International Journal of Advancements in Research & Technology, Volume 2, Issue 11, November-2013
ISSN 2278-7763
132
Heat and mass transfer effect of chemically reactive fluid on flow over
an accelerated vertical surface in presence of radiation with constant
heat flux
DR. ARPITA JAIN
Head, Department of Mathematics,
JECRC UDML College of Engineering,
Kukus, Jaipur, Rajasthan, India
E-mail*: arpita_252@rediffmail.com
ABSTRACT. This paper aims to investigate the influence of chemical reaction and the
IJOART
combined effects of heat and mass transfer on laminar boundary layer flow over a
moving vertical plate in presence of radiation with heat is supplied to the plate at
constant rate. The governing equations for this investigations are solved analytically by
Laplace-transform
technique.
Graph
results
are
presented
for
temperature,
concentration, velocity, skin friction, Nusselt number and Sherwood number. The
effects of various parameters on flow variables are illustrated graphically and the
physical aspects of the problem are discussed.
KEY WORDS: Free convection, heat and mass transfer, chemical reaction, radiation,
accelerated plate.
1. INTRODUCTION:
The analysis of free convection flow near a vertical plate has been carried out as an
important application in many industries. Numerous investigations are performed by
using both analytical and numerical methods. The first exact solution of the NavierStokes equation was given by Stokes [1] and explains the motion of a viscous in-
Copyright © 2013 SciResPub.
IJOART
International Journal of Advancements in Research & Technology, Volume 2, Issue 11, November-2013
ISSN 2278-7763
133
compressible fluid past an impulsively started infinite horizontal plate in its own
plane. This is known as Stokes’s first problem in the literature. If the plate is in a
vertical direction and gives an impulsive motion in its own plane in a stationary
fluid, then the resulting effect of buoyancy force was first studied by Sundalgekar
[2] by Laplace transformation technique and the effects of heating or cooling of the
plate by free convection cur- rents were discussed. Natural convection has been
analyzed extensively by many investigators. Some of them are Revankar [3], Li et.al.
[4].
Furthermore, the free convection flows together with heat and mass transfer are of
great importance in geophysics, aeronautics, and engineering. In several process
IJOART
such as drying, evaporation of water at body surface, energy transfer in a wet
cooling tower, and flow in a desert cooler, heat and mass transfer occurs
simultaneously A number of investigations have already been carried out with
combined heat and mass transfer under the assumption of different physical
situations. The illustrative examples of mass transfer can be found in the book of
Cussler [5] .Combined heat and mass transfer flow past a surface analyzed by
Chaudhary et. al. [6], Muthucumaraswamy et. al. [7,8] and Rajput et.al.[9 ]with
different physical conditions. Chaudhary et. al. [10] pioneered unsteady heat and mass
transfer flow past a surface by Laplace Transform method.
Combined heat and mass transfer problems with chemical reaction are of
importance in many processes and have, therefore, received a considerable amount of
attention in recent years. Chemical reaction can be codified as either homogeneous or
heterogeneous processes. A homogeneous reaction is one that occurs uniformly through
a given phase. In contrast, a heterogeneous reaction takes place in a restricted region or
Copyright © 2013 SciResPub.
IJOART
