Research Journal of Applied Sciences, Engineering and Technology 3(9): 887-898, 2011
ISSN: 2040-7467
© Maxwell Scientific Organization, 2011
Submitted: June 14, 2011 Accepted: August 08, 2011 Published: September 20, 2011
1 Anjana Bhattacharya and 2 R.K. Deka
1 Department of Mathematics, B. Borooah College, Gauhati University,
Guwahati-781007, India
2 Department of Mathematics, Gauhati University, Guwahati-781014, India
Abstract:
The unsteady free convection flow of a viscous incompressible stably stratified fluid past an infinite vertical oscillating plate with variable temperature and mass diffusion is presented here, taking into account the homogeneous chemical reaction of first order. The plate temperature as well as the concentration level near the plate is raised linearly with respect to time. The non-dimensional governing equations are solved in closed form by the Laplace’s transform technique, when the plate is oscillating harmonically in its own plane. The effects of different parameters like phase angle, chemical reaction parameter, thermal Grashof number, mass
Grashof number, Schmidt number, stratification parameter, and time on velocity, temperature, concentration, skin-friction, Nusselt number and Sherwood number are studied.
Key words: Chemical reaction, heat and mass transfer, oscillating plate, stratified fluid
INTRODUCTION
The study of heat transfer phenomenon has attained increasing interest in a number of engineering and technological disciplines such as chemical, nuclear, metallurgy, biochemical and many other branches of chemical technology dealing with polymers, ceramics, vegetable oils, pharmaceuticals and fine chemicals. The convective heat transfer phenomenon in nature is often accompanied by mass transfer, that is, by the transport of certain substances that act as components in the fluid mixture. In many operations in the chemical industry, convective heat and mass transfer process forms the backbone.Again in many chemical engineering processes, chemical reactions take place between a foreign mass and the fluid. These reactions take place in industrial applications such as food processing, manufacturing of ceramics and polymer production etc. Chemical reaction can be codified as either heterogeneous or homogeneous processes depending on whether it takes place at an interface or as a single phase volume reaction. In well mixed system, homogeneous chemical reaction takes place in solution. In most cases of real chemical reactions, the reaction rate depends on the concentration of the species itself. A chemical reaction is said to be of first order, if the rate of reaction is directly proportional to the concentration.
Mass diffusion effects on flow past a vertical surface was investigated by Muthucumaraswamy (2006 a). Mass diffusion and natural convection flow past a flat plate was studied by many researchers like Chandrasekhara et al .
(1992) and Panda et al . (2003). Effects of mass transfer on free convection flow past a semi-infinite vertical isothermal plate was first studied by Gebhart, (1971).
Soundalgekar et al . (1984) considered the effects of mass transfer on the flow past an impulsively started infinite vertical plate with variable temperature.Flows past vertical plates oscillating in its own plane have got applications in many industrial fields. Soundalgekar
(1979) was the first to consider flow of a viscous, incompressible fluid past an infinite isothermal vertical plate oscillating in it’s own plane. Revankar (2000) solved the same problem for an impulsively started oscillating plate. The effect of mass transfer on flow past a vertical oscillating plate was investigated by Lahurikar et al . (1995). Soundalgekar et al . (1994) studied the same problem in presence of constant heat flux. The combined effect of concentration and temperature differences on flow past a vertical oscillating plate was studied extensively by Soundalgekar (1983).
The analytical solution for mass transfer with a chemical reaction of first order was investigated by
Apelblat (1980). Das et al . (1994) have studied the effect of homogeneous chemical reaction of first order on the flow past an impulsively started infinite vertical plate with uniform heat flux and mass transfer. Muthucumaraswamy
(2002) have discussed the heat and mass transfer effects on a continuously moving isothermal vertical surface with uniform suction, taking into account the homogeneous chemical reaction of first order. Muthucumaraswamy,
Corresponding Author:
Anjana Bhattacharya, Department of Mathematics, B. Borooah College, Gauhati University, Guwahati-
781007, India
887

Res. J. Appl. Sci. Eng. Technol., 3(9): 887-898, 2011
(2006b) carried out theoretical study of chemical reaction effects on vertical oscillating plate with variable temperature. Vijayalakshmi et al . (2010) carried out study on radiation and chemical reaction effects on isothermal moving vertical plate with variable mass diffusion.
Recently researchers in this field like Park and Hyun
(1998), Park (2001), Shapiro and Fedorovich (2004),
Magyari et al . (2006) and Deka (2009) etc. has modified the one dimensional natural convection flow along vertical plates by including the pressure work term in the thermodynamic energy equation, which was absent in all the previous studies discussed above. This term is combination of two terms, the pressure work and the vertical temperature advection. Since for the one dimensional model both the terms are of the same form, so both processes are combined into a single advection term and included in the energy equation. This leads to temperature stratification which allows the unsteady flows to approach a steady state in many flows. But till date study of stratification effects on combined heat and mass transfer flows in presence of chemical reaction effects are very limited. The aim of the present paper is to study the chemical reaction effects on vertical oscillating plate immersed in a stably stratified fluid. Here we have extended the work of Muthucumaraswamy (2006b) to a stratified fluid.
