International Journal of Application or Innovation in Engineering & Management (IJAIEM)
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Volume 3, Issue 2, February 2014
ISSN 2319 - 4847
DOUBLE DIFFUSIVE MAGNETOMARANGONI CONVECTION IN A
COMPOSITE LAYER
Dr. R. Sumithra
Department of Mathematics, Government Science College
Bangalore-560 001,Karnataka
INDIA
ABSTRACT
The Hydrothermal growth of crystals is mathematically modeled as the onset of double diffusive magneto-marangoni convection
in a two-layer system comprising an incompressible two component, electrically conducting fluid saturated porous layer over
which lies a layer of the same fluid in the presence a vertical magnetic field. The upper boundary of the fluid layer is made free
at which act the surface tension effects depending on both the temperature and species concentration and the lower boundary of
the porous layer is rigid. Both the boundaries are insulating to both 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 eigenvalue problem is solved by regular perturbation technique. The critical thermal Rayleigh number, which is the
criterion for stability of the system is obtained. The effects of different physical parameters on the onset of double diffusive
magneto-marangoni-convection are investigated in detail which enables to control convection during the growth of crystals in
order to obtain pure crystals.
Key words: Double Diffusive Magneto Marangoni Convection, Darcy Brinkman model, Regular Perturbation Method
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. The apparatus in which the hydrothermal growth is carried out consists of an autoclave which has two layers.
The bottom layer consists of nutrient chips along with the suitable mineralizer to increase solubility of the nutrient in a
suitable solvent. The bottom layer is heated to obtain the hydrothermal conditions. In the top layer the seeds of the
crystals to be grown are suspended by means of a metal frame. A perforated metal disc called a baffle is often placed
within the autoclave separating the two layers to aid in localizing the temperature differential. Convections carries hot
saturated fluid from nutrient zone to the growth zone, where the fluid relieves its super saturation by deposition onto the
crystal, which grows on the seed crystals placed in that region. This situation exactly simulates double diffusive
convection in a composite layer. The idea of using magnetic field is to dampen melt turbulence and thereby improve
microscopic homogeneity of the crystal has been first introduced independently byUtech and Flemings [13] and Chedzey
and Hurle [1]. In addition to damping out the turbulence and thereby removing the dopant striations, the magnetic field
can be used to control the growth conditions at various stages in the growth process.
Single component convection in composite layers is investigated by Many of the researchers started by Nield[5] Rudraiah
[7], Taslim and Narusawa [12], McKay [4] and Chen [3].
Shivakumar et al [8] 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 agreement. Recently Sumithra and Manjunatha (2012) have discussed the problem of surface tension driven
single component magneto convection in a composite layer and obtained an exact solution of the problem.
Multicomponent convection in composite layers is prominent in crystal growth and solidification of alloys. Chen and
Chen [2] 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 [6]. Venkatachalappa et al [14] have
investigated the double diffusive convection in composite layer conducive for hydrothermal growth of crystals with the
lower boundary rigid and the upper boundary free with deformation. Here, in the present investigation the onset of
double diffusive convection in a composite layer horizontally bounded by rigid walls in the presence of vertical magnetic
field is considered. Recently Sumithra [10] has investigated the onset of double diffusive magnetoconvection in a
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Volume 3, Issue 2, February 2014
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composite layer with the both the walls bounded by rigid boundaries by using the regular perturbation technique, also
considered the problem (Please see [9]) of triple diffusive convection in a composite layer in the absence of magnetic field
and the resulting eigenvalue problem is solved exactly. The problem considered here also has many engineering
applications like the moisture migration in thermal insulation and stored grain, underground spreading of chemical
pollutants, waste and fertilizer migration in saturated soil and petroleum reservoirs.
2. FORMULATION OF THE PROBLEM
We consider a horizontal two - component , electrically conducting fluid saturated isotropic sparsely packed porous layer
of thickness d m underlying a two component fluid layer of thickness d with an imposed magnetic field intensity H 0 in
the vertical z – direction. The lower surface of the porous layer is bounded by a rigid wall and the upper surface of the
fluid layer is made free without deformation, at which the surface tension effects depending on both temperature and
species concentration (salinity) are considered.
Both the boundaries are kept at different constant temperatures and
salinities. A Cartesian coordinate system is chosen with the origin at the interface between porous and fluid layers and
the z – axis, vertically upwards. For the fluid layer, the continuity, solenoidal property of the magnetic field, momentum,
energy, species concentration, magnetic induction and the equation of state respectively are,

q  0
(1)

