Fl35967977

J. Prathap Kumar et al. Int. Journal of Engineering Research and Application
ISSN : 2248-9622, Vol. 3, Issue 5, Sep-Oct 2013, pp.967-977

RESEARCH ARTICLE
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Effect of first order chemical reaction in a vertical double-passage
channel
J. Prathap Kumar, J.C. Umavathi, and Jagtap Sharadkumar
Department of Mathematics, Gulbarga University, Gulbarga, Karnataka – 585 106, India

ABSTRACT:
Fully developed laminar mixed convection flow in a vertical channel in the presence of first order chemical
reactions has been investigated. The channel is divided into two passages by means of a thin, perfectly plane
conducting baffle and hence the velocity, temperature and concentration will be individual in each stream. The
coupled, nonlinear ordinary differential equations are solved analytically using regular perturbation method
valid for small values of Brinkman number. The effects of thermal Grashoff number, mass Grashoff number,
Brinkman number and chemical reaction parameter on the velocity, temperature and concentration fields at
different positions of the baffle are presented and discussed in detail. The increase in thermal Grashof number,
mass Grashoff number and Brinkman number enhances the flow, whereas the chemical reaction parameter
suppress the flow at all baffle positions.

I.

INTRODUCTION:

Convective heat transfer in a vertical
channel has practical importance in many
engineering systems. Examples are solar energy
collection, the design of heat exchangers and the
cooling of electronic systems. A convection
situation, in which the effects of both forced and
free convection are significant is commonly referred
to as mixed convection or combined convection.
The effect is especially pronounced in situations
where the forced fluid flow velocity is moderate
and/or the temperature difference is very large. In
mixed convection flows, the forced convection
effects and the free convection effects are of
comparable magnitude. Thus, in this situation, both
the forced and free convection occurs
simultaneously, that is mixed convection occurs in
which the effect of buoyancy forces on a forced
flow or the effect of forced flow on buoyant flow is
significant. Merkin [1, 2] has studied dual solutions
occurring in the problem of the mixed convection
flow over a vertical surface through a porous
medium with constant temperature for the case of
opposing flow. Aly et al. [3] and Nazar and Pop [4]
have investigated the existence of dual solutions for
mixed convection in porous medium.
The techniques of mixed convection to
enhance heat transfer is an important main objective
in numerous engineering applications of the micro
and nano fluids including pumping systems,
chemical catalytic reactors, electronic system
cooling, plate-fin heat exchangers, etc. Moreover,
the arrangement of baffles may be used to cool
down the temperature in the passage for the
thermally developed flow. When the channel is
divided into several passages by means of plane
baffles, as usually occurs in heat exchangers or
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electronic equipment, it is quite possible to enhance
the heat transfer performance between the walls and
fluid by the adjustment of each baffle position and
strengths of the separate flow streams. In such
configurations, perfectly conductive and thin baffles
may be used to avoid significant increase of the
transverse thermal resistance. Stronger streams may
be arranged to occur within the passages near the
channel wall surfaces in order to cool or heat the
walls more effectively. Even though the subject of
channel flow have been investigated extensively,
few studies have so far evaluated for these effects.
Recently, Prathap Kumar et al. [5, 6] and Umavathi
[7] analyzed the free convection in a vertical double
passage wavy channel for both Newtonian and nonNewtonian fluids.
The growing need for chemical reactions in
chemical and hydro metallurgical industries requires
the study of heat and mass transfer in the presence
of chemical reactions. There are many transport
processes that are governed by the simultaneous
action of buoyancy forces due to both thermal and
mass diffusion in the presence of chemical reaction
effect. These processes are observed in nuclear
reactor safety and combustion systems, solar
collectors as well as chemical and metallurgical
engineering. Das et al. [8] have studied the effects of
mass transfer on the flow started impulsively past an
infinite vertical plate in the presence of wall heat
flux and chemical reaction. Muthucumaraswamy
and Ganeshan [9, 10] have studied the impulsive
motion of a vertical plate with heat flux/mass and
diffusion of chemically reactive species. Seddeek
[11] has studied the finite element method for the
effects of chemical reaction, variable viscosity,
thermophoresis, and heat generation/absorption on a

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boundary layer hydro magnetic flow with heat and
mass transfer over a heat surface.
When the previous studies are reviewed,
one can see there was no much study performed on
heat and mass transfer on a vertical enclosure with
baffle. Therefore, an analysis is made to understand
heat and mass transfer in a vertical channel by
introducing a perfectly thin baffle for viscous fluid
in the presence of first order chemical reaction.