The energy equation is modified by including the pressure work term, which leads to temperature stratification.The non-dimensional governing equations are solved by Laplace’s transform technique for the case when Prandtl number is unity. The effect of various physical parameters such as thermal Grashof number
(Gr), mass Grashof number (Gc), Schmidt number (Sc), chemical reaction parameter (K), stratification parameter
( ( ), phase angle ( T t) and time (t) on velocity, temperature and concentration profiles are shown on graphs and discussed. Also, the effects of different parameters involved are shown graphically on skin friction ( J ),
Nusselt number (Nu) and Sherwood number (Sh).
Governing equations: Here the first order chemical reaction effects on unsteady flow of a viscous incompressible stratified fluid past an infinite vertical oscillating plate with variable temperature and mass diffusion is considered. It is assumed that there is a first order chemical reaction between the diffusing species and the fluid. To investigate the flow situation we consider a co-ordinate system, in which the x / axis is taken along the plate in vertically upward direction and the y / axis is taken normal to the plate. Initially the plate and the fluid are at the same temperature T t !
>0, the plate starts oscillating in it’s own plane with frequency T /
4
and concentration C /
4
. At time
and the temperature of the plate is raised linearly with respect to time t and the concentration level is raised with respect to t. Since the plate is infinite in extent, so all the flow variables are independent of x / and depends only on y / and t / . Thus we have a one dimensional flow field with only one non zero velocity component u / in the vertical direction. Then by usual
Boussinesqs’ approximation, the unsteady flow is governed by the following equations:
σ u
/
σ t
/
= g
β *
T
g
β
(
C C
∞
) v
σ 2 u
/
σ y
/ 2
(1)
∂
∂
T t
′
= k
ρ
C p
∂
∂ 2
T y
′ 2
γ (2)
∂
∂
C
′ t
′
=
D
∂ 2
C
′
∂ y
′ 2
− ′ (3) with the following initial and boundary conditions : t t
/
/ u /
# 0 : u
> 0 : u /
=0 where,
A
/ = 0,
= u
=
U
2
0 v
0 cos T / t / , and the term
T = T
4
T = T
4
T ÷ T
4
,
+(T w
- T
4
)At /
γ ' = dT
' ∞ dx
'
+ g
C p
C
C
C
/
/
/
= C
= C
÷ C
/
4
/
4
+(C
/
4
/ w
- C /
4
) At / for all y / at y / as y
= 0
/ ÷ 4
is the thermal stratification parameter. Here dT ' ∞ dx '
(4) is the vertical temperature advection termed as thermal stratification. Also, g/C p
is the rate of reversible work done on the fluid particles by compression, known as work of compression. As, the work of compression is very small; the parameter ( !
will be termed as thermal stratification parameter in our study. The work of compression is retained as additive one to thermal stratification for validating numerical models.
888

Res. J. Appl. Sci. Eng. Technol., 3(9): 887-898, 2011
On introducing the following non-dimensional quantities: u
Pr
=
= u
′
, u
0
µ
C p t
, k
=
Sc v
=
2
0 , v
D
, y
=
K
= v
0 , vK u
2
0 l
θ =
, ω =
T T
Tw T
,
Gr
ω ′ v u
2
0
, γ =
=
β (
T u
3
0
γ ′ v
(
T
)
)
,
C
=
C C
C w C
,
Gc
= g
β *
(
C u
3
0
)
Equations (1) to (3) yields:
∂ u
∂ t
=
Gr
θ +
GcC
+
∂ 2 u
∂ y
2
∂θ
∂ t
=
1 pr
∂ θ
∂ y
2
− γ u
(5)
(6)
(7)
∂
∂
C
=
1 t Sc
∂ 2
C
∂ y
2
−
KC
(8) the non-dimensional form of the initial and boundary conditions (4) are: t # 0: t > 0: u = 0, u = cos u = 0,
T t,
2 = 0, C = 0
2 = t, C = t
2 ÷ 0, C ÷ 0 for all y, at y = 0 at y ÷ 4
(9)
All the physical variables are defined in nomenclature. The solutions are obtained for hydrodynamic flow field in presence of first order chemical reaction.