 H  0
(2)






 q
 0    q   q   P   2 q   gkˆ   p H  H (3)
 t

T

  q   T   2T
(4)
t
C

(5)
  q    C  D 2C
t

H
 
(6)
   q  H  m 2 H
t
  0 1   t T  T0    s  C  C0  
(7)


The corresponding equations for the porous layer are,

m  qm  0
(8)

(9)
m  H  0

 

 
 1 qm 1 
0 
 2  qm  m  qm     m Pm   2 qm  qm   m gkˆ
K
  t 


 qm
 
  p  H  m  H 
Cb qm qm
K
(10)
Tm

2
(11)
A
  qm   m  Tm   m mTm
t
C

(12)
 m   qm  m  Cm  Dm  m2 Cm
t


H

(13)

  m  qm  H m   em  m2 H m
t
 m   0 1   tm Tm  T0    sm  Cm  C0  
(14)


Where the symbols in the above equations have the following meaning. q   u , v , w  is the velocity vector, H is the
magnetic field, t is the time,  is the fluid viscosity, P  p 
Volume 3, Issue 2, February 2014
pH 2

is the total pressure,  0 is the fluid density, g
2
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Volume 3, Issue 2, February 2014
ISSN 2319 - 4847
is the acceleration due to the gravity,  p is the magnetic permeability,
A
 C 
 C 
0
p m
p
is the ratio of heat capacities,
f
C p is the 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 or the salinity field, D is the solute diffusivity of the fluid,
magnetic
viscosity, 
porosity, em 
is
the electrical conductivity,  t  
m 
1
is the
 p
1   
1   
.

 ,s   
  T  P,T
  C  P ,C

is the
m
is the effective magnetic viscosity and 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,

u , v , w, P , T , C , H   0, 0, 0, Pb  z  , Tb  z  , Cb  z  , H 0  z  
(15)
um , vm , wm , Pm , Tm , Cm   0, 0, 0, Pmb  zm  , Tmb  zm  , Cmb  zm 
(16)
in the fluid layer.
In the porous layer
Where the subscript ‘b’ denotes the basic state.
Tmb  zm  , and Cb  z  , Cmb  zm  ,
The temperature and species concentration distributions Tb  z  ,
respectively are found to be
Tb  z   T0 
T0  Tu  z
Tmb  zm   T0 
in
d
Tl  T0  zm
0zd
in 0  zm  dm
dm
 C  Cu  z in 0  z  d
Cb  z   C0  0
d
Cmb  z m   C0 
Where T0 
 Cl  C0  zm
dm
in
0  zm  d m
(17)
(18)
(19)
(20)
 d mTu   m dTl
Dd mCu  Dm dCl
and C0 
are the interface temperature and species concentration
 dm   m d
Dd m  Dm d
respectively.
In order to investigate the stability of the basic solution, infinitesimal disturbances are introduced in the form,




 q , P , T , C , H   0, Pb  z  , Tb  z  , Cb  z  , H 0  z     q , P,  , S , H  
(21)
And




 qm , Pm , Tm , Cm , H   0, Pmb  zm  , Tmb  zm  , Cmb  z m  , H 0  zm     qm , Pm , m , S m , H  (22)




Where the primed quantities are the perturbed ones over their equilibrium counterparts. Now (21) and (22) are
substituted into (1) to (14) and are linearised in the usual manner. Next, the pressure term is eliminated from (3) and
(10) by taking curl twice on these two equations and only the vertical component is retained. The variables are then
nondimensionalised using d ,
d2 
,
, T0  Tu , C0  Cu and H 0 as the units of length, time velocity, temperature,
 d
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Volume 3, Issue 2, February 2014
ISSN 2319 - 4847
species concentration and the magnetic field in the fluid layer and
dm ,
dm2  m
,
, Tl  T0 , Cl  C0
 m dm
as the
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
d
ratios dˆ  m . Thus obtained dimensionless equations for the perturbed variables are then subjected to normal mode
d
expansion in the usual manner to get an eigenvalue problem given consisting of the following ordinary differential
equations, (for details see Sumithra [10] )
In
0  z 1,
n  2
 2
2
2
2
2
2
2
 D  a    D  a  W  Ra   Rs a   Q fm  D  a  H
Pr