International Journal of Advancements in Research & Technology, Volume 2, Issue 11, November-2013
ISSN 2278-7763
134
within the boundary of a phase. A reaction is said to be first order, if the rate of
reaction is directly proportional to the concentration itself which has many applications
in different chemical engineering processes and other industrial applications such as
polymer production, manufacturing of ceramics or glassware and food processing. Das
et al. [11] considered the effects of first order chemical reaction on the flow past an
impulsively started infinite vertical plate with constant heat flux and mass transfer.
Muthucumarswamy et. al. [12] studied study of chemical reaction effects on vertical
plate with variable temperature.
In the above mentioned studies the effects of radiation on flow has not been
considered. Actually, many processes in new engineering areas occur at high
IJOART
temperature and knowledge of radiation heat transfer becomes imperative for the
design of the pertinent equipment. Nuclear power plants, gas turbines and the various
propulsion devices for aircraft, missiles, satellites, and space vehicles are examples of
such engineering areas. The effects of radiation on free convection on the accelerated
flow of a viscous incompressible fluid past an infinite vertical plate has many
important technological applications in the astrophysical, geophysical and engineering
problem. Unsteady Free Convection flow past a Vertical plate with chemical reaction
under different temperature condition on the plate is elucidated by Bhaben Ch. Neog
et.al.[13] and Rajesh V. [14] et.al. Thermal radiation effect on flow past a vertical plate
with mass transfer is examined by Muralidharan M. et. al. [15] and Rajput et.al. [16].
Natural convective flow past a plate with constant mass flux in the presence of
radiation is studied by Chaudhary et. al. [17]
However, it seems less attention was paid on free convection flows near a vertical
plate subjected to a constant heat flux boundary condition even though this situation
Copyright © 2013 SciResPub.
IJOART
International Journal of Advancements in Research & Technology, Volume 2, Issue 11, November-2013
ISSN 2278-7763
135
involves in many engineering applications. In many problems particularly those
involving the cooling of electrical nuclear components, the wall heat flux is specified.
In such problems, over heating burns out and melt down are very important issues.
From practical stand point, an important wall model is considered with constant
heat flux. Ogulu et. al. [18] and Narahari, M. et. al. [19] examined the flow past a
surface with constant heat flux. Free convection effects on flow past an infinite vertical
accelerated with constant heat flux is pioneered by Chaudhary et. al. [20 ].
Hence, the objective of this paper is to study the influence of chemical reaction and
the combined effects of heat and mass transfer on flow over a moving vertical plate in
presence of radiation with heat is supplied to the plate at constant rate.
IJOART
2. MATHEMATICAL ANALYSIS: We consider a two-dimensional flow of an
incompressible and electrically conducting viscous fluid along an infinite vertical plate.
The x'-axis is taken on the infinite plate and parallel to the free stream velocity and y'axis normal to it. Initially, the plate and the fluid are at same temperature T ' with
concentration level C ' at all points. At time t' > 0, the plate concentration is changed
to C'w with heat supplied at a Constant rate to the plate and it accelerates with a
velocity
U R 3t '
in its own plane. It is assumed that there exist a homogeneous