II. MATHEMATICAL FORMULATION:
Consider a steady, two-dimensional
laminar fully developed mixed convection flow in
an open ended vertical channel filled with pure
viscous fluid and is concentrated. The X-axis is
taken vertically upward, and parallel to the direction
of buoyancy, and the Y-axis is normal to it (see
Fig. 1). The walls are maintained at a constant
temperature. The fluid properties are assumed to be
constant. The channel is divided into two passages
by means of thin, perfectly conducting plane baffle
and each stream will have its own pressure gradient
and hence the velocity will be individual in each
stream.
X

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d 2C2
(6)
 K 2C2  0
dy 2
subject to the boundary conditions on velocity,
temperature and concentration as
U1  0 , T1  TW1 , C1  C1 at Y  h
D

U 2  0 , T2  TW2 , C2  C2 at Y  h
dT1 dT2
, C1  C1 ,

dY dY

(7)
C2  C2 at Y  h*
Introducing the following non-dimensional
variables
Ti  TW2
U
g T Th3
i 
,
,
ui  i ,
Gr 
TW1  TW2
2
U1

U1  0 , U 2  0 , T1  T2 ,

Gc1 

Br 

1 

g C1C1h3

2

, Gc2 

g C 2 C2 h3

2

, Re 

U1h



,

U12 
h 2 P
, p
, i  1, 2 , T  TW2  TW1 ,
U1 X
K T

C  C01
,
C1  C01

2 


C2  C2  C02 , Y 

C  C02
,
C1  C02

C1  C1  C01 ,

C C
h*
y
, y* 
, n1  11 01 ,
C1  C01
h
h

C2  C02
.
(8)
C21  C02
One obtains the momentum, energy and
concentration equations in the non-dimensional
form in stream-I and stream-II as
Stream-I
d 2u1
(9)
 GRT 1  GRC11  p  0
dy 2
n2 

g
Stream-I

Stream-II
Y

2

 du 
d 21
 Br  1   0
dy 2
 dy 

Y h

Y  h
Figure 1. Physical configuration.

The governing equations for velocity, temperature
and concentrations are
Stream-I
d 2U1
P
 g T T1  TW2   g C1 (C1  C01 ) 

0
X
dY 2
(1)





2

d 2T1
 dU 
 1   0
dY 2
 dY 
2
d C1
D
 K1C1  0
dY 2
Stream-II
K

 g T T2  TW    g C 2 (C2  C02 ) 
2

(2)
(3)
d 2U 2
P

0
X
dY 2
(4)

2

K

d 2T2
 dU 2 

 0
dY 2
 dY 

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(5)

(10)

d 21
 121  0
dy 2
Stream-II
d 2 u2
 GRT  2  GRC 22  p  0
dy 2

(11)

(12)

2

 du 
d 2 2
 Br  2   0
2
dy
 dy 
2
d 2
2
  2 2  0
dy 2
subject to the boundary conditions,
u1  0 , 1  1 , 1  1 at y  1
u2  0 ,  2  0 , 2  n2 , y  1

u1  0 ,

u2  0 ,

2  1 at y  y *

1   2 ,

(13)
(14)

d1 d 2
, 1  n1 ,

dy
dy
(15)

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K2
K1
Gr
, 2  h
, GRT 
and
D
D
Re
Gc
Gc
 1 , GRC 2  2 .
Re
Re

where 1  h

GRC1

II.

SOLUTIONS:

Equations (9), (10), (12) and (13) are
coupled non-linear ordinary differential equations.
Approximate solutions can be found by using the
regular perturbation method. The Brinkman number
is chosen as the perturbation parameter. Adopting
this technique, solutions for velocity and
temperature are assumed in the form
ui  y   ui 0  y   Brui1  y   Br 2ui 2  y   ...
(16)

i  y   i 0  y   Bri1  y   Br i 2  y   ...
2

(17)

Substituting Equations (16), (17) in
Equations (9), (10), (12) and (13) and equating the
coefficients of like power of Br to zero and one, we
obtain the zeroth and first order equations as
Stream-I
Zeroth order equations
d 210
(18)
0
dy 2
d 2u10
 GRT 10  GRC11  p  0
dy 2
First order equations

d 2u20
 GRT  20  GRC 22  p  0
dy 2
2

d11 d 21
, at y  y *

dy
dy
(27)
The solutions of zeroth and first order equations
(18) to (25) using the boundary conditions as given
in equations (26) and (27) are given as follows
Stream-I
(28)
10  c1 y  c2

u11  0 , u21  0 , 11   21 ,

u10  A1  A2 y  r1 y 2  r2 y 3  r3Cosh 1 y 
 r4 Sinh 1 y 

(29)