D
2
 a2  n  W  0
  D 2  a 2   n    W  0


 fm  D 2  a 2   n  H  DW  0


In
(23)
(24)
(25)
(26)
0  zm  1
 2
n 2 
2
ˆ 2  m  1  Dm2  am2  Wm  Rm am2 m  Rsmam2 m
 Dm  am  
Prm


(27)
Qm mm  2 Dm  Dm2  am2  H m
D
 am2  nm  m  Wm  0
(28)
 pm  Dm2  am2   nm   m  0


2
2
 mm  Dm  am   nm  H mz  DWm  0


(29)
2
m
(30)
3
3
Where, for the fluid layer Pr   is the Prandtl number, R  g t T0  Tu  d is the Rayleigh number, R  g s  C0  Cu  d
s



2
is the Solute Rayleigh number, Q   p H 0 d
 fm
ratios. For the porous layer, Prm 
 m
m
viscosity ratio, R  g t T0  Tu  d m K  RDa
m
 mv
D
and  
are the diffusivity



K
is the Prandtl number,  2  2  Da is the Darcy number, ˆ  m is the
dm

g s  Cl  C0  d m K
is the Rayleigh – Darcy number,
is the
2
is the Chandrasekhar number,
 fm 
Rsm 
 m
 m
 Rs Da
 em
D
, and  pm  m are
m
m
the diffusivity ratios in the porous layer, a and am are the nondimensional horizontal wavenumbers, n and nm are the
2
2
Solute Rayleigh – Darcy number, Q   p H 0 d m  Q dˆ 2 is the Chandrasekhar number  mm 
m
 m mm
frequencies. Since the dimensional horizontal wavenumbers must be the same for the fluid and porous layers, we must
have
a am
ˆ . D and D denote the differential operators  and  respectively.
and hence am  da

m
d dm
z
zm
It is known that the principle of exchange of instabilities holds for double diffusive magneto 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 otherwords, it is assumed that the onset of convection is in the
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Volume 3, Issue 2, February 2014
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form of steady convection and accordingly we take
(26) and (30) we get, in
n  nm . And eliminating the magnetic field in (23) and (27) from
0  z 1
D
2
2
 a 2  W  Ra 2   Rs a 2   QD 2W
D
2
(31)
(32)
 a2    W  0
  D 2  a2    W  0
In
(33)
0  zm  1
 Dm2  am2  
ˆ 2  1  Dm2  am2  Wm  Rm am2 m  Rsm am2  m  Qm Dm2Wm


2
(35)
D
 m  am2  m  Wm  0
 pm  Dm2  am2  m  Wm  0
(34)
(36)
Thus we note that, in total we have a sixteenth order ordinary differential equation and we need sixteen boundary
conditions to solve it.
3. BOUNDARY CONDITIONS
All the sixteen dimensionless boundary conditions are subjected to normal mode expansion and are given by
W (1)  0, D 2W (1)  Ma 2 (1)  M s a 2 (1)  0, D(1)  0, D (1)  0
ˆ (0)  W (1),
TW
m
ˆˆ
TdDW
(0)  DmWm (1),
ˆ ˆ 2  D 2  a 2  W (0)  ˆ  D 2  a 2  W (1)
Td
m
m
m
ˆ ˆ 2  2  D 3W (0)  3a 2 DW (0)    D W 1  
ˆ 2  Dm3 Wm 1  3am2 DmWm 1 
Td
m m
(0)  Tˆ m (1),
(0)  Sˆ  (1),
D(0)  Dm m (1),
D(0)  Dm  m (1),
Wm (0)  0, DmWm (0)  0, Dm m (0)  0, Dm m (0)  0,
m
Where
(37)
 T0  Tu  d is the thermal Marangoni number,
  C0  Cu  d is the solutal Marangoni number,
M 
Ms  
T

