MHD flow past accelerated vertical plate with chemical reaction and radiation

K.Rajasekhar, G.V.Ramana Reddy and B.D.C.N.Prasda / International Journal of Engineering
Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 5, September- October 2012, pp.907-913
Chemical Reaction And Radiation Effects On MHD Free
Convective Flow Past An Exponentially Accelerated Vertical
Plate With Variable Temperature And Variable Mass Diffusion
K.Rajasekhar *, G.V.Ramana Reddy ** and B.D.C.N.Prasad***
*(Department of Mathematics, Usha Rama College of Engineering, Telaprolu, Krishna (India)-521109)
** (Department of Mathematics, RVR & JC College of Engineering, Guntur, India -522 019)
***(Department of Mathematics, PVPSIT, College of Engineering, Vijayawada, India-520007)

ABSTRACT
An analytical study is performed to MHD bearings etc. The phenomenon of mass transfer
investigate the effects of chemical reaction and is also very common in theory of stellar structure and
radiation on unsteady MHD flow past an observable effects are detectable, at least on the solar
exponentially accelerated vertical plate with surface. The study of effects of magnetic field on free
variable temperature and variable mass diffusion convection flow is important in liquid metals,
in the presence of applied transverse magnetic electrolytes and ionized gases. The thermal physics of
field. It is assumed that the effect of viscous hydro -magnetic problems with mass transfer is of
dissipation is negligible in the energy equation and interest in power engineering and metallurgy.
there is a first order chemical reaction between the
diffusing species and the fluid. The flow is Diffusion rates can be altered tremendously
assumed to be in x’ ‐ direction which is taken by chemical reactions. The effect of a chemical
along the infinite vertical plate in the upward reaction depend whether the reaction is homogeneous
direction and y’ ‐ axis is taken normal to the plate. or heterogeneous. This depends on whether they
At time t’ > 0, the both temperature and species occur at an interface or as a single phase volume
concentration levels near the plate are raised reaction. In well-mixed systems, the reaction is
linearly with time t. A general exact solution of the heterogeneous, if it takes place at an interface and
governing partial differential equations is obtained homogeneous, if it takes place in solution. In most
by usual closed analytical method. The velocity, cases of chemical reactions, the reaction rate depends
temperature and concentration fields are studied on the concentration of the species itself. A reaction is
for different physical parameters like thermal said to be of the order n, if the reaction rate is
Grashof number (Gr), mass Grashof number proportional to the nth power of the concentration. In
(Gm), Schmidt number (Sc), Prandtl number (Pr), particular, a reaction is said to be first order, if the
radiation parameter (R), magnetic field parameter rate of reaction is directly proportional to the
(M), accelerated parameter (a), heat absorption concentration itself. A few representative fields of
parameter, chemical reaction parameter (K) and interest in which combined heat and mass transfer
time (t) graphically. along with chemical reaction play an important role in
chemical process industries such as food processing
Keywords – MHD, thermal radiation, chemical and polymer production.
reaction, variable temperature, variable mass
diffusion Chambre and Young [1] have analyzed a
first order chemical reaction in the neighbourhood of
INTRODUCTION a horizontal plate. Gupta [2] studied free convection
Free convection flows are of great interest in on flow past a linearly accelerated vertical plate in the
a number of industrial applications such as fiber and presence of viscous dissipative heat using
granular insulation, geothermal systems etc. perturbation method. Kafousias and Raptis [3]
Buoyancy is also of importance in an environment extended this problem to include mass transfer effects
where differences between land and air temperatures subjected to variable suction or injection. Free
can give rise to complicated flow patterns. Magneto convection effects on flow past an exponentially
hydro-dynamic has attracted the attention of a large accelerated vertical plate was studied by Singh and
number of scholars due to its diverse applications. In Naveen kumar [4]. The skin friction for accelerated
astrophysics and geophysics, it is applied to study the vertical plate has been studied analytically by Hossain
stellar and solar structures, interstellar matter, radio and Shayo [5]. Basant kumar Jha [6] studied MHD
propagation through the ionosphere etc. In free convection and mass transform flow through a
engineering it finds its applications in MHD pumps, porous medium. Later Basant kumar Jha et al. [7]
analyzed mass transfer effects on exponentially
accelerated infinite vertical plate with constant heat