Stream-II
20  c3 y  c4

(30)

u20  A4  A3 y  r5 y  r6 y  r7 Cosh  2 y 
2

3

 r8 Sinh  2 y 

(31)

Stream-I
11  E1 y  E2  p1 y 2  p2 y 3  p3 y 4  p4 y 5  p5 y 6

 p6Cosh  21 y   p7 Sinh  21 y   p8Cosh 1 y 
 p9 Sinh 1 y   p10 yCosh 1 y   p11 ySinh 1 y 
 p12 y 2Cosh 1 y   p13 y 2 Sinh 1 y 

(32)
u11  E5 y  E6  R1 y 2  R2 y 3  R3 y 4  R4 y 5  R5 y 6
 R6 y 7  R7 y 8  R8Cosh  21 y 
 R9 Sinh  21 y   R10 Cosh 1 y   R11 Sinh 1 y 

2

d 11  du10 

 0
dy 2  dy 
d 2 u11
 GRT 11  0
dy 2
Stream-II
Zeroth order equations
d 2 20
0
dy 2
2

(19)

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(20)

 R12 yCosh 1 y   R13 ySinh 1 y 
 R14 y 2 Cosh 1 y   R15 y 2 Sinh 1 y 

(33)

(21)
Stream-II
 21  E4  E3 y  s1 y 2  s2 y 3  s3 y 4  s4 y 5  s5 y 6

 s6 Cosh  2 2 y   s7 Sinh  2 2 y   s8Cosh  2 y 

(22)

 s9 Sinh  2 y   s10 yCosh  2 y   s11 ySinh  2 y 
 s12 y 2 Cosh  2 y   s13 y 2 Sinh  2 y 

(23)

d 2 21  du20 

(24)
 0
dy 2  dy 
d 2 u21
(25)
 GRT  21  0
dy 2
The corresponding zeroth order and first order
boundary conditions are
u10  0 , 10  1 at y  1
u20  0 , 20  0 at y  1
(26)
d10 d 20

at y  y *
u10  0 , u20  0 , 10   20 ,
dy
dy

(34)
u21  E8  E7 y  w1 y 2  w2 y 3  w3 y 4  w4 y 5  R5 y 6
 w6 y 7  w7 y 8  w8Cosh  2 2 y   w9 Sinh  2 2 y 
 w10 Cosh  2 y   w11 Sinh  2 y 
 w12 yCosh  2 y   R13 ySinh  2 y 
 w14 y 2 Cosh  2 y   w15 y 2 Sinh  2 y 

(35)
Solutions of equations (11) and (14) can be obtained
directly and are given as follows
1  B1Cosh 1 y   B2 Sinh 1 y 
(36)

2  B3Cosh(2 y)  B4 Sinh(2 y)

(37)

u11  0 , 11  0 at y  1
u21  0 , 21  0 at y  1

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969 | P a g e

III.

RESULTS AND DISCUSSIONS:

The velocity, temperature, and concentration
fields for a viscous fluid in a vertical channel in the
presence of first order chemical reaction containing a
thin conducting baffle is studied analytically. The
non-linear coupled ordinary differential equations
governing the flow have been solved by regular
perturbation method. The thermal Grashoff number
GRT , mass Grashoff numbers GRC   GRC1  GRC 2  ,
pressure gradient p , chemical reaction parameter 
are fixed as 5, 5,-5, and 1 respectively, for all the
graphs except the varying one.
The effect of thermal Grashoff number GRT
on the velocity and temperature is shown in Figures
2a,b,c and 3a,b,c respectively at three different baffle
positions. As the thermal Grashoff number increases,
the velocity increases near the left (hot) wall, and
also at the right (cold) wall at all the baffle positions
in both the streams. The maximum point of velocity
is in stream-II for the baffle position at y*  0.8 ,
where as it is in stream-I for the baffle position at
y*  0.0 and 0.8. From Figure 3a,b,c it is seen that
when the baffle position is near the hot wall, the
temperature increases in stream-I where as in
stream-II it increases up to y  0.5 and decreases
from y  0.5 as seen in Figure 3a. The temperature
increases continuously in both the streams when the
baffle is at the center of the channel and at the right
wall as seen in Figure 3b and 3c respectively. The
increase in temperature with increase in thermal
Grashoff number is obvious. As the increase in
thermal Grashof number increases the buoyancy
force and hence increases the velocity and
temperature fields.
The effect of mass Grashoff number
GRC   GRC1  GRC 2  on the flow is shown in Figure
4a,b,c and Figure 5a,b,c at all the baffle positions.
The effect of GRC is to increases the velocity and
temperature in both the streams at all baffle positions.
When the baffle position is near the left and right
walls, there is no much effect of mass Grashoff
number on the flow field. There is a distinct increase
in velocity and temperature field in stream-II when
the baffle position is near the left wall, in stream-I
and in stream-II when the baffle position is in the
middle of the channel and in stream-I when the baffle
position is near the right wall. From Figure 5a,b,c it is
seen that there is reversal of temperature field in
stream-II when the baffle position is near the left and
middle of channel, where as there is no such reversal
of the temperature field when the baffle position is at
the right wall. This is due to the fact that the left wall
is at higher temperature when compared to right wall.
Further increase in mass Grashoff number implies
increase in concentration buoyancy force and hence
increases the flow field. The similar result was also