Method of Solution: The non-dimensional governing equations (6) to (8), subject to boundary conditions (9), are solved by usual Laplace’s transform technique for tractable case of Pr = 1. The expression for concentration, temperature and velocity are obtained as:
C e
{
= t
− 2 η
2 e
{
ScKt
− 2 η
ScKt erfc
(
η
Sc
−
Kt
} )
+ e
2 η erfc
( η
Sc
−
Kt
) − e
2 η
ScKt
ScKt erfc
(
η
Sc
+
Kt erfc
( η
Sc
+
Kt
)
}
)
−
η
2
Sct
K (10)
θ =
γ
4 iA
[ ( , ω ) − f
1
( − iA i
ω ) + ( , − ω ) − f
1 ( − iA i
ω )] +
1
2
⎜
⎛
⎝
1 +
( Sc
γ 2
− 1 ) 2
⎟
⎞
⎠
{
( ) + f
3
( − iA
)
}
⎡
⎣⎢
+
2 (
γ
Gc
− 1 )
[
C
1
{
( ) − f
2
( − iA )
}
+
C
2
{ f iA
− f
3
( − iA )
}
+ ( C
3
− iC
4
)
{
4
( , + iB
1
)
− f
4
( − , +
1
)
}
+ ( C
3
+ iC
4
)
{
4
( , −
1
) − f
4
( − , −
1
)
} ]
+
γ
Gc
2 ( Sc
− )
[ { ( )
+ f
2
(
− iA
) }
+
(
D
3
+ iD
4
) { (
, +
1
)
+ f
4
(
− , +
1
) }
+
(
D
3
− iD
4
) { (
, −
1
)
+ f
4
(
− , −
1
) ] }
−
γ
Gc
(
Sc
− 1
) 2
2
1 (
D
1
+
{
)
− 2 η erfc
(
η
Sc
−
Kt
)
+ e
2 η erfc
(
η
Sc
+
Kt
} )
−
η
D
2
.
2
K
889

Res. J. Appl. Sci. Eng. Technol., 3(9): 887-898, 2011
{
+
(
− 2 η
(
−
3
+
)
( , +
1
)
−
2 η
(
+
3
−
4
)
5
(
, −
1
] )
} )
(11) u
=
1
4
[ ( , ω ) + f
1
( − , ω ) + f iA i
ω ) + f
1
( − , − ω )] + iA
2 γ f iA
− f
3
( − iA )}
+
2 (
Gc
Sc
− 1 )
C f iA
+ f
2
( − iA )} +
C
2 f iA
+ f 3 ( − iA
+
C
3
− iC
4 f iA B
+ iB
1
)
+ f
4
( − , +
1
)
} ]
+ ( C
3
+ iC
4
−
1
) + f
4
( − , −
1
)}] + iAGc
1
)}]
2 ( sc
− 1 ) 2
[ {
( ) − f
2
( − iA ) − f
2
( − iA )
}
+
D
2
{ ( ) − f
3
( − iA
+
− f
4
( − , +
1
D
3
− iD
4
−
1
) − f
4
( − , −
D
3
+ iD
4
+
1
)
−
−
(
Gc sc
− 1 )
η
C
2 sc t
2 K
⎡
⎣⎢
1
2
( C
1
+
2
)
{
{ e
− 2 η
− 2 η erfc
(
η sc
−
Kt
)
+ e
2 η erfc
(
η sc
−
Kt
)
− e
2 η
+ ( C
3
− iC
4
) ( , +
1
+
C
3
+ iC
4
) ( , −
1
)
] erfc
(
η sc
+
Kt erfc
(
η sc
+
Kt
) }
} ) where,
η =
C
4
=
2 y t
,
A
=
(
BB
1
B
2 +
B
1
)
γ
, D
1
,
= −
=
(
KSc
Sc
− 1
,
B
1
=
2 B
B
2 +
B
1
)
, D
A
Sc
−
2
=
1
,
C
1
=
1
B
2 +
B
1
2
(
,
B
B
2
1
2
D
3
+
−
=
B
B
(
2
1
)
,
B
C
2
B
2 +
B
1
2
,
C
3
B
B
2 +
B
1
)
, D
4
=
2
=
2 (
B
2 −
B
B
2 +
B
1
2
1
)
B
2 −
B
1
2
( 2 +
B
1
)
,
(12)
Also the f i
' s are inverse Laplace’s transforms given by:
( , ) =
L
− 1
⎜
⎝
⎜
⎛
( ,
1
+ iq
2