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flux and uniform mass diffusion. Das et al. [8] have t  0 : u   0, T  T , C   C
  forall y
studied the effect of homogeneous first order
u   u0 exp(a t ), T   T  Tw  T  At ,
  
chemical reaction on the flow past an impulsively 
started infinite vertical plate with uniform heat flux t   0,  C '  C  (Cw  C ) At  at y   0
  
and mass transfer. Again, mass transfer effects on u   0, T   T  , C   C  as y   
  
moving isothermal vertical plate in the presence of
chemical reaction studied by Das et al. [9]. The (4)
2
dimensionless governing equations were solved by u
where A  , u  - velocity of the fluid in
0
the usual Laplace Transform technique.
Muthucumaraswamy and Senthil Kumar [10] studied 
heat and mass transfer effects on moving vertical the x  direction, y  - the dimensional distance
plate in the presence of thermal radiation. Recently along the co-ordinate axis normal to the plate,
Muthucumaraswamy et al. [11] studied mass transfer g  the gravitational acceleration, ρ - the fluid
effects on exponentially accelerated isothermal
vertical plate. density,  and   are the thermal and concentration
The object of the present paper is to study expansion coefficients respectively, B0 - the
the effects chemical reaction and a uniform transverse
magnetic induction, T  - temperature of the fluid in
magnetic field (fixed relative to the plate) on the free
convection and mass transform flow past an the x  direction, C  the corresponding
exponentially accelerated vertical plate with variable concentration, C  concentration of the fluid far
temperature and mass diffusion. The dimensionless away from the plate, σ is the electric conductivity,
governing equations are solved using the closed
C p - the specific heat at constant pressure, D -the
analytical method.
diffusion coefficient, q r - the heat flux, Q0 -the
2. Formulation of the Problem
dimensional heat absorption coefficient, and K r is
We consider the unsteady hydro magnetic
radiative flow of viscous incompressible fluid past an the chemical reaction parameter.
exponentially accelerated infinite vertical plate with The local radiant for the case of an optically thin gray
variable temperature and also with variable mass gas is expressed by
qr
 4a T4  T 4 
diffusion and heat source parameter in the presence of
chemical reaction of first order and applied transverse (5)
y
magnetic field. Initially, the plate and the fluid are at
 It is assumed that the temperature differences within
the same temperature T in the stationary condition
the flow are sufficiently small and that T  may be
4

with concentration level C at all the points. At expressed as a linear function of the temperature. This
time t   0 , the plate is exponentially accelerated is obtained by expanding T  in a Taylor series about
4

with a velocity u  u0 exp( at ) in its own plane 
T and neglecting the higher order terms, thus we get
and at the same time the temperature of the fluid near T 4  4T 3T   3T3

 (6)
the plate is raised linearly with time t and species From equations (5) and (6), equation (2) reduces to
concentration level near the plate is also increased
T   2T 
linearly with time. All the physical properties of the C p   16a T3 T  T  
  (7)
fluid are considered to be constant except the y  y 2
influence of the body‐force term. A transverse On introducing the following non-dimensional
magnetic field of uniform strength B0 is assumed to quantities
be applied normal to the plate. Then under usual u t u 2 y u0 T   T
 C   C

u  ,t  0 , y  ,  ,  ,
Boussinesq’s approximation, the unsteady flow is u0   
Tw  T  
Cw  C
governed by the following set of equations. a   
g  (Tw  T )  
g   (Cw  C )
u   2u   B0
2 a , Gr  , Gm  ,
 g  T   T   g    C   C  
   u
2 3 3
u0 u0 u0
t  y 2 
C p   B02 
16a 2 T3
Pr  , Sc  ,M  ,R  ,
(1) k D  u02
 u02

T   2T  q K r  Q0
 CP  k 2  r  Q0 T   T 
 (2) Kr  2
,Q 
 C p u0
2
,
t  y  y  u 0

(8)
C   2C 
 D 2  K r  C   C 
 (3) we get the following governing equations which are
t  y  dimensionless
with the following initial and boundary conditions:

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2
u  u
  Gr  Gm  Mu (9)

u( y, t )  A3e N3 y  A1e N2 y  A2e N1 y t (23) 
 
t y 2
  y, t   te N1 y (24)
 1  2 R
    Q (10)  ( y, t )   t e  N2 y  (25)
t Pr y 2 Pr
It is now important to calculate the physical
2
 1   quantities of primary interest, which are the local wall
  K r (11) shear stress, the local surface heat, and mass flux.
t Sc y 2 Given the velocity field in the boundary layer, we can
The initial and boundary conditions in dimensionless now calculate the local wall shear stress ( i.e., skin-
form are as follows: friction) is given by
t  0 : u  0,   0,   0 forall y   u  
w     , and in dimensionless form, we
u  exp(at ),   t ,   t at y  0 (12)  y   y  0
t  0, 
u  0,   0,   0 as y   obtain
where  u 
Cf      N 3 A3  N 2 A1  N1 A2  t
M , Gr , Gm, R, Pr, Q, Sc and K r denote the  y  y  0
magnetic field parameter, thermal Grashof number, From temperature field, now we study the
mass Grashof number, radiation parameter, Prandtl rate of heat transfer which is given in non -
number, heat absorption parameter, Schmidt number dimensional form as:
and chemical reaction parameter respectively.
  
3. SOLUTION OF THE PROBLEM: Nu      tN1
In order to reduce the above system of  y  y  0
partial differential equations to a system of ordinary From concentration field, now we study the
equations in dimensionless form, we may represent rate of mass transfer which is given in non -
the velocity, temperature and concentration as dimensional form as:
u( y, t )  u0 ( y)eit (13)
  
it Sh     tN 2
 ( y , t )   0 ( y )e (14) 
 y  y  0
it
 ( y, t )  0 ( y)e (15)
where

Substituting Eqns (13), (14) and (15) in Eqns (9), (10) N1  Sc  K r  i  , N 2  Pr i  R ,
and (11), we obtain:

u0  N3u0   Gr0  Gm0 
2

N3  M  i , A1  Gr / N 2  N3 ,
2

  (16)

0  N 2 0  0
2
(17)
 
A2  Gm / N12  N3 , A3  eat  t  A1  A2 
2


C0  N12C0  0 (18) Results and Discussion
Here the primes denote the differentiation with In order to get physical insight into the
respect to y. problem, we have plotted velocity profiles for
The corresponding boundary conditions can be different values of the physical parameters
written as a(accelerated parameter), M(magnetic field
parameter) R(radiation parameter), Gr(thermal
u0  e a i t ,  0  te  it , 0  te  it at y  0 Grashof number), Gm(mass Grashof number),
(19)
u0  0,  0  0, 0  0 as y   Pr(Prandtl number), Sc(Schmidt number), t(time) and
The analytical solutions of equations (16) – (18) with the chemical reaction parameter (K) is presented
satisfying the boundary conditions (19) are given by graphically. The value of Sc (Schmidt number) is

 
u0 ( y)  A3e N3 y eit  A1e N2 y  A4e N1 y teit
taken to be 0.6 which corresponds to the water‐vapor.
Also, the value of Pr (prandtl number) are chosen
(20) such that they represent air (Pr=0.71).
0  y   te   N2 y
e it
(21)
The effect of the magnetic parameter M is shown in
Fig.1. It is observed that the tangential velocity of the
0 ( y )   t e  N1 y  e  it (22)
fluid decreases with the increase of the magnetic field
number values. The decrease in the tangential
In view of the above solutions, the velocity, velocity as the magnetic parameter M increases is
temperature and concentration distributions in the because the presence of a magnetic field in an
boundary layer become electrically conducting fluid introduces a force called