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observed by Muturaj and Srinivas [12] without
inserting the baffle in a vertical wavy channel.
The effect of Brinkman number Br on the
flow field is shown in Figures 6a,b,c and 7a,b,c. As
the Brinkman number increases both the velocity and
temperature increases in both the streams at all the
baffle positions. However the increase in velocity and
temperature is highly significant in stream-I when the
baffle position is near the left wall, in both the
streams when the baffle position is in the middle e of
the channel and in stream-I when the baffle position
is at the right wall. This is due to the fact that as the
Brinkman number increases the viscous dissipation
also increases which results in the increase of the
flow field.
The effect of first order chemical reaction
parameter    1   2  on the velocity, temperature
and concentration is shown in Figures 8a,b,c, 10a,b,c,
and 11a,b,c respectively. As  increases the velocity,
temperature and concentration decreases in both the
streams at all the baffle positions. This is due to the
facts that increase in  increase the concentration of
the fluid which results in the reduction of heat and
mass transfer. The increase of chemical reaction
parameter was to reduce the velocity, concentration
field as observed by Shivarih and Anand Rao [13] for
unsteady MHD free convection flow past a vertical
porous plate.

IV.

CONCLUSIONS:

The effect of first order chemical reaction on
heat and mass transfer of viscous fluid in a vertical
double-passage channel by inserting perfectly thin
conducting baffle was analysed. According to the
results following conclusions can be drawn.
[1] Increasing the values of thermal Grashoff
number increases the velocity and temperature in
both the streams at all the baffle positions.
[2] Increasing the value of mass Grashoff number
increases the velocity in both the streams at all
the baffle positions. The temperature increases in
stream-I where as it reveres its direction near the
right wall when the baffle position is at the left
wall. The temperature field increase in both the
streams as mass Grashoff number increases,
when the baffle position is in the middle of the
channel and at the right wall.
[3] With the increase in Brinkman number, increases
the flow field in both the streams at all the baffle
positions.
[4] The increase in the chemical reaction parameter
decreases the velocity, temperature and
concentration in both the streams at all the baffle
positions.
[5] By inserting the baffle the heat transfer can be
enhanced depending on the position of the baffle.

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[7] J. C. Umavathi (2011) Mixed convection of
micropolar fluid in a vertical double-passage
channel. Int. J. Eng. Sci. Techno., 3: 197-209.
[8] Das UN, Deka RK, Soundalgekar VM (1994)
Effects of mass transfer on the flow past an
impulsively started infinite vertical plate with
constant heat flux and chemical reaction.
Forsch Ingenieurwes 60:284–287.
[9] Muthucumaraswamy R, Ganesan P (2000)
On impulsive motion of a vertical plate with
heat flux and diffusion of chemically reactive
species. Forsch Ingenieurwes 66:17 23.
[10] Muthucumaraswamy R, Ganesan P (2001)
First order chemical reaction on flow past an
impulsively started vertical plate with
uniform heat and mass flux. Acta Mech
147:45 57.
[11] Seddeek MA (2005) Finite element method
for the effects of chemical reaction, variable
viscosity,
thermophoresis
and
heat
generation/absorption on a boundary layer
hydromagnetic flow with heat and mass
transfer over a heat surface. Acta Mech
177:1–18.
[12] R.Muthuraj, S. Srinivas heat and mass
transfer in a vertical wavy channel with
traveling thermal waves and porous medium.
Computers
and
mathematicas
with
Applications 59 (2010)3516-3528.
[13] S. Shivaiah, J. Anand Rao .Chemical reaction
effect of an unsteady MHD free convection
flow past a vertical porous plate in the
presence of suction or injection. Appl.Mech.,
Vol.39, No.2, pp. 185-208, Belgrade 2012.