chemical reaction of first order with constant rate K l between the diffusing species
and the fluid. Since the plate is infinite in extent therefore the flow variables are the
functions of y' and t' only. The fluid is considered to be gray absorbing-emitting
radiation but non scattering medium . The radiative heat flux in the x'-direction is
considered negligible in comparison with of y'-direction. Then neglecting viscous
dissipation and assuming variation of density in the body force term (Boussinesq’s
Copyright © 2013 SciResPub.
IJOART
International Journal of Advancements in Research & Technology, Volume 2, Issue 11, November-2013
ISSN 2278-7763
136
approximation), the problem can be governed by the following set of equations:
T '
 2T'
1 q r


'
2
 C p y'
 C p y '
t
…(1)
C'
 2 C'

D
 k l C'
'
2
t
y'
…(2)
u'
 2 u'
  2  g  (T '  T' )  g c (C '  C ' ')
t'
y'
…(3)
with following initial and boundary conditions
u'  0 , T '  T' , C '  C ' for all y', t'  0
u' 
…(4)
IJOART
U R 3 t ' T '
,

y '



'
'
as y   , t  0 
q
 - , C'  C 'w at y'  0, t'  0

u '  0, T '  T' , C '  C '
The radiative heat flux term, by using the Rosseland’s approximation is given by
qr  
4 T'4
3 y'
where UR is reference
velocity,
…(5)
g is gravitational acceleration, C p is specific
heat at constant pressure, D is mass diffusivity,  is thermal expansion coefficient,
C is concentration expansion coefficient,  is density,  is thermal conductivity
of fluid,  is mean absorption coefficient,  is electrical conductivity of fluid,
 is kinematic viscosity and , qr is radiative heat
flux, σ is Stefan-Boltzmann
constant.
We assume that the temperature differences within the flow are such that T' 4
may be expressed as a linear function of the temperature T'. This is accomplished by
expanding T' 4 in a Taylor series about T ' and neglecting higher-order terms
Copyright © 2013 SciResPub.
IJOART
International Journal of Advancements in Research & Technology, Volume 2, Issue 11, November-2013
ISSN 2278-7763
137
T'4  4 T'3 T'  3T'4
…(6)
By using equations (5) and (6), equation (1) gives
 Cp
T'
 2 T' 16 T'3  2 T'


 t'
y'2
3 y'2
…(7)
Introducing the following dimensionless quantities
t'
, y 
tR
t 
 Cp
Pr 
C

y'
,u 
LR
, Sc 
U2 k
u'
, k  R2 l ,
UR

T '  T'

g q 2
,
,G 
q
U R 4
D
U R
 g C (C 'w  C ' )
C '  C '
k
,
Gm
, k  2l , R 

'
'
3
C w  C
UR
UR
 
4 T'3
IJOART
T  Tw'  T' , U R  (gt)1/3 ,
 g  T 
LR   2 
  
1  3
 t R   g T) 2  3 1  3
.
…(8)
where LR is reference length, tR is reference time, Gm is modified
Grashof
number,
Pr
is
Prandtl number, Sc is Schmidt number and
u is dimensionless velocity component,  is dimensionless temperature,
C is dimensionless concentration,  is viscosity of
fluid, t is
time
in
dimensionless coordinate, R is radiation parameter and k is chemical reaction
parameter.
The governing Equations (1) to (3) reduce to the following non-dimensional form
Copyright © 2013 SciResPub.
IJOART
International Journal of Advancements in Research & Technology, Volume 2, Issue 11, November-2013
ISSN 2278-7763
Pr
138

4  2
 (1 
)
t
3R y 2
...(9)
1  2C
C

 kC
 t Sc y 2
…(10)
u  2 u

 G  Gm C
 t y 2
…(11)
with the following initial and boundary conditions
u  0 ,   0, C  0
for all y, t  0
…(12)


 1, C  1 at y  0, t  0

y

u  0,   0, C  0
as y  , t  0 
u  t,
…(13)
IJOART
On taking Laplace-transform of Equations (9) to (11) and Boundary conditions
(12, 13), we get
2
4 d 
(1 
)
 p Pr   0
3R dy 2
…(14)

d2

k+p
Sc
C
0


dy 2
…(15)

d 2u
 p u   G  ( y,p) -G m C
2
dy
…(16)
u 
1 d
1
1
,
  ,C 
2
p
dy
p
p
u 



a s y   , t  0 
at y  0, t  0
0,   0, C  0
…(17)
Where p is the Laplace -transform parameter.
Solving Equations (14) to (16) with the help of Boundary condition (17), we get
 (y,p) 
Copyright © 2013 SciResPub.
H
Pr
exp (  y
p3/ 2
pPr
)
H
...(18)
IJOART
International Journal of Advancements in Research & Technology, Volume 2, Issue 11, November-2013
ISSN 2278-7763
C (y,p) 
exp (  y (kSc+p Sc))
p
u (y,p) 
exp (  y p)
p2
…(19)
Gm exp (  y p )
H G exp (  y p )

Pr
kSc
Pr
p 5 / 2 ( -1)
p (Sc-1)(p+
)
H
Sc  1


139
Gm
exp (  y pSc  kSc )
kSc 

p (Sc  1)  p +

Sc  1 



H
Pr

p pr
)
H
p 5 / 2 (Pr/ H  1)
G exp (  y
IJOART
…(20)
On taking inverse Laplace-transform of Equations (18) to (20), we get


tH
- 2 Pr
Pr 
erfc (
)   t erfc  

H
H 
 Pr

…(21)
1
C  {exp(2 kSc t )erfc ( Sc  kt )  exp( 2 kSc t )erfc ( Sc  kt )}
2
…(22)
For Pr =Sc  1

