoni convection is delayed and hence the
system is stabilized.
500
200
250
180
400
160
=3
140
=4
200
=5
120
=3
=5
=4
=3
300
200
=4
150
=5
100
100
80
100
50
60
0
2
4
6
8
10
0
0
Fig. (a)
2
4
6
Fig. (b)
8
10
0
0
2
ˆ 
m

6
8
10
Fig. (c)
Fig.3. The effects of ˆ  3, 4,5 on the Thermal Marangoni number
The effects of the viscosity ratio
4
Mt
which is the ratio of the effective viscosity of the porous medium to the
fluid viscosity are displayed in Figures (a), (b) and (c) respectively for fixed values of Da  10, Sˆ  1, Tˆ  1,
  1,
 pm  1, a  5, M s  10. The line curve is for   3 , the big dotted curve is for   4 and the small dotted line
curve is for   5 . The curves for the parabolic profile are converging at both the ends, (fig. 3(b)) that is, the effect
of the viscosity ratio is only for the values of depth ratio 0.2  dˆ  10 , so the effect of the viscosity ratio is limited to
this range of depth ratio. In this range, for a fixed value of depth ratio, the increase in the value of viscosity
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Volume 3, Issue 10, October 2014
ratio ˆ

ISSN 2319 - 4847
m
increases the thermal Marangoni number. Whereas the effect of the viscosity ratio is opposite to that for

linear and inverted parabolic salinity profile. The curves for the linear and inverted parabolic profile are diverging and
the effect of the viscosity ratio is larger for larger values of the depth ratio. For a fixed value of depth ratio, the increase
in the value of viscosity ratio decreases the thermal Marangoni number and so destabilizes the system and hence the
onset of the double diffusive Marangoni convection is earlier.
200
200
Ms = 5
600
Ms = 10
500
150
150
Ms = 5
Ms = 15
Ms = 15
400
100
Ms = 10
100
Ms = 10
300
Ms =15
Ms = 5
200
50
50
100
0
0
2
4
6
8
10
0
2
4
Fig. (a)
6
8
10
0
0
Fig. (b)
Fig.4. The effects of
2
6
8
10
M s  5,10,15 on the Thermal Marangoni number M t
The effects of the Solute Marangoni number
M s are displayed in Fig. a ,b and c for the linear, parabolic and inverted
parabolic salinity profiles respectively for fixed values of Da  10, Sˆ  1, Tˆ  1,  1,
line curve is for
4
Fig. (c)
 pm  1, a  5,   5. The
M s  5 , the big dotted curve is for M s  10 and the small dotted line curve is for M s  15 .The
curves for the parabolic profile are converging at both the ends, that is, the effect of the viscosity ratio is only for the
range of values of depth ratio
number
1  dˆ  9 .The increasing values of M s increases the value of the Thermal Marangoni
M t i.e., to stabilize the system, so that, onset of surface driven double diffusive convection is delayed.
Whereas the curves for the linear and inverted parabolic profile, are converging for larger values of depth ratio and
the increase in the solute Marangoni number has no much effect of the thermal Marangoni number for larger values
of the depth ratio. For a fixed value of depth ratio
d̂ , the increase in the values of of the solute Marangoni number
decreases the thermal Marangoni number, so the double diffusive Marangoni convection sets in earlier and hence
destabilizes the system.
=1
140
700
=0.5
120
=1
600
150
100
= 0.25
500
= 0.75
80
= 0.25
= 0.5
=1
400
100
300
= 0.5
60
40
200
20
50
100
0
2
4
6
8
10
Fig. (a)
Fig.5. The effects of 
Volume 3, Issue 10, October 2014
0
2
4
6
Fig. (b)
8
10
0
0
2
4
6
8
10
Fig. (c)
 0.5, 0.75,1 on the Thermal Marangoni number M t
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Volume 3, Issue 10, October 2014
The effects of the diffusivity ratio
in fluid layer are displayed in Figures (a), (b) and (c) for linear, parabolic and
Da  10, Sˆ  1, Tˆ  1,
 1, a  5, M s  10,   5. The line curve is for   0.5 , the big dotted curve is for   0.75 and the small
inverted
 pm