ACKNOWLEDGEMENT:
One of the authors J. Prathap Kumar would like to
thank UGC-New Delhi for the financial support
under UGC-Major Research Project (Project No. 41774/2012 (SR)).

REFERENCES:
[1] Merkin JH (1980) Mixed convection
boundary layer flow on a vertical surface in a
saturated porous medium. J Eng Math
14:301–303.
[2] Merkin JH (1985) On dual solutions
occurring in mixed convection in a porous
medium. J Eng Math 20:171–179.
[3] Aly EH, Elliot L, Ingham DB (2003) Mixed
convection boundary layer flow over a
vertical surface embedded in a porous
medium. Eur J Mech B, Fluids 22:529–543.
[4] Nazar R, Pop I (2004) Mixed convection
boundary layer flow over a vertical surface in
a porous medium with variable surface heat
flux. In: Recent advances and applications in
heat and mass transfer. IMECE, Kuwait, pp
87–97.
[5] J. Prathap Kumar, J.C. Umavathi, Ali J.
Chamkha and H. Prema (2011) Free
convection in a vertical double passage wavy
channel filled with a Walters fluid (model
B’). Int. J. Energy & Technology, 3: 1–13.
[6] J. Prathap Kumar, J.C. Umavathi, and H.
Prema (2011) Free convection of Walter’s
fluid flow in a vertical double-passage wavy
channel with heat source. Int. J. Eng. Sci.
Techno., 3: 136-165.
5

18

GRc=5




15

16
4

14

15

20

p=-5
n1=1
n2=1

15

16
12
3
10

10

10
10

u

u

u

12

8
2
6

5

5

8
5
GRT=1

4

GRT=1

1
4

GRT=1

2

0
-1.0 -0.5

0.0

0.5

1.0

0
-1.0 -0.5

0.0

0.5

1.0

0
-1.0 -0.5

0.0

0.5

1.0

y
y
(c)
(b)
Figure2 . velocity distribution for different values of thermal Grashoff number GR T
at (a) y*=-0.8 (b) y*=0.0 (c) y*=0.8
(a)

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y

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2.0

4.0

GRc=5


p=-5
n1=1
n2=1

15

3.5
15

3.0

15

2.5

1.5

3.0
10

10

2.5

10

2.0

5

5



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 1.0

2.0

5

GRT=1


1.5

1.5

GRT=1

GRT=1

1.0
0.5

1.0

0.5
0.5

0.0
-1.0 -0.5 0.0

0.5

1.0

y

(a)

0.0
-1.0 -0.5 0.0

0.5

1.0

0.0
-1.0 -0.5 0.0

y

(b)

0.5

1.0

y

(c)

Figure3.Temperature profile for different values of Thermal Grashoff number GRT
at (a) y*=0.8 (b) y*=0 (c) y*=0.8

4.0

14

3.5

15

12

14
GRT=5

p= -5
n1=1
n2=1


15

15

12

3.0
10

10

10
10

10

2.5
8

8

u

u

2.0

5

u

5

5

6

6
1.5
GRc=1

GRc =1

GRc=1

1

4

4
1.0

2

Sem1
t a -I5
r I

S e m 15
tr a I

2

0.5

1
0

1
0

5

5

1

- .0
10

- .9
05

- .9
00

0
-1.0 -0.5 0.0

(a)

y

Gc 1
R
=

- .8
05

- .8
00

0.5

1.0

0.0
-1.0 -0.5 0.0

y

00
.
8

0.5

1.0

05
.
8

00
.
9

05
.
9

0
-1.0 -0.5 0.0

10
.
0

0.5

1.0

y
(c)
Figure4.Velocity profile for different values of mass Grashoff number GRC
at (a) y*=-0.8 (b) y*=0 (c) y*=0.8

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(b)

972 | P a g e

5

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4

3
GRT=5


p=-5
n1=1
n2=1

15
15

4

15

3
2
10

10

3
10

 2





5

5

5

2
1

GRc=1

GRc=1

GRc=1

1
1

0
-1.0 -0.5 0.0

0.5

y

(a)

0
-1.0 -0.5 0.0

1.0

0.5

0
-1.0 -0.5 0.0

1.0

y

(b)

0.5

1.0

y

(c)

Figure5.Temperature profile for different values of mass Grashoff number
GRC at (a) y*=-0.8 (b) y*=0 (c) y*=0.8