2exp (  2 ) 
2
u  t (1+2 )erfc() 




H G t 3/2  4

(1  2 ) exp (  2 )    6  42  erfc() 

Pr
Pr 3 ( -1)  

H
Copyright © 2013 SciResPub.
IJOART
International Journal of Advancements in Research & Technology, Volume 2, Issue 11, November-2013
ISSN 2278-7763

140
H G t 3/2  4
Pr
Pr
Pr
Pr
Pr  Gm
(1  2 ) exp(  2 )  
(6  42 ) erfc(
) 
erfc()

Pr 3 (1  Pr )  
H
H
H
H
H  k Sc
H

Gm
kSc t
kSc t
kSc t
kSc t
kSc t
{exp(
) ((exp(2
)erfc( 
)  exp( 2
)erfc( 
)
2k Sc
1  Sc
1  Sc
1  Sc
1  Sc
1  Sc

Gm
kSc t
kSc t
kt
k Sct
kt
exp(
) (exp(2
)erfc( Sc 
)  exp( 2
)erfc( Sc 
)
2k Sc
1  Sc
1  Sc
1  Sc
1  Sc
1  Sc

Gm
exp(2 k Sct erfc( Sc  k t )  exp(2 k Sct erfc( Sc  k t )
2 k Sc




IJOART
Where H  1 
…(23)
4
3R
In expressions, erfc (x1+i y1) is complementary error function of complex argument
which can be calculated in terms of tabulated functions in Abramowitz et al. [21].The
tables given in Abramowitz et al. [21] do not give erfc (x1 + i y1) directly but an
auxiliary function W1(x1 + iy1) which is defined as
erfc (x1  iy1   W1   y1  i x 1  exp {  x 1  iy1  2 
Some properties of W1 (x1 + iy1) are
W1   x 1  iy1   W2  x 1  i y1 
W1  x 1  i y1   2 exp {-(x1  iy1  2   W2  x 1  i y1 
where W2  x 1  iy1  is complex conjugate of W1  x 1  iy1 
Copyright © 2013 SciResPub.
IJOART
International Journal of Advancements in Research & Technology, Volume 2, Issue 11, November-2013
ISSN 2278-7763
141
3. SKIN-FRICTION:
From velocity field , skin-friction at the plate in non dimensional form is expressed as:
 u 
   
 y  y 0
H G t 
Pr 
1


Pr ( Pr -1) 
H 
H
2t

t

Gm 
kSc t 
kSc t
kSc Gm
)  erf (
)

kSc erf kt
 exp(
k Sc 
1  Sc 
1  Sc 1  Sc k Sc

Gm 
kSc t 
kt
kSc
)  erf (
)
 exp(
k Sc 
1  Sc 
1  Sc 1  Sc
IJOART
…(24)
4. NUSSELT NUMBER
From temperature field, the rate of heat transfer in non-dimensional form is
expressed as
Nu  
1 
(0) y


y0
1
(0)
 Pr
tH
…(25)
5. SHERWOOD NUMBER
From the concentration field, the rate of concentration transfer, which when
expressed in non-dimensional form, is given by
Sh  
Copyright © 2013 SciResPub.

y
y 0
IJOART
International Journal of Advancements in Research & Technology, Volume 2, Issue 11, November-2013
ISSN 2278-7763