ISSN 2319 - 4847
parabolic
salinity
profiles
respectively
for
fixed
values
of
dotted line curve is for 
 1 . The curves for the parabolic profile are converging at both the ends, that is, the effect of
the viscosity ratio is only for the values of depth ratio 1  dˆ  10 , whereas the curves for the linear and inverted
parabolic profile are converging. The increase in the values of  increases the value of the Thermal Marangoni
number M t for the linear and inverted parabolic salinity profile, to stabilize the system, so the onset of surface driven
double diffusive convection is delayed, where as the same decreases the corresponding thermal Marangoni number for
the parabolic salinity profile.
250
500
Da = 10
Da =10
Da = 10
800
200
400
300
Da = 9
Da = 9
Da = 9
600
Da = 8
Da = 8
150
400
100
200
200
100
0
50
0
0.0
0.2
0.4
0.6
0.8
1.0
0
0.0
Fig. (a)
Fig.6. The effects of
inverted
parabolic
salinity
0.2
0.4
0.6
Fig.(b)
The effects of the Darcy number Da 
Da = 8
2 
profiles
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Fig.(c)
Da  8,9,10 on the Thermal Marangoni number M t
K
, are displayed in Figures (a), (b) and (c) for linear, parabolic and
d m2
respectively
for
fixed
values
of
Sˆ  1, Tˆ  1,   1,
 pm  1, a  5, M s  10,   5 The line curve is for Da  8 , the big dotted curve is for Da  9 and the small
dotted line curve is for Da  10 . The effect of The Darcy number is visible only for small values of depth ratios. The
effect of Darcy number is same for all the profiles. For a fixed value of depth ratio, increase in the value of Darcy
number increases the thermal Marangoni number for all the profiles, that is this stabilizes the system, so the onset of
surface driven double diffusive convection is delayed, this may be due to the presence of second diffusing component.
6. CONCLUSIONS
1. The parabolic salinity profile is the most stable one and the inverted salinity profile is the unstable one.
2. For various variations of the parameters, the effect of the parabolic salinity profile is opposite to those of linear and
inverted parabolic salinity profiles except for that of Darcy number.
3. The variation of Darcy number has same effect on the onset on the double diffusive Marangoni convection for all
the profiles.
4. The increase the values of horizontal wave number a , the viscosity ratio and solute Marangoni number and the
decrease in the values of diffusivity ratio in the fluid layer stabilizes the system for parabolic salinity profile,
where as the same destabilizes the system for the linear and inverted parabolic salinity profile.
Acknowledgements
I express my gratitude to Prof. N. Rudraiah and Prof. I.S. Shivakumara, UGC-CAS in Fluid mechanics, Bangalore
University, Bangalore, for their help during the formulation of the problem.
REFERENCES
[1] I.S. Shivakumara, N. Rudraiah and C.E. Nanjundappa “Effect of non-uniform basic temperature gradients on
Rayleigh-Bernad- Marangoni convection in ferrofluids” Journal of Magnetism and Magnetic Materials, 248,379395, 2002.
Volume 3, Issue 10, October 2014
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org
Volume 3, Issue 10, October 2014
ISSN 2319 - 4847
[2] I.S.Shivakumara, S.Sureshkumar and Devaraju N “Effect of non-uniform temperature gradients on the onset of
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[5] Siti Suzillian Putri Mohamed Isa , Norihan Md. Arifin, Mohd Noor Saad and Roslinda Mohd Nazar “Effects of
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Onset of Double-Diffusve Convection in Micropolar Fluid”, Scientific Research an academic publisher,
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[7] Sumithra R and Manjunath N ‘Effects of parabolic and inverted parabolic temperature profiles on magneto
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5435-5450, March 2014.
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