24

5.0
GRc=5
GRT=5

p=-5
n1=1
n2=1

4.5

1

1

20
4.0

1

24

20
3.5
16
3.0

16

0.5

0.5

0.5

u

12

u

u

2.5

12
0.1

2.0
Br=0

8
0.1
Br=0

8

1.5

1.0

0.1
Br=0

4

4
1

S e mI
tr a -

S
tream 1
-II

0.5

0
.5

0.5

0
.1
0.1
0

0

- .0
1 0

0
-1.0 -0.5

(a)

- .9
0 5

- .9
0 0

0.0

y

- .8
0 5

- .8
0 0

0.5

1.0

0.0
-1.0 -0.5

(b)

0.80

0.0

y

0.5

1.0

0.85

0
-1.0 -0.5

(c)

0.90

0.95

0.0

1.00

0.5

1.0

y

Figure6.Velocity for different values of Brinkman number Br
(a)y*=-0.8 (b)y*=0 (c)y*=0.8

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973 | P a g e

16

7

14

6

GRc =5
GRT=5
p= -5
n1=1
n2=1

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12

1

14

1




12

1
1

5

10

10
4





8
0.5

8
0.5



3

2

0.5

6

4

6

4
0.1

1

0.1

2

2

Br=0

0.1
Br=0

Br=0

0
-1.0 -0.5 0.0

0.5

1.0

0
-1.0 -0.5 0.0

y

(a)

(b)

0.5

0
-1.0 -0.5 0.0

1.0

y

0.5

1.0

y

(c)

Figure7.Temperature profile for different values of Brinkman number Br
at (a) y*=-0.8 (b) y*=0 (c) y*=0.8

8

3

8
GRc=5
GRT=5

p=-5
n1=1
n2=1

7

7

6

6
2

5

5


u

4

4
u





u

3

3
1

2

Sem
t ar I

2

Sem
t a -I
r I

0 ,0 ,11
. . , .
1 5 5

0 ,0 ,11
. . , .
15 5

1

1

-. 0
1
0

-. 5
0
9

-. 0
0
9

0
-1.0 -0.5 0.0

(a)

y

-. 5
0
8

-. 0
0
8

0.5

1.0

0
-1.0 -0.5 0.0
y
(b)

00
.
8

0.5

1.0

05
.
8

00
.
9

0
-1.0 -0.5 0.0
y
(c)

05
.
9

10
.
0

0.5

1.0

Figure8.Velocity profile for different values of chemical reaction parameter 
at (a) y*=-0.8 (b) y*=0 (c) y*=0.8

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974 | P a g e

2.5

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2.0
GRc=5
GRT=5

p=-5
n1=1
n2=1

1.4

2.0

1.2

1.5
1.0




1.5



0.8



 1.0


1.0

0.6

0.4

0.5

0.5
0.2

0.0
-1.0 -0.5 0.0
y

0.5

1.0

(a)

0.0
-1.0 -0.5 0.0
y
(b)

0.5

0.0
-1.0 -0.5 0.0
y
(c)

1.0

0.5

1.0

Figure9.Temperature profile for different values of chemical reaction parameter 
at (a) y*=-0.8 (b) y*=0 (c) y*=0.8



1.0



1.0
1.0

0.5


0.5
0.5

0.9

0.9

0.8

0.8
0.9

1

0.7

1

1

0.7






0.6

0.6

0.8

1.5

0.5

0.4



Stream-I
0.3
-1.0 -0.5 0.0 0.5 1.0
y
(a)
-1.00

-0.95

-0.90

-0.85

-0.80

1.5

GRc=5
GRT=5

p= -5
n1=1
n2=1

0.7
-1.0 -0.5 0.0 0.5 1.0
y
(b)

1.5

0.5

0.4


Stream
-II
0.3
-1.0 -0.5 0.0 0.5 1.0
y
(c)
0.80

0.85

0.90

0.95

1.00

Figure10.Concentration profile for different values of chemical reaction parameter 
at (a) y*=-0.8 (b)y*=0 (c) y*=0.8

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975 | P a g e

NOMENCLATURE:
Br
Brinkman number
Cp
specific heat at constant pressure

u1 , u2

D
g

h
h

1

y

diffusion coefficients
acceleration due to gravity

 h3 g T 
Grashoff number 
Gr

2
 

 g  C 1 C1 h3
Grashoff number 
GrC1

2

GrC1 , GrC 2 modified Grashoff number

k
p

U1

baffle position

GREEK SYMBOLS
chemical reaction parameters
coefficients of thermal expansion

1 , 2








non-dimensional concentrations
density
viscosity

coefficients of concentration expansion
in
temperatures
and
T , C difference
concentration
i
non-dimensional temperature
kinematics viscosity