k Sc
1 Sc
(2 
exp( k t))
2
2 t
142
…(26)
6. DISCUSSION: In order to get physical insight into the problem, the values of
Schmidt number are chosen to represent the presence of species by Hydrogen (0.22),
Water vapor (0.60) and Carbon dioxide (0.96) at 250C temperature and 1 atmospheric
pressure, the values of Pr are chosen 0.71, 7 which represent air and water respectively
at 200C temperature and 1 atmospheric pressure. The values of other parameters are
chosen arbitrary.
Figure 1 reveals temperature profiles against  (distance from the plate). It is
obvious from the figure that the magnitude of temperature is maximum at the plate and
then decays to zero asymptotically. Further, the magnitude of the temperature for air is
IJOART
greater than that of water. This is due to the fact that thermal conductivity of fluid
decreases with increasing Pr, resulting a decrease in thermal boundary layer thickness.
It is also seen that it decreases steeply for Pr = 7 than that of Pr = 0.71. Furthermore, it
is noticed that an increase in radiation parameter to 4, 15 the temperature decreases
but it increases with increasing time.
The species concentration profiles verses  is
plotted in F igure 2. It is
observed that the concentration at all points in the flow field decreases with  and tends
to zero as   . Furthermore, an increase in the value of Sc leads to a fall in the
concentration. Physically, it is true since increase of Sc means decrease of molecular
diffusivity which results in decrease of concentration boundary layer. Hence, the
concentration of species is higher for small values of Sc. It is also observed that
concentration decreases with an increase in time.
Copyright © 2013 SciResPub.
IJOART
International Journal of Advancements in Research & Technology, Volume 2, Issue 11, November-2013
ISSN 2278-7763
143
Figure 3 illustrates the influences of Sc, t, and Pr on the velocity against η. It is
noticed that at the plate, fluid velocity is equal to value of time then it increases and
attains maximum velocity in the vicinity of the plate(η<0.6) after that it decreases and
vanish far away from the plate for both Pr = 7 and 0.71 whereas with an increase in Sc it
decreases continuously to asymptotic value. Further, it also increases with an increase
in time at each point in the flow field for Hydrogen gas. The effect of time on velocity
boundary layer is more dominant than Sc for both water and air. Finally, we observed
that the velocity for Hydrogen(Sc=0.22) is higher than that of carbon di oxide (Sc=0.96)
for both air and water. Physically, it is possible since an increase in Sc means increase
in kinematic viscosity or viscosity of fluid due to which the velocity of fluid decreases.
IJOART
It is also obvious from figure that increase or decrease in thickness of velocity boundary
layer with an increase in parameters value is greater near the plate then away from the
plate and on moving away from the plate
this difference is negligible
Figure 4
illustrates the influences of Gm, G and Pr on the velocity. This Figure reveals that the
maximum velocity attains near the plate then decreases and faded away from the plate.
Further velocity for Pr=7 is less than that of Pr=0.71 since increasing Pr means
increasing viscosity which in turn reduces the velocity of flow. Moreover, it is observed
that an increase in G, Gm leads to an increase in velocity for both Pr=7 and Pr =0.71
when Hydrogen gas is present in the flow field. The reason is that the values of Grashof
number and modified Grashof number has the tendency to increase the mass buoyancy
effect. The increase in velocity due to increase in G and Gm is more near the plate than
away from the plate. Figure 5 elucidates the effects of k, R and Pr on velocity profile. It
is found that the velocity decreases with an increase in chemical reaction parameter k
from 0.2 to 10 for both Pr=7 and 0.71. The same phenomenon is observed with an
Copyright © 2013 SciResPub.
IJOART
International Journal of Advancements in Research & Technology, Volume 2, Issue 11, November-2013
ISSN 2278-7763
144