) = e
− s
+ iq
L
− 1 ⎜⎜
⎛
⎝
⎟
⎠
⎟
⎞
, ( ) =
L
− 1
⎜
⎝
⎜
⎛
− e
+ + iq 2 e
− s
⎟⎟
⎞
⎠
, f
5
( ,
1
+ iq
2
) =
⎟
⎠
⎟
⎞
, ( ) =
L
− 1
⎜
⎝
⎜
⎛ e
− s
2
L
− 1
⎜
⎝
⎜
⎛ − + e s
+ q
1
+ iq
2
)
⎟
⎠
⎟
⎞
⎠
⎟
⎞
⎟
In order to get the physical insight into the problem, the numerical values of C, 2 and u are computed from Eq. (10),
(11) and (12) for different values of the parameters. Since the above expressions involves complex arguments of the error function, hence we separate them into real and imaginary parts by using well known formulae given by Abramowitz and Stegun (1970) as: erf a
+ ib ) = ( ) +
+ 2 exp(
π
− a
2
) exp( − a
2 )
2 a
π
[ 1 − cos( 2 ab ) + i sin( 2 ab ) n
∞
∑
= 1 exp( − n
2 4 n
2 + 4 a
2
[ ( , ) + ( , )] + ε a b
890
(13)

Res. J. Appl. Sci. Eng. Technol., 3(9): 887-898, 2011 where, fn
= 2 a
− 2 a g n
= 2 a
ε ≈≈ nb nb
2 ab )
2
+
10 − 16 | ( + ib
)| ab n
) + n nb nb
2 ab )
2 ab )
For Sc = 1, the solutions are obtained as:
C * =
−
2 t
2
η
{ e
− 2 η t
Kt
{ e
− 2 η erfc
(
η −
Kt
)
+ e
2 η
Kt
Kt erfc
(
η −
Kt
)
− e
2 η erfc
(
η +
Kt
) }
Kt erfc
(
η +
Kt
) }
K
(14)
θ * =
γ
4 iA
{
1
( ω ) − f
1
( − , ω ) + (
, − ω ) − f
1
( − , − ω ) }
+
⎝
⎜
⎜
⎛
1
+
2 2
(
Gc
γ
K
2 +
A
2
)
⎟
⎠
⎟
⎞
{
η
2
⎣⎢
⎡ t
2
{
( ) + f
3
( − iA
) } e
− 2 η
Kt
−
γ
GcK
(
2 +
A
2
) erfc
(
η −
Kt
)
+ e
2 η
Kt t
K
{ e
− 2 η
Kt erfc
(
η −
Kt
)
− e
2 η
{ ( ) − f
3
( − iA
) } erfc
(
η +
Kt
) }
Kt
− erfc
(
η +
Kt
)
⎦⎥
⎤
}
−
γ
Gc
(
K
2 +
A
2
)
(15) u * =
4
1 [ f iA i
ω ) + f
1
( − iA i
ω ) + f iA i
ω ) + f
1
( − iA i
ω )] + iA
2
⎜
⎛
⎝ γ
1
+
Gc
K
2 +
A
2
⎟
⎞
⎠ f iA
− f
3
( − iA
)} −
⎡
⎣⎢ t
2
{ e
− 2 η
−
η
2
Kt erfc
( η − t
K
{ e
− 2 η
2 (
KGc
K
2 +
A
2 )
Kt
) + e
2 η
Kt erfc
( η − f iA
+ f
3
( − iA
)} −
Kt erfc
( η +
Kt
) − e
2 η
Kt
)
}
Kt erfc
( η +
KGc
K
2 +
A
2
Kt
)
}
⎦⎥
⎤
(16)
Skin-Friction: Knowing the velocity field we now find the skin friction profile ( J ), which is the measure of shear stress on the wall. In non dimensional form skin-friction is given by:
τ = du dy y
= 0
From the expression of velocity given by (12), the expression for skin friction is derived as:
τ = cos At
π t
+
1
2
+ ( r 3 + r 4 ω
A
−
2
ω
[( r 1 − r 2 ) cos ω t
+ ( r 1 + r 2