U h 
Reynolds number  1 
  



Re

temperatures of the boundaries

2

*

1 , 2
T
C

channel width
width of passage
thermal conductivity of fluid
non-dimensional pressure gradient

*

nondimensional velocities in stream-I,
stream-II
dimensional temperature distributions

T1 , T2
Tw , Tw

C1 , C2 the concentration in stream-I and stream-II
C01 , C02 reference concentrations

www.ijera.com

SUBSCRIPTS
i refer quantities for the fluids in stream-I and
stream-II, respectively.

reference velocity

U1 ,U 2 dimensional velocity distributions
APPENDIX:
B1 
B3 

c4 

Sinh 1 y *  n Sinh 1 

Sinh  2   n2 Sinh  2 y *

Sinh  2  Cosh  2 y *  Sinh  2 y * Cosh  2 

1
,
2

r7  

r1 

 p  GRT c2  ,

GRC 2 B3

2

2

2

, r8  

GRC 2 B4

 22

r2  

GRT c1
,
6

B4 

,

r3  

1  B3Cosh  2 y *

GRC1 B1

Sinh  2 y *

r4  

12

GRC1 B2

12

,

c1  

1
,
2

c2 

1
,
2

c3  

1
,
2

GRT c4  p
GRT c3
, r6 
,
2
6

r5 

, A2  A1  r1  r2  r3 Cosh 1   r4 Sinh 1  ,

2

3

1

1

4

1

1

1 y *

 r   y * 1  r   y * 1  r Cosh  y *  Cosh     r  Sinh  y *  Sinh    
3

5

6

7

2

2

8

2

2

y * 1

A4   A3  r5  r6  r7 Cosh  2   r8 Sinh  2  ,

4 r


2
1

p8  

 6 A1 r2 
12

2 A r 

p11  

s3 

Sinh 1 y * Cosh 1   Sinh 1  Cosh 1 y *

3

1

2

p3

n Cosh 1   Cosh 1 y *

 r  y * 1  r  y * 1  r Cosh  y *  Cosh     r  Sinh  y *  Sinh    
2

A1  
A3 

B2 

Sinh 1 y * Cosh 1   Sinh 1  Cosh 1 y *

1 4

2
1

 8 r1 r3 1  36 r2 r4 

13

12

2
1

 r4212  r3212 
4

,

p2  

2 A1 r1
,
3

 r32  r42  ,
r r
3 r22
p7   3 4 ,
p6  
,
p5  
4
8
10
2
 2 A1 r3 1  8 r1 r4 1  36 r2 r3  , p    4 r1 r4 1  24 r2 r3  ,
p9  
10
13
12

3r r
p4   1 2 ,
5

,

 4 r1 r3 1  24 r2 r4 

p1

2 A


, p12  

,

6 r2 r4

1

, p13  

6 r2 r3

1

, s1  

(2 A32  r8222  r7 222 )
2 A3 r5
, s2 
,
4
3

r r
3r r
3r6
r 2  r82
(2 A3 r8 2 2  8r5 r7 2  36r6 r8 )
(4r5  6 A3 r6 )
,
s6   7
, s7  7 8 , s8  
, s4   6 5 , s5 
4
 23
5
10
8
12
2

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2

976 | P a g e

(2 A3 r7 2 2  8r5 r8 2  36r6 r7 )

s9  

s10 

(4r5 r8 2  24r6 r7 )

2

, s11 

(4r5 r7 2  24r6 r8 )

2
G9    p1  p2  p3  p4  p5  p6 Cosh  2 1   p7 Sinh  2 1   p8 Cosh 1 
2

3

2

2

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, s12  

6r6 r8

2

, s13  

6r6 r7

2 ,

 p9 Sinh 1   p10 Cosh 1   p11 Sinh 1   p12 Cosh 1   p13 Sinh 1  

G10    s1  s2  s3  s4  s5  s6 Cosh  2  2   s7 Sinh  2  2   s8 Cosh  2 
 s9 Sinh  2   s10 Cosh  2   s11 Sinh  2   s12 Cosh  2   s13 Sinh  2  ,
G11  G4  G3 y  s1 y *2  s2 y *3  s3 y *4  s4 y *5  s5 y 6  s6Cosh  21 y *  s7 Sinh  21 y *  s8Cosh 1 y *
 s9 Sinh 1 y *  s10 y * Cosh 1 y *  s11 ySinh 1 y *  s12 y *2 Cosh 1 y *  s13 y *2 Sinh 1 y *
 p1 y *2  p2 y *3  p3 y *4  p4 y *5  p5 y *6  p6 Cosh  2 1 y *  p7 Sinh  2 1 y *  p8 Cosh 1 y *