increase in radiation parameter from 4 to 20. It is obvious since the radiation parameter
defines the relative contribution of conduction heat transfer to the thermal radiation
transfer. Moreover, the variation of the velocity across the boundary layer with R is
found to be negligibly small.
Figure 6 elucidates the effects of Sc, k, Pr, G, Gm on skin friction which is plotted
against time t. The maximum value of skin friction occurs for smaller values of t and
then it decreases rapidly with increasing t. It is clear from the Figure that skin-friction
increases with an increase in Sc, Pr and k but reverse effect is observed with an increase
in G and Gm. The Figure depicts that skin friction is higher for Sc=0.96 in comparison
to for Sc=0.22. Physically, it is correct since an increase in Sc serves to increase
IJOART
momentum boundary layer thickness. Further, increasing Pr means increasing viscosity
of fluid which increases the magnitude of skin friction. Moreover increasing value of
chemical reaction parameter reflects decrease in kinematic viscosity or viscosity of fluid
which results increase in the value of skin friction
The boundary layer separation
occurs in the fluid flow. The variation in the values of skin friction is more dominant
with parameters Pr and Gm in comparison with other parameters.
Figure 7 exhibits the Nusselt number against time. It is concluded from the
Figure that there is a increase in it with an increase in the value of radiation parameter.
Further, the value of Nusselt number for water is greater than air. It is consistent with
the fact that smaller values of Pr are equivalent to increasing thermal conductivities and
therefore heat is able to diffuse away from the plate more rapidly than higher values of
Pr, hence the rate of heat transfer is reduced.
Figure 8 depicts the effect of chemical reaction parameter k and Schmidt
number Sc on Sherwood number. It is observed that Sherwood number increase with an
Copyright © 2013 SciResPub.
IJOART
International Journal of Advancements in Research & Technology, Volume 2, Issue 11, November-2013
ISSN 2278-7763
145
increase in k and Sc. Since increase in Sc means decrease in molecular diffusivity which
in turn gives rise to increase in Sherwood number as Sherwood number is the ratio of
convective and diffusive mass transfer coefficient. Chemical reaction parameter is the
interfacial mass transfer so Sherwood number increases with an increase in k.
REFERENCES:
[1]
Stokes GG, On the Effect of Internal Friction of Fluids on the Motion of
Pendulums, Mathematical and Physical Papers 1851; 3; 8-106.
[2]
Soundalgekar VM, Free Convection Effects on Stokes’s Problem for a
v ertical Plate, American Society of Mechanical Engineers Journal of Heat
Transfer 1977; 99C; 499-501.
IJOART
[3] Revankar ST, Free convection effect on a flow past an impulsively started or
oscillating infinite vertical plate, Mechanics Research Comm. 2000; 27; 241-246.
[4]
Jian L, Ingham DB and Pop I, Natural convection from a vertical flat plate
with a surface temperature oscillation, Int. J. Heat Mass Transfer 2001; 44; 23112322.
[5]
Cussler EL, Diffusion Mass Transfer in Fluid Systems, Cambridge:
Cambridge University Press; 1998.
[6]
Chaudhary RC and Jain A, Combined heat and mass transfer effect on
MHD free convection flow past an oscillating plate embedded in porous
medium, Rom.Journ.Phys.(Romania) 2007; 52;505-524.
[7]
Muthucumaraswamy R and Vijayalakshmi A, Effects of heat and mass
transfer on flow past an oscillating vertical plate with variable temperature., Int.
J. of Appl. Math and Mech. 2008; 4; 59 – 65.
Copyright © 2013 SciResPub.
IJOART
International Journal of Advancements in Research & Technology, Volume 2, Issue 11, November-2013
ISSN 2278-7763
146
[8] Muthucumaraswamy R, Sundar RM and Subramanian VSA, Unsteady flow past
an accelerated infinite vertical plate with variable temperature and uniform mass
diffusion, Int. J. of Appl. Math and Mech. 2009.;5; 51 – 56.
[9]
Rajput US and Kumar S, MHD flow past an impulsively
started vertical plate with variable temperature and mass diffusion,
Applied Mathematical Sciences 2011; 5; 149 –157.