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the Lorentz force, which acts against the flow if the the fluid and therefore, heat is able to diffuse away
magnetic field is applied in the normal direction, as in from the heated surface more rapidly for higher
the present study. This resistive force slows down the values of Pr . Hence in the case of smaller Prandtl
fluid velocity component as shown in Fig.1. number as the thermal boundary later is thicker and
Fig.2 shows the effect of the accelerated parameter a the rate of heat transfer is reduced.
on the velocity distribution. It is found that the
velocity increases with an increase in an accelerated For different values of thermal radiation R
parameter a . the temperature profiles are shown in Fig.11. It is
For various values of the thermal Grashof observed that an increase in the thermal radiation
number Gr and mass Grashof number Gm , the parameter, the decrease of the temperature profiles.
velocity profiles ‘ u ’ are plotted in Figs. (3) and (4). The effect of the Schmidt number Sc on the
The thermal Grashof number Gr signifies the concentration is shown in Fig.12. As the Schmidt
relative effect of the thermal buoyancy force to the number increases, the concentration decreases. This
viscous hydrodynamic force in the boundary layer. As causes the concentration buoyancy effects to decrease
expected, it is observed that there is a rise in the yielding a reduction in the fluid velocity. A reduction
velocity due to the enhancement of thermal buoyancy in the concentration distribution is accompanied by
force. Also, as Gr increases, the peak values of the simultaneous reduction in the concentration boundary
layers. The effect of chemical reaction
velocity increases rapidly near to the plate and then
decays smoothly to the free stream velocity. The mass parameter K r on the concentration profiles as shown
Grashof number Gm defines the ratio of the species in Fig.13. It is observed that an increase in leads to a
buoyancy force to the viscous hydrodynamic force. decrease in the values of concentration.
As expected, the fluid velocity increases and the peak
1.4
value is more distinctive due to increase in the species
buoyancy force. The velocity distribution attains a
1.2 Sc=0.60;Pr=0.71;R=1;Kr=5;
distinctive maximum value in the vicinity of the plate t=0.1;a=0.2;Gr=5;Gm=5; =1;
and then decreases properly to approach the free
1
stream value. It is noticed that the velocity increases
with increasing values of the mass Grashof number.
Fig.5 shows the effect of the time t on the velocity 0.8
Velocity

field. It is found that the velocity increases with an
increase in time t . Fig.6. illustrates the velocity 0.6
M=1, 2, 3, 4

profiles for different values of radiation parameter R ,
0.4
the numerical results show that the effect of
increasing values of radiation parameter result in a
0.2
decreasing velocity. For different values of the
Prandtl number Pr the velocity profiles are plotted in
0
Fig.7. It is obvious that an increase in the Prandtl 0 0.5 1 1.5 2 2.5
y
number Pr results in decrease in the velocity within
the boundary layer. Fig.8 illustrates the behavior Fig. 1. Effects of magnetic parameter on velocity
velocity for different values of chemical reaction profiles.
parameter K r . It is observed that an increase in leads 2

to a decrease in the values of velocity. For different 1.8 R=5;Sc=0.60;M=4;Pr=0.71;
t=0.4;Kr=5;Gr=20;Gm=5; =1;
values of the Schmidt number Sc the velocity 1.6

profiles are plotted in Fig.9. It is obvious that an 1.4

increase in the Schmidt number Sc results in 1.2 a = 0.2, 0.5, 0.9, 1.5
decrease in the velocity within the boundary layer.
Velocity

For different values of frequency of excitation  on
1

0.8
the temperature profiles are shown in Fig.9. It is
noticed that an increase in the frequency of excitation 0.6

results a decrease in the temperature field. Fig.10 0.4

shows the behavior of the temperature for different 0.2

values Prandtl number Pr . It is observed that an 0
0 0.5 1 1.5 2 2.5
increase in the Prandtl number results a decrease of y
the thermal boundary layer thickness and in general Fig. 2. Effects of accelerated parameter on velocity
lower average temperature within the boundary layer. profiles.
The reason is that smaller values of Pr are
equivalent to increase in the thermal conductivity of

910 | P a g e

1.4

1.2
1.4
R=5;Sc=0.60;M=4;Pr=0.71;
1 t=0.4;Kr=5;a=0.2;Gm=5; =1;
1.2
Sc=0.60;Pr=0.71;M=4;Kr=5;
0.8 t=0.1;a=0.2;Gr=20;Gm=5; =1;
Velocity