,

 p9 Sinh 1 y *  p10 y * Cosh 1 y *  p11 y * Sinh 1 y *  p12 y *2 Cosh 1 y *  p13 y *2 Sinh 1 y *

G12  2s1 y * 3s2 y *2 4s3 y *3 5s4 y *4 6s5 y *5 2 2 s6 Sinh  2  2 y *  2 2 s7 Cosh  2  2 y *  s8 2 Sinh  2 y *
 2 s9 Cosh  2 y *  s10  y *  2 Sinh  2 y *  Cosh  2 y *   s11  y *  2 Cosh  2 y *  Sinh  2 y * 
 s12  2 y * Cosh  2 y *   2 y *2 Sinh  2 y *   s13  2 y * Sinh  2 y *   2 y *2 Cosh  2 y *   2 p1 y *

3 p2 y *2 4 p3 y *3 5 p4 y *4 6 p5 y *5 21 p6 Sinh  2 1 y *  21 p7 Cosh  2 1 y *  p81 Sinh 1 y *
 p9 Cosh 1 y *  p10  y * 1 Sinh 1 y *  Cosh 1 y *   p11  y * 1 Cosh 1 y *  Sinh 1 y * 
 p12  2 y * Cosh 1 y *   y *2 Sinh 1 y *   p13  2 y * Sinh 1 y *   y *2 Cosh 1 y * 

E1  

E3

2
 G10  G11  G12 1  y * 
9

 G


R6  
R11

 y * G12  G9  G10  G11  G12  ,
2

9

R15  

2
1

 2 p101  6 p13  GRT

14

GRT p13



2
1

, w1  

G

9

 G 1 0G  1 1 1  y * 
G
1 2
2
GRT p1
,
12

, E4  G10  E3 , R1  

R2  

GRT E1
6

R8  

,

GRT p4
GR p
, R7   T 5 ,
42
56

p


E2 

GRT p6
GR p
, R9   T 2 7 , R10  
2
41
41

, R12  

R3  

 p101  4 p13  GRT


3
1

, R13  

GRT E2
,
2

GR p
GRT p2
, R5   T 3 ,
20
30
2
 p81  2 p111  6 p12  GRT

R4  

14

 p111  4 p12  GRT


3
1

, R14  

GRT p12

12

,

GRT E4
GR s
GR s
GR E
GR s
GR s
, w2   T 3 , w3   T 1 , w4   T 2 , w5   T 3 w6   T 4 ,
2
12
42
6
20
30

 s9 22  2s10 2  6s13  GRT , w    s10 2  4s13  GRT ,
GRT s6
GRT s7
GRT s5
, w8  
, w9  
, w11  
w7  
12
4
3
2
2
2
2
4 2
4 2
56
w13  

 s11 2  4s12  GRT

, w14  

GRT s12

, w15  

GRT s13

,

2
2
G13    R1  R2  R3  R4  R5  R6  R7  R8 Cosh  2 1   R9 Sinh  2  1   R10 Cosh  1 
3
2

2
2

 R11 Sinh 1   R12 Cosh 1   R13 Sinh 1   R14 Cosh  1   R15 Sinh  1  

,

G15  R1 y *2  R2 y *3  R3 y *4  R4 y *5  R5 y *6  R6 y *7  R7 y *8  R8 Cosh  2 1 y *  R9 Sinh  2 1 y *
 R10 Cosh 1 y *  R11 Sinh 1 y *  R12 y * Cosh 1 y *  R13 y * Sinh 1 y *  R14 y *2 Cosh 1 y *
 R15 y *2 Sinh 1 y *

G14    w1  w2  w3  w4  w5  w6  w7  w8 Cosh  2  2   w9 Sinh  2  2   w10 Cosh  2 
 w11 Sinh  2   w12 Cosh  2   R13 Sinh  2   w14 Cosh  2   w15 Sinh  2  

,

,

G16  w1 y *2  w2 y *3  w3 y *4  w4 y *5  w5 y *6  w6 y *7  R7 y *8  w8 Cosh  2  2 y *  w9 Sinh  2  2 y *
 w10 Cosh  2 y *  w11 Sinh  2 y *  w12 y * Cosh  2 y *  w13 y * Sinh  2 y *  w14 y *2 Cosh  2 y * ,
 w15 y *2 Sinh  2 y *

E5   G15  G13  1  y * , E7   G14  G16  1  y * , E6  G13  E5 , E8  G14  E7
.
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