+
γ
A
⎢
⎡
⎣ ⎢ t
π sin
ω +
1
2
A
+
2
ω
[( r 3 − r 4 )cos ω t
A
2
( r 5 + r 6 ) +
2
1 sA
( r 5 − r 6 )
⎤
⎦
⎥ +
2 (
Gc
Sc
− 1 )
891

Res. J. Appl. Sci. Eng. Technol., 3(9): 887-898, 2011
[(
+
C
1
7
+
2 e
9
+
C t )
⎧
⎩
2 cos
B t
+ r
8
π t
At sin
+
1
)(
2 ( 5 − r 6 )
⎫
⎭
1 3
−
Q C
B t
− r
10 sin
1
)(
2 3
−
Q C
−
+
C
2
2
A
( r 5 + r 6 ) + 2 e
−
Bt
8
B t
− r
7 sin
1
)(
1 3
−
PC
)
P C
+
Q C
)( r
10 cos
B t
+ r
9 sin
B t
)}
1
2
+
2
1
π t
{
C
3 cos(
B t
+ 2
PQ t
) −
C
4 sin(
B t
+ 2
PQ t
} +
2 e
B t
− 2
P Q t
) −
C
4 sin(
B t
− 2
P Q t
)}] −
2 (
AGc
Sc
− 1 ) 2
⎢
⎡
⎣ ⎢
⎧
⎨
⎩
−
π
2
2
+ t sin
At
π t
+
3
2
2
A r
5 + r
6 )
⎫
⎬
⎭
(
D
1
+
D t
) +
D
2
π t
( r
6 − r
5 ) + 2 e
−
Bt {(
D r
7 −
D r
8 )(
Q
1 cos
B t
−
P
1 sin
B t
+
D r
8
( cos B t
+
Q
1 sin B t
+
D r 9 +
D r 10 )( Q
2 cos B t
+
P
2 sin B t
−
D r 9 −
D r 10 )
+
D r
7 )
( cos
B t
−
Q
2 sin
B t
)} +
2 e
π t
{
D
4 cos(
B t
+ 2
PQ t
) − where,
2 e
D
3 sin(
B t
+ 2
PQ t
)} + {
D
3 sin(
B t
− 2
P Q t
) −
D
4 cos(
B t
− 2
P Q t
)}]
π t
+
Gc
Sc
− 1 ⎢
⎣
⎢
⎡
⎩⎪
− e
Kt
− 2 e
−
Bt
Sc
{(
C P
Sc
π t
−
−
C Q
Sc Kerf
( )
⎭⎪
(
C
1
+
C t
)( r
11 cos
) −
C
2
Sc
2
K erf
( )
B t
− r
12 sin
B t
) (
C Q
+
C P
)( r
11 sin
B t
+ r
12 cos
B t
)} −
2 e
π t
{
3 cos
(
B t
− 2
P Q t
)
−
C
4 sin
(
B t
− 2
P Q t
) }
⎥
⎦
⎥
⎤
B
2
=
(
B
2
P
3
+ iQ
3
=
+
+ (
A
−
B
1
B
3 iQ
1 t
)
= r
7 +
=
K
− +
(
( ir
8 ,
B
2
+ (
A
+
B
1
−
(
ω
2
) t
2 t
)
= r
1 + ir
2 , erf
(
(
+ iQ
2 t p
1
)
=
= r
9
B
2 −
B
+ ir
10 ,
,
+ erf P
3 t
B
2 +
B
+ iQ
3 t
B
3
−
B
Q
1
ω
= p
2
=
= r
11 + ir
12
,
Q
2
=
(
2
) t
)
= r
3 + ir
4 , erf iAt
= r
5 + ir
6
)
2
( )
B
3
+
B
2
,
(17)
For Sc =1, the expression for skin-friction is derived from (16) as:
τ * = cos
At
π t
+
1
2
⎢
⎡
⎣ ⎢
−
⎢
⎡
⎣ ⎢
π t sin
At
+ t t
π e
−
Kt + ⎜
⎛
⎝ 2
A
−
2
ω
[( r
1 − r
2 ) cos ω t
+ ( r
1 + r
2 ω +
1
2
A
+ ω
2
[( r
3 − r
4 )cos wt
+ ( r
3 + r
4 )sin ω t
−
A
(
γ
1
+
A
2
( r
5 + r
6 ) +
Kt
K
⎟
⎞
⎠ erf
( )
⎥
⎤
⎦ ⎥
1
2 2
A
( r
6 − r
5 ) ⎥
⎤
⎦ ⎥
−
(
GcK
K
2 +
A
2 )
[ t
π cos
At
+ t
A
2
( r
5 − r
6 ) +
1
2 2
A
( r
5 + r
6 )] −
Gc
K
2 +
A
2
)
GcK
(
K
2 +
A
2 )
(18)
The complex arguments of the error functions are calculated by using well known formulae given by (13).