C

C  C0 .
T T
Tˆ  l 0 and Sˆ  l
T0  Tu
C0  Cu
The Eqs.(31) to (36) are to be solved with respect to the boundary conditions (37).
4. SOLUTION BY REGULAR PERTURBATION TECHNIQUE
For the constant heat and mass flux boundaries, convection sets in at small values of horizontal wavenumber ‘a’,
accordingly, we expand
W j 
Wmj 
W  
Wm  
2j 

(38)
    a 2 j    and   
ˆ
da

m
 j
 mj 
  

j 0
j 0
 j 
 mj 
  
  m 
 


 
Substituting (38) into (31) to (36) and into the boundary conditions (37) yields, a sequence of equations for the unknown
functions Wi  z  , i  z  , i  z  , Wmi  z m  ,  mi  z m  ,  mi  z m 
for i  0,1, 2,...
2
At the leading order in a , (31) to (36) and the corresponding Boundary conditions are,
for the fluid layer
(39)
D 4W0  QD 2W0  0
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D 2  0  W0  0
(40)
2
 D  0  W0  0
(41)
for the porous layer,
ˆ 2 Dm4Wm 0  Dm2Wm 0  Qm Dm2Wm 0  0

(42)
2
m
D  m 0  Wm 0  0
(43)
2
m
 m D  m 0  Wm 0  0
(44)
The corresponding boundary conditions are,
W0 (1)  0, D 2W0 (1)  0, D0 (1)  0, D 0 (1)  0
ˆ (0)  W (1), TdDW
ˆˆ
TW
(0)  D W (1),
0
m0
0
m
m0
ˆ ˆ 2 D 2W (0)  ˆ D 2W (1)
Td
0
m m0
ˆ ˆ 2  2 D3W (0)   D W 1  
ˆ 2 Dm3Wm0 1
Td
0
m m0
0 (0)  Tˆ m 0 (1), D0 (0)  Dm m 0 (1),
0 (0)  Sˆ m 0 (1), D0 (0)  Dm m 0 (1),
Wm 0 (0)  0, DmWm 0 (0)  0, Dm m 0 (0)  0, Dm  m 0 (0)  0
(45)
The solution to the zeroth order of equations (39) to (44) subjected to the Boundary conditions (45) is given by
W0  z   0, 0  z   Tˆ , 0  z   Sˆ , Wm 0  zm   0, m0  zm   1, m 0  z m   1.
(46)
2
At the first order in a , (31) to (36), using the solution of zero order (46) reduces to,
D 4W1  QD 2W1  RTˆ  Rs Sˆ  0
D 2 1  Tˆ  W1  0
(47)
(48)
 D 1   Sˆ  W1  0
2
2
4
m
2
m
2
(49)
2
m
ˆ D Wm1  D Wm1  Qm  D Wm1  Rm  Rsm  0

(50)
(51)
2
m
D  m1  1  Wm1  0
 m Dm2  m1   m  Wm1  0
(52)
The corresponding boundary conditions are,
W1 (1)  0, D 2W1 (1)  M 0 (1)  M s  0 (1)  0, D1 (1)  0, D1 (1)  0,
ˆ W (1), TD
ˆ (0)  dˆ 2W (1), TDW
ˆ
ˆ 2W (0)  ˆ D 2W (1),
TW
(0)  dD
1
m1
1
m
m1
1
m
m1
ˆ ˆ 2  2 D3W (0)   D W 1  
ˆ 2 Dm3Wm1 1 ,
Td
1
m m1
ˆ ˆ 2 (1), D (0)  dˆ 2 D  (1),
 (0)  Td
1
m1
1
ˆ ˆ 2 (1),
1 (0)  Sd
m1
m
m1
D1 (0)  dˆ 2 Dm m1 (1),
Wm1 (0)  0, DmWm1 (0)  0, Dm m1 (0)  0, Dmm1 (0)  0.
The solutions of (47) and (52),
(53)
W1 and Wm1 respectively are important in obtaining the eigen values and they are found
to be,
W1  z   a1  a2 z  a3Cosh