[10] Chaudhary RC and Jain A, An exact solution of magnetohydrodynamic
convection flow past an accelerated surface embedded in porous medium,
International Journal of Heat and Mass Transfer 2010; .53; 1609- 1611.
[11]
Das UN, Deka RK and Soundalgekar VM, Effects of mass transfer
IJOART
on flow past an impulsively started in- finite vertical plate with constant heat
flux and chemical reaction, Forschung im Ingenieurwesen 1994; 60; 284-287.
[12]
Muthucumaraswamy R, and Meenakshisundaram S, Theoretical study of
chemical reaction effects on vertical oscillating plate with variable
temperature, Theoret. Appl. Mech. 2006 ; 33; 245-257.ts reaction on moving i
[13] Bhaben CN, Das RK, Unsteady Free Convection MHD Flow
past a vertical plate with variable temperature and chemical reaction,
International Journal of Engineering Research & Technology 2012; 1; 1-5.
[14] Rajesh V, Radiation effects on MHD free convection flow near a vertical
plate with ramped wall temperature, Int. J. of Appl. Math and Mech. 2010;
6; 60 – 77.
[15] Muralidharan M and Muthucumaraswamy R, Thermal radiation on
linearly accelerated vertical plate with variable temperature and uniform mass
flux, Indian Journal of Science and Technology 2010; 3; 398-401.
Copyright © 2013 SciResPub.
IJOART
International Journal of Advancements in Research & Technology, Volume 2, Issue 11, November-2013
ISSN 2278-7763
147
[16] Rajput US and Kumar S, Radiation Effects on MHD flow past an
impulsively started vertical plate with variable heat and mass transfer, Int.
J. of Appl. Math. and Mech. 2012; 8; 66-85.
[17] Chaudhary RC and Jain A, Unsteady free convection flow past an
oscillating plate with constant mass flux in the presence of radiation, Acta
Technica CSAV 2007; 52; 93-108.
[18] Ogulu A and Makinde OD, Unsteady hydromagnetic free convection flow of
a dissipative and rotating fluid past a vertical plate with constant heat flux,
Chem. Eng. Commun.2009; 196; 454–462 .
[19] Narahari M and Debnath L, Unsteady magnetohydrodynamic free convection
IJOART
flow past an accelerated vertical plate with constant heat flux and heat
generation or absorption, Z. Angew. Math. Mech. 2013; 93; 38 – 49.
[20] Chaudhary RC and Jain A, Free convection effects on MHD flow past
an infinite vertical accelerated plate embedded in porous media with constant
heat flux, Matematicas 2009; XVII; 73-82.
[21]
Abramowitz BM and Stegun IA, Handbook of Mathematical Functions, New
York: Dover Publications; 1970.
Copyright © 2013 SciResPub.
IJOART
International Journal of Advancements in Research & Technology, Volume 2, Issue 11, November-2013
ISSN 2278-7763
148
IJOART
Figure1: Temperature profile
Copyright © 2013 SciResPub.
IJOART
International Journal of Advancements in Research & Technology, Volume 2, Issue 11, November-2013
ISSN 2278-7763
149
IJOART
Figure 2: Concentration profile for k=0.2
Copyright © 2013 SciResPub.
IJOART
International Journal of Advancements in Research & Technology, Volume 2, Issue 11, November-2013
ISSN 2278-7763
150
IJOART
Figure 3 : Velocity profile for R=4, k=0.2, G=5, Gm=2
Copyright © 2013 SciResPub.
IJOART
International Journal of Advancements in Research & Technology, Volume 2, Issue 11, November-2013
ISSN 2278-7763
151
IJOART
Figure 4: Velocity profile R=4, k=0.2, SC=0.22, t=0.2
Copyright © 2013 SciResPub.
IJOART
International Journal of Advancements in Research & Technology, Volume 2, Issue 11, November-2013
ISSN 2278-7763
152
IJOART
Figure 5: Velocity profile for sc=0.22, t=0.2, G=5, Gm=2
Copyright © 2013 SciResPub.
IJOART
International Journal of Advancements in Research & Technology, Volume 2, Issue 11, November-2013
ISSN 2278-7763
153
IJOART
Figure 6: Skin-friction for R=4.
Copyright © 2013 SciResPub.
IJOART
International Journal of Advancements in Research & Technology, Volume 2, Issue 11, November-2013
ISSN 2278-7763
154
IJOART
Figure 7: Nusselt number
Copyright © 2013 SciResPub.
IJOART
International Journal of Advancements in Research & Technology, Volume 2, Issue 11, November-2013
ISSN 2278-7763
155
IJOART
Figure 8 : Sherwood number
Copyright © 2013 SciResPub.
IJOART