Nusselt Number: In non-dimensional form plate heat flux (Nusselt number) is given by: Nu
= d
θ dy y
= 0
892

Res. J. Appl. Sci. Eng. Technol., 3(9): 887-898, 2011
From the expression of temperature ( 2 ), given by (11), the expression for Nusselt number is derived as:
Nu
= −
γ
A
⎢
⎡
⎣ ⎢ sin At
π t
+
ω −
2 2
A
{( r 2 + 1 ) ω + ( r 1 − r 2 ω t
−
ω −
2 2
A
{( r 3 + r 4 ) cos ω t
+ ( r 3 − r 4 t
]
+ t
A
2
( r 5 − r 6 ) + t
π cos At
+
1
2 2 A
( r 5 + r 6 )
−
−
2
γ
Gc
− 1 )
[( C
1
+
C t ) ⎨
⎩
2 sin
π t
At
− 2 A r 5 + r 6 ) ⎬
⎭
−
C
2
2 A
( r 6 − r 5 ) +
2 exp(( −
Bt
−
1
2
P t
−
Q
1
2
π t
))
(
( C
3 sin( B t
+ 2 PQ t )) +
C
4
C
3 sin
( cos( B t
+ 2 PQ t )
B t
+ 2
PQ t
) )
+
C
4 cos
(
−
2 exp(
B t
+ 2
PQ t
)
− 2 e
−
Bt
−
Bt
−
P
2
2 −
2
2
Q t
π t
( {
+
C Q
) ( r
8 cos
B t
− r
7 sin
B t
)
( C P
−
C P r B t
+ r 8 sin )}] − 2 e
−
Bt {( C P
−
C Q )( r 10 cos B t
+ r 9 sin )
+
−
−
(
C Q
+
C P r B t
− r
10 sin
B t
)}] −
γ
2 (
Sc
Gc
− 1 ) 2
⎢
⎡
⎣ ⎢
− (
D
1
+
D t
)
⎩
⎧
⎨ 2
A r
5 − r
6 +
2 cos
At
π t
D
2
2
A
( r
5 + r
6 ) −
2 exp( −
Bt
−
P t
+
Q t
π t
(
2 exp( −
Bt
−
P t
+
Q t
π t
3
(
1
+ 2
PQ t
) +
D
4 sin(
B t
+ 2
PQ t
))
(
D Cos
(
B t
− 2
P Q t
) +
D
4 sin(
B t
− 2
P Q t
)) − 2 e
−
Bt {(
D P
−
D Q
)
⎫
⎬
⎭
( r 9
7 B t
+ r 8 sin B t
−
D P 1 +
D Q r cos B t
− r 10 sin B t
)
+
(
D P
−
D Q
( ) r 10 cos
B t
− r 7
B t
+ r 9 sin B t )} sin
)
−
] }
2 e
−
Bt {( D P
+
D Q )
+
γ
Gc
( Sc
− 1 ) 2
[(
D
1
+
D t
)
⎩⎪
−
Sc
π t e
−
Kt − .
( )
( D
3 cos(
+ ( D P
B t
−
− 2
D Q
P Qt
)( r 12
+
D cos
4 sin( B t
− 2 P Q t ))
B t
+ r 11 sin )}]
− 2 e
−
Bt
⎭⎪
−
D
2
2
Sc erf
( )
K
−
2 exp( −
Bt
−
P
3
2 t
+
3
2
Q t
π t
Sc {( D P
+
D Q )( r 11 cos B t
− r 12 sin )
For Sc = 1, the expression for Nusselt number is obtained using (15) as:
−
⎢
⎡
⎣ ⎢ t
⎢
⎡
⎣ ⎢
Nu * = −
γ
A
⎢
⎡
⎣ ⎢ sin At
π t
+
ω −
2 2
A
{( r 2 + r 1 )cos ω t
+ ( r 1 − r 2 ω t
π
ω
2 2
A
2
−
( r
A
5 −
{( r r 3 + r 4
6 ) + t
π
) cos ω cos t
At
+ ( r 3
1
2 2
−
A r 4 ω t
⎤
⎦
⎥
⎥
+ ( 1 +
( r
5 + r
6 ) ⎥
⎤
⎦ ⎥
−
γ
A K
2
K
GcK
+
2
γ
A
Gc
+
2 )
A
2
2 sin
At
− t
2
A r
5 + r
6 ) −
1
2
A
( r
6 − r
5 ) ⎥
⎤
⎦ ⎥
−
K
2
γ
Gc
+
A
2
⎢
⎡
⎣ ⎢
) t
π e
−
Kt +
2
Kt
2
+
K
1 erf
( )
⎥
⎤
⎦ ⎥ where all the constants involved are defined earlier.
893
(19)
(20)

Res. J. Appl. Sci. Eng. Technol., 3(9): 887-898, 2011
Sherwood number: The Sherwood number, which is measured as the rate of mass transfer, in non-dimensional form is given by:
Sh
= dC dy y
− 0
From the expression of concentration C , given by (10), the expression for Sherwood number is derived as:
Sh
=
Sc
K
⎝⎜
⎛ 2 tK
2
+ 1
⎠⎟
⎞ erf
( )
+
π e
−
Kt
It can be observed from the above result that, Sherwood number increases with time.
(21)
0.4
0.35
0.3
0.25
0.2
C
0.15
0.1
Sc = .6, K = .2
Sc = .6, K = 2
Sc = 1,
Sc = 2.01,
K = .2
K = .2
0.05
0
0 0.5
η
1 1.5
2
Fig.