Q z  a4 Sinh

z2
Q z  RTˆ  Rs Sˆ
2Q
Wm1  zm   b1  b2 zm  b3Cosh  zm   b4 Sinh   z m    Rm  Rsm 
Volume 3, Issue 2, February 2014
 
zm 2
2 1  Qm  2 

(54)
(55)
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Where
 
and
1  Qm  2
ˆ 2

a1 , a2 , a3 , a4 and b1 , b2 , b3 , b4 are constants to be determined using the velocity boundary
conditions of (53) and are as follows,
b1 
 3 5   2  6 ,
 2  4  15
b2 
  3  4  1 6  ,
 2  4  15
 
ˆ Cosh dˆ
2R
 b3  

ˆ
Q
TQ

2
a1  

dˆ Sinh
a2  b3   A Q 

Tˆ

a3 
A
2
 Cosh
Tˆ
b3  b1 , b4   b2 ,

 
 1 
ˆ Sinh dˆ

  b4  
ˆ
TQ


2
2
 
,

 Sinh
Tˆ


dˆ (Cosh  1)  
2 R dˆ 
C Q  m ,
  b4   B Q 

 
Tˆ
Tˆ 


 
2 R ˆ

b  2 Cosh  b4 2 Sinh  2 Rm  , a4  b3 A  b4 B  C ,
ˆ  3
Q TQ
ˆ 2 2  1 Sinh
  
,
ˆ ˆ  2Q Q
Td
B
ˆ 2 2Cosh  Cosh  1
  
,
ˆ ˆ  2Q Q
Td
C
2 Rm
.
ˆ ˆ  2Q Q
Td
4.1 Solvability condition
Integrating (48) with respect to z between the limits z = 0 and 1 and integrating (51) with respect to
limits
zm between the
10
zm  0 and 1 and adding the resulting equations and using the Boundary conditions  53  , we get
1
1
(56)
2
2
 W1dz  dˆ  Wm1dzm  Tˆ  dˆ
0
0
Similarly integrating (49) with respect to z between the limits z = 0 and 1 and integrating (52) with respect to
between the limits
zm
12
zm  0 and 1and adding the resulting equations and using the Boundary conditions  53 , we get
1
1

 pm  W1dz   dˆ 2  Wm1dzm    pm Sˆ  dˆ 2
0

(57)
0
Now adding (56) and (57) we obtain the solvability condition
1
1
1     W dz  1    dˆ  W
2
pm
1
m1
0

dzm  Tˆ  dˆ 2    pm Sˆ  dˆ 2
W1 and Wm1 into (58) and integrating,
Substituting the expressions

(58)
0
the thermal critical Rayleigh number
obtained in the form,
Rc 
And



2Q   26  Rs  27 30   Rs Sˆ  29  2 MTˆ  M s Sˆ  28

 29Tˆ  2Q 27 30
(59)
are given by
2
1  
ˆ 2  2  1  Sinh d Sinh
ˆ 2Cosh d  Cosh  1  




,
ˆ
ˆ ˆ 2
TQ
TQd
T
T
2  
ˆ 2  2Cosh  Cosh  1  dˆ 2  Sinh    dˆ   Cosh  1
ˆ 2 Sinh  




ˆ
ˆ ˆ 2 Q
Tˆ
Tˆ
TQ
TQd
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3 
2ˆ Rm MT  M s S 2 Rm dˆ
2 Rm
dˆ 2 Rm




 R,
2
ˆ
ˆ ˆ
Q
TQ
Tˆ
Tˆ
TQd
4 
ˆ 2  2  1  Sinh
ˆ 2Cosh Cosh Q QSinh Q  


,
ˆ ˆ  2Q Q
Tˆ
Td
5 
ˆ 2  2Cosh  Cosh  1  QSinh Q
ˆ 2 Sinh Cosh Q  


,
ˆ ˆ 2 Q
Tˆ
TQd
 6  2 RCosh Q 
2 ˆ Rm Cosh Q 2 Rm QSinh Q

 2 R  M T  M s S,
ˆ ˆ 2 Q
Tˆ
TQd
2
d
2
2d
2QSinh Q 2Cosh Q
  

,
8 

,
2

ˆ Q d 2
T
Tˆ
TQ
T d Q T
TQ
2
ˆ 2 Cosh dˆ  Cosh  1

dˆ 2 Sinh   
9  

,     Sinh  
,
10
ˆ
ˆ
TQ
T
ˆ
TQ
Tˆ
7 
2
11   A Q 
14 
d Sinh
T

,
 5  21 Cosh Q  1
13
   
17  1 8 4 7 ,
13
 20   9 
15 
13   2  4 1 5 ,
,
T

, 
 4  2 1 Cosh Q  1
13
  Q 1
18  5
,
13
19 

16

 7  5 8  1
,
13
 4  Q 1
13

11  2 Sinh QCosh A Cosh Q  1


,
2
Q
TQ Q


12  2 Sinh QSinh B Cosh Q  1


,
2
Q
TQ Q
 21  10 
 22 
d  Cosh  1
12   B Q 
,
Cosh 1 
  ,

 2
 23 
Sinh
 1,

 24  
2 2Sinh Q 1


Q
3
Q Q

2 Sinh Q 2 Cosh Q  1
1
 25  
  


,
ˆ ˆ 2
TQ T T TQd
TQ Q
T d 2Q 2
2
2
d

d



2
2
 26  T  d   pm S  d ,
2
1

 27  1   pm    25  16 20  17  21   d 1     17  22  16  23   ,
3

2
  1            d 1          ,
28
pm
19
21
18
20
19
22
18
23
2
 29  1   pm    24  14 20  15 21   d 1    15  22  14  23  ,
 30 
1
.
2 1  Qm  2 
5. RESULTS AND DISCUSSION
The critical thermal Rayleigh number Rc obtained as a function of the parameters is drawn versus the depth ratio d̂ and
the results are represented graphically showing the effects of the variation of one physical quantity, fixing the other
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parameters. The fixed values of the parameters are Tˆ  Sˆ  ˆ    1.0 ,
Q  5.0, M  10.0,
   pm  1.0 , Da  0.1 , Rs  10,
M s  10. The effects of the parameters Da, M , M s , ˆ ,  , Q, Rs ,  ,  pm on the critical
thermal Rayleigh number are obtained and portrayed in the figures 1 to 9 respectively.
600
500
400
300
200
100
0
0
2
4
6
8
10
Fig.1. The effects of Darcy number Da on the Critical Thermal Rayleigh number
The effects of the Darcy number Da  K on the critical thermal Rayleigh number
2
Rc .
Rc is exhibited in the Figure 1.
dm
The graph has three diverging curves. The line curve is for Da  0.01 , the big dotted curve is for 0.1 and the small
dotted line curve is for 100.0. Since the curves are drastically diverging, it indicates that the increasing values of Da
dm
, that is for porous layer dominant
d
composite systems. From the curves it is evident that for a fixed value of d̂ , increase in the value of Da is to increase
the value of the critical thermal Rayleigh number Rc i.e., to stabilize the system, so
the onset of double diffusive
will affect the onset of convection only for larger values of the depth ratio dˆ 
magneto-marangoni convection is delayed. In other words increasing the permeability of the porous matrix one can
stabilize the fluid layer system, this may due to the presence of both the species concentration and magnetic field.
Figure 2 displays the effects of thermal Marangoni number M    T0  Tu  d
T
graph has three curves.
on the Critical Rayleigh number

Rc . The
The line curve is for M  0.0 , the big dotted curve is for M  10.0 and the small dotted
line curve is for M  20.0 . From the curves it is evident that for a fixed value of d̂ , increase in the value of
decrease the value of the critical thermal Rayleigh number
Rc i.e., to destabilize the system, so
M is to
the onset of double
diffusive magneto-marangoni convection is faster.
250
200
150
100
50
0
2
4
6
8
10
Fig.2. The effects of Thermal Marangoni number M on the Critical Thermal Rayleigh number
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250
200
150
100
50
0
2
4
Fig.3. The effects of Solutal Marangoni number
6
8
10
M s on the Critical Thermal Rayleigh number Rc
Figure 3 displays the effects of thermal Marangoni number M     C0  Cu  d on the Critical Rayleigh number Rc .
s
C

The effects are same as that of thermal Marangoni number. The graph has three curves. The line curve is for
M s  0.0 , the big dotted curve is for M s  10.0 and the small dotted line curve is for M s  20.0 . From the curves
it is evident that for a fixed value of d̂ , increase in the value of
Rayleigh number
Rc i.e., to destabilize the system, so
M s is to decrease the value of the critical thermal
the onset of double diffusive magneto-marangoni convection is
faster.
250
200
150
100
50
0
2
4
6
8
10
Fig.4. The effects of viscosity ratio ̂ on the Critical Thermal Rayleigh number
The effects of the viscosity ratio ˆ 
m