1: Concentration profile for different values of Sc and K at time t = 0.4
1.6
1.4
1.2
1
γ
= .01
γ = .05
γ = .1
γ
= .5
0.8
U
0.6
0.4
0.2
0
0 0.5
1
η
1.5
2 2.5
3
Fig. 2: Effect of ( on velocity profile when T t = 0
DISCUSSION
In order to understand physical meaning of the problem, we have computed numerical values of velocity,
0.5
0.4
0.3
U
0.2
0.1
γ = .01
γ
= .1
γ
= .5
0
-0.1
-0.2
-0.3
0
0.5
1
η
1.5
2 2.5
Fig. 3: Effect of ( on velocity profile when T t = B /2
3 temperature, skin-friction and Nusselt number from their solutions obtained in the proceeding sections, for different values of the physical parameters Gr, Gc, Sc, T t, ( , K and time t and are plotted in Fig. 1 to 14.
Figure 1 shows the effect of Sc and K on the concentration at time t = 0.4. Here we have observed that concentration decreases as both Sc and K increases.
Increase in Sc means decrease in mass diffusion, which leads to decrease in concentration. Also the concentration is highest on the surface and falls monotonically.
Figure 2 and 3 represents velocity profile for different values of ( when T t = 0 and T t = B /2, respectively. We have observed that velocity increases with time. Also velocity decreases as ( increases for both the cases (i.e.,
T t = 0 and T t = B /2). This is due to the reason that stratification produces a resistive type of force in the flow which opposes the motion of the fluid. Moreover in both the cases for higher value of ( , a velocity first decrease near the plate and then increases and finally decreases to a zero value far away from the plate. Figure 4 to 7 represent respectively the effects of T t (0, B /4, B /3, B /2),
K(0.2, 1, 1.5), Sc(0.6, 1, 2.01), Gr (5, 10) and Gc (5, 10)
894

Res. J. Appl. Sci. Eng. Technol., 3(9): 887-898, 2011
2.5
2
1.5
U
1
0.5
ω t = 0
ω t = π /4
ω t = π /3
ω t = π /2
0
0 0.5
1
η
1.5
2 2.5
3
Fig. 4: Effect of T t on velocity profile for Gr = 5, Gc = 5, Sc =
0.6, ( = 0.01,K = 0.2
1.6
1.4
1.2
1
U 0.8
0.6
0.4
0.2
0
0 0.5
1
η
1.5
2
K = 0.2
K = 1.0
K = 1.5
2.5
3 3.5
Fig. 5: Effect of K on velocity profile for Gr = 5, Gc = 5, Sc =
0.6, T t = B /3, ( = 0.01
1
U
0.8
0.6
0.4
0.2
1.6
1.4
1.2
0
0 0.5
η
1 1.5
2
Sc - 0.6
Sc - 1.0
Sc - 2.01
2.5
3
Fig. 6: Effect of Sc on velocity profile for Gr = 5, Gc = 5, K =
0.2, T t = B /3, ( = 0.01 on velocity field. From the figures we have observed that velocity decreases as T t, K, Sc and Gr increases, but increases as Gc increases. Increase in Sc means decrease
895
3
2.5
2
1.5
U
1
Gr = 5 Gc = 10
Gr = 5 Gc = 5
Gr = 10 Gc = 5
0.5
0
0.5
1 1.5
2 2.5
3
Fig. 7: Effect of Gr and Gc on velocity profile for Sc = 0.6, T t
= B /3, ( = 0.01, K = 0.2
0.4
0.35
0.3
0.25
0.2
θ 0.15
0.1
0.05
0
0 0.5
η
1
K = 0.2
K = 0.2
γ = 0.05
γ
= 0.01
K = 2.0
γ = 0.01
1.5
2
Fig. 8: Effect of K and ( on temperature profile for Gr = 5, Gc
= 5, Sc = 0.6, T t = B /3
0.6
0.5
0.4
0.3
θ
0.2
t = 0.6
t = 0.4
t = 0.2
0.1
0
0 0.5
η
1 1.5
2
Fig. 9: Temperature profile at different time in kinematic viscosity, which leads to decrease in fluid velocity. Another observation is that velocity increases near the plate and then gradually decreases to a zero value at larger distance from the plate.