Rc
which is the ratio of the effective viscosity of the porous matrix to the fluid
viscosity are displayed in Figure 4. The line curve is for ˆ  1 , the big dotted curve is for 1.5 and the small dotted line
curve is for 2. From the curves it is evident that for a fixed value of d̂ , increase in the value of
value of the critical thermal Rayleigh number
̂ is to increasee the
Rc i.e., to stabilize the system, so the onset of double diffusive magneto-
marangoni-convection is delayed. In other words when the effective viscosity of the porous medium
than the fluid viscosity
m
is made larger
 , the onset of the convection in the fluid layer can be made delayed.
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200
150
100
50
0
2
4
6
8
10
Fig.5 . The effects of Porosity  on the Critical Thermal Rayleigh number
The effects of the porosity
thermal Rayleigh number
Rc
 , which is the ratio of the void volume to the total volume of the porous layer, on the critical
Rc is exhibited in the Figure 5. The graph has three diverging curves. The line curve is
for   0.8 , the big dotted curve is for   0.9 and the small dotted line curve is for   1.0 Since the curves are
diverging, it indicates that the increasing values of  will affect the onset of convection only for larger values of the
dm
, that is for porous layer dominant composite systems. From the curves it is evident that for a fixed
d
value of dˆ , increase in the value of  is to increase the value of the critical thermal Rayleigh number Rc i.e., to
depth ratio dˆ 
stabilize the system, so
the onset of double diffusive magneto-marangoni convection is delayed.
500
400
300
200
100
0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
Fig.6. The effects of by the Chandrasekhar number Q on the Critical Thermal Rayleigh number
Figure 6 exhibits the effects of the magnetic field on the critical Rayleigh number
Q
2
0
 pH d
2
.
Rc
Rc by the Chandrasekhar number
The line curve is for Q  5 , the big dotted curve is for 10 and the small dotted line curve is for 20. As
 fm
the curves are widely diverging the effect of the magnetic field is very large for even a small change in the value of the
depth ratio. From the curves it is evident that for a fixed value of d̂ , increase in the value of Q is to increase the value
of the critical thermal Rayleigh number
Rc i.e., to stabilize the system, so
the onset of double diffusive magneto-
marangoni convection is delayed.
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The effects of the second diffusing grandients on the onset of convection are recorded by the Solute Rayleigh
3
number R  g s  C0  Cu  d which is the balance between the generation of energy by the solute gradients and the energy
s

dissipation due to viscosity and heat are given in Figure 7. The line curve is for
Rs  10 , the big dotted curve is for 50
and the small dotted line curve is for 100. It is very important to note that all the three curves are converging. So for
larger values of the depth ratio, there is no effect of any variation in the values of Rs . The effect of Rs is prominent for
fluid layer dominant composite systems. For a fixed value of depth ratio the increase in the value of the solute Rayleigh
number is to increase the value of the critical thermal Rayleigh number Rc . Increasing values of solute Rayleigh number
Rs makes the system stable and hence delay convection.
250
200
150
100
50
0
2
4
6
8
10
Fig.7. The effects of Rs on the Critical Thermal Rayleigh number
Rc
D
, which is the solute to thermal diffusivity ratio of the fluid, are shown in

Figure 8. The graph has three diverging curves. The line curve is for   0.25 , the big dotted curve is for 0.5 and
the small dotted line curve is for 1.0. Since the curves are diverging, it indicates that the increasing values of  will
d
have effect only for larger values of the depth ratio dˆ  m , that is for porous layer dominant composite systems. From
d
the curves one can see that for a fixed value of d̂ , increase in the value of  is to increase the value of the critical
The effects of the diffusivity ratio  
thermal Rayleigh number i.e., to stabilize the system by delaying the onset of convection.
200
150
100
50
0
2
4
6
8
10
Fig.8 . The effects of  on the Critical Thermal Rayleigh number
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200
150
100
50
0
2
4
6
8
10
Fig.9 . The effects of  pm on the Critical Thermal Rayleigh number
Figure 9 displays the effects of variations of the value of pm 
Rc
Dm
, which is the ratio of the solute diffusivity to
m
thermal diffusivity ratio of the porous layer.
The graph has three slightly diverging curves. The line curve is for
 pm  0.25 , the big dotted curve is for 0.5 and the small dotted line curve is for 1.0. Since the curves are diverging, it
indicates that the increasing values of
d
 pm will have effect only for larger values of the depth ratio dˆ  m , that is for
d
porous layer dominant composite systems. From the curves one can see that for a fixed value of d̂ , increase in the value
of
 pm is to increase the value of the critical thermal Rayleigh number Rc i.e., to stabilize the system, so the onset of
the convection delayed.
6. CONCLUSIONS
1. For Porous layer dominant composite systems, increasing the values of Da ,
 , Q,  ,  pm one can delay the
convection during the growth of crystals
2. For Fluid layer dominant composite systems, it is possible to control convection by increasing the values of
Rs .
3. For any composite layer systems, by increasing the values of the viscosity ratio and decreasing the values of
M s , convection can be controlled.
M and
4. By suitably choosing the parameters one can control the convection during the growth of crystals and hence avoid
turbulence during the process and obtain pure crystals.
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] H.A. Chedzey, and D.T.J. Hurle, Journal of Applied Physics, 210, 933,1966 (journal style)
[2] F. Chen and C.F. Chen, “Onset of Finger convection in a horizontal porous layer underlying a fluid layer” J. Heat
transfer, 110, 403,1988.(journal style)
[3] F. Chen , “Throughflow effects on convective instability in superposed fluid and porous layers”, J. Fluid mech., 23,
113,-133,1990. (journal style)
[4] Mc Kay, “Onset of buoyancy-driven convection in superposed reacting fluid and porous layers”, J. Engg. Math., 33,
31-46,1998.(journal style)
[5] D.A. Nield “ Onset of convection in a fluid layer overlying a layer of a porous medium”, J. Fluid Mech., 81, 513522,1977. (journal style)
[6] D. Poulikakos and M. Kazmierczak “Transient double-diffusive convection experiments in a horizontal fluid layer
extending over a bet of spheres”, Phy. Of fluids –A, 1, 480. 1989. (journal style)
Volume 3, Issue 2, February 2014
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 3, Issue 2, February 2014
ISSN 2319 - 4847
[7] N. Rudraiah “Flow past porous layers and their stability in sullry flow Technology”, Encyclopedia of Fluid
mechanics (Ed. Cheremisinoff, N. P.), Gulf Publishing Company, USA, Chapter 14, 567,1986. (Book style)
[8] I.S. Shivakumara, Suma, B. Krishna “ Onset of surface tension driven convection in superposed layers of fluid and
saturated porous medium”, Arch. Mech., 58, 2, pp. 71-92, Warszawa, 2006. (journal style).
[9] R. Sumithra Exact solution of Triple diffusive Marangoni convection in a composite layer International Journal of
Engineering Research & Technology (IJERT) Vol. 1 Issue 5, July – 2012. (online)
[10] R. Sumithra, “ Mathematical modeling of Hydrothermal growth of crystals as double diffusive convection in
composite layer bounded by rigid walls”, International Journal of Engineering Science and Technology,
Vol.04,No.02, 2012, 779-791, 2012.(online)
[11] R. Sumithra and N. Manjunatha, “Analytical study of Surface tension driven Magnetoconvection in a composite layer
bounded by Adiabatic Boundaries”, International Journal of Engineering and Innovative Technology (IJEIT) Volume
1, Issue 6, June 2012.(Online)
[12] M.E. Taslim and V. Narusawa, “Thermal stability of horizontally superposed porous and fluid layers” , ASME J.
Heat Transfer, 111, 357-362, 1989. (journal style).
[13] H.P. Utech and M.C. Flemings, Journal of applied Physics,37, 2021,1966. (journal style).
[14] M. Venkatachalappa, M, Prasad, V., I.S. Shivakumara and R.Sumithra, (1997), Hydrothermal growth due to double
diffusive convection in composite materials, Proceedings of 14 th
National Heat and Mass Transfer Conference
and 3rd ISHMT –ASME Joint Heat and Mass transfer conference, December 29-31, 1997.
(Book style)
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