Res. J. Appl. Sci. Eng. Technol., 3(9): 887-898, 2011
0.4
0.35
0.3
0.25
0.2
θ
0.15
0.1
0.05
0
0
Gr = 5 Gc = 15
Gr = 5
Gr = 15
Gc = 5
Gc = 5
0.5
η
1 1.5
2
Fig. 10: Effect of Gr and Gc on temperature profile for ( =
0.01, K = 0.2, Sc = 0.6, T t = B /3
0.4
0.35
0.3
0.25
0.2
O
0.15
0.1
0.05
0
0
Sc = 0.4
Sc = 0.6
Sc = 0.1
0.5
η
1 1.5
2 2.5
Fig. 11: Effect of Sc on temperature profile for Gr = 5, Gc = 5,
K = 0.2, ( = 0.01, T t = B /3
0
-2
-4
-6
-8
τ
-10
-12
-14
-16
0
γ
0.01
0.05
0.01
0.01
K Sc
0.2
0.2
0.6
0.2
0.6
0.5 0.6
1.0
t
0.5
1 1.5
2
Fig. 12: Effect of ( , Sc and K on skin-friction for Gr = 5, Gc =
5, T t = B /3
Effects of different parameters on temperature profile are represented in Fig. 8 to 11. It is observed that temperature increases with time as well as with increase in both ( and Gc. Also temperature decreases with increase in K, Sc and Gr. Furthermore temperature is
896
0
-2
-4
-6
-8
-10
-12
τ
-14
-16
-18
-20
0
ω t
Π /3
Gr Gc
10 5
Π
Π
Π
/4
/3
/3
5
5
5
5
5
10 t
0.5
1 1.5
2
Fig. 13: Effect of Gr, Gc and T t on skin-friction for ( = 0.01,
Sc = 0.6, K = 0.2
1.8
1.4
1.2
1
0.8
0.8
Nu
0.4
0.2
0
0 t
0.5
Sc = 0.3, K = 0.5
Sc = 0.1, K = 0.2
Sc =0.3, K = 0.2
Sc = 0.16, K=0.2
1 1.5
2
Fig. 14: Effect of Sc and K on Nusselt number for Gr = 5, Gc
= 5, ( = 0.01, T t = B /3
Nu
0.8
0.6
0.4
0.2
0
0
2
1.8
1.6
γ = 0.05, Gc = 5
γ
= 0.01, Gc = 10
1.4
1.2
γ = 0.01, Gc = 5
1 t
0.5
1 1.5
2
Fig. 15: Effect of Gc and ( on Nusselt number for Gr = 5, Sc =
0.6, K = 0.2, T t = B /3 highest at the plate and decreases monotonically to a zero value at larger distance from the plate.
Figure 12 and 13 represent effects of various parameters on skin-friction profile. We observe that skinfriction decreases gradually with time. Also skin-friction

2
1.6
1.2
0.8
Sh
0.4
0
0
Sc = 0.16 K = 0.3
Sc = 0.16 K = 0. 5
Sc = 0.3 K = 0.3
Sc = 1.0 K = 0.3
t
0.5
1
Res. J. Appl. Sci. Eng. Technol., 3(9): 887-898, 2011
1.5
Fig. 16: Effect of Sc and K on Sherwood number
2 increases as K, ( , Sc, Gc and T t increases. Whereas it decreases as Gr increases.
Figure 14 and 15 shows effects of various parameters on Nusselt number. Figures show an increasing trend of
Nusselt number with time. Also the value of Nusselt number increases with increase in Gc, Sc, K, and ( .
Effects of Gr and T t on Nusselt number is very negligible and hence not shown in figure.
Figure 16 represents the Sherwood number against time for different values of Sc(0.16, 0.3, 1) and K(0.2,
0.5). It is observed that Sherwood number increases with time. Also Sherwood number increases as both Sc and K increases.
C dimensionless concentration
C p specific heat at constant pressure
D mass diffusion co-efficient
Gc mass Grashof number
Gr thermal Grashof number g acceleration due to gravity k thermal conductivity
K l chemical reaction parameter
K dimensionless chemical reaction parameter
Pr Prandtl number
Sc Schimdt number
T temperature of the fluid near the plate t !
time t dimensionless time u !
velocity of fluid in x u
0 velocity of the plate
/ direction u dimensionless velocity
Greek symbols:
$ volumetric co-efficient of thermal expansion
$ * concentration
: co-efficient of viscosity with
< kinematic viscosity
D density of the fluid
J dimensionless skin-friction
2 dimensionless temperature
T t phase angle
0 similarity parameter
( !
dimensional stratification parameter
( dimensionless stratification parameter
CONCLUSION
From the above study, the following conclusions have been made:
C Concentration decreases as both Sc and K increases.
C Concentration is highest at the plate.
C Velocity decreases as ( , T t, K, Sc and Gr increases, but increases as Gc increases.
C Temperature increases with time as well as with increase in both ( and Gc. But temperature decreases with increase in K, Sc and Gr.
C Temperature is highest at the plate and decreases to a zero value at larger distance from the plate.
C Skin-friction decreases gradually with time as well as with an increase in Gr. Whereas skin-friction increases as K, ( , Sc, Gc and T t increases.
C Nusselt number increases with time. Also the values of Nusselt number increases with increase in Gc, Sc,
K, and ( .
C Sherwood number increases with time as well as with increase in Sc and K.
A constant
NOMENCLATURE
C !
species concentration in the fluid
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