1
Gr = 5, 10, 15, 20
0.6
0.8

Velocity
0.4
0.6

0.2 R=1, 2, 3, 4
0.4

0
0 0.5 1 1.5 2 2.5 0.2
y

Fig. 3. Effects of Thermal Grashof number on 0
0 0.5 1 1.5 2 2.5
velocityprofiles. y

Fig.6. Effects of radiation parameter on velocity
1.4
profiles.
R=5;Sc=0.60;M=4;Pr=0.71;
1.2 t=0.4;Kr=5;a=0.2;Gr=20; =1;
1.4

1 R=5;Sc=0.60;M=4;Kr=5;
1.2
t=0.4;a=0.5;Gr=20;Gm=5; =1;
Gm = 5, 10, 15, 20
0.8
Velocity

1

0.6
0.8
Velocity

Pr = 0.71, 1.00 3.00, 7.00
0.4
0.6

0.2
0.4

0
0 0.5 1 1.5 2 2.5 0.2
y

Fig. 4. Effects of mass Grashof number on velocity 0
0 0.5 1 1.5 2 2.5
profiles. y

Fig.7. Effects of Prandtl number on velocity profiles.
1.5
R=5;Sc=0.60;M=4;Pr=0.71;
Gm=5;Kr=5;a=0.2;Gr=20; =1; 1.4

R=5;Sc=0.60;M=4;Pr=0.71;
1.2 t=0.4;a=0.5;Gr=20;Gm=5; =1;
t = 0.2 0.4 0.6 0.8
1
1
Velocity

0.8
Velocity

Kr =1, 5, 10, 15
0.5 0.6

0.4

0.2
0
0 0.5 1 1.5 2 2.5
y 0
0 0.5 1 1.5 2 2.5
Fig.5. Effects of time on velocity profiles. y

Fig.8. Effects of chemical reaction parameter on
velocity profiles.

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1.4 0.2

R=5;Pr=0.71;M=4;Kr=5; 0.18
Pr=0.71;Sc=0.60;M=4; =1;
1.2 t=0.4;a=0.5;Gr=20;Gm=5; =1;
Gm=5;Kr=5;a=0.2;Gr=20;t=0.2;
0.16

1 0.14

0.12

Temperature
0.8
Velocity

Sc = 0.22, 0.30, 0.60, 0.78 0.1
0.6 R=1, 2, 3, 4
0.08

0.4 0.06

0.04
0.2
0.02

0 0
0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5
y y

Fig.9. Effects of Schmidt number on velocity profiles. Fig.12. Effects of radiation parameter on temperature
profiles.
0.2

0.18 0.2
R=1;Sc=0.60;M=4;Pr=0.71;
Gm=5;Kr=5;a=0.2;Gr=20;t=0.2; 0.18 Pr=0.71;Sc=0.60;M=4;R=5;
0.16
Gm=5;Kr=5;a=0.2;Gr=20;t=0.2;
0.14 0.16

0.12 0.14
Temperature

0.12
Concentration

0.1
=1, 2, 3,4
0.08 0.1

0.06 0.08
=1, 2, 3, 4
0.04 0.06

0.02 0.04

0 0.02
0 0.5 1 1.5 2 2.5
y 0
0 0.5 1 1.5 2 2.5
Fig.10. Effects of frequency of excitation on y

temperature profiles. Fig.13. Effects of frequency of excitation on
concentration profiles.
0.2

0.18 R=1;Sc=0.60;M=4; =1; 0.2
Gm=5;Kr=5;a=0.2;Gr=20;t=0.2;
0.16 0.18
Pr=0.71;Kr=5;M=4;R=5;
Gm=5; =1;a=0.2;Gr=20;t=0.2;
0.14 0.16

0.12 0.14
Temperature

0.12
Concentration

0.1

0.08 Pr = 0.71, 1.00, 3.00, 7.00 0.1
Sc= 0.22, 0.30, 0.60, 0.78
0.06 0.08

0.04 0.06

0.02 0.04

0 0.02
0 0.5 1 1.5 2 2.5
y 0
0 0.5 1 1.5 2 2.5
Fig.11. Effects of Prandtl number on temperature y

profiles. Fig.14. Effects of Schmidt number on concentration
profiles.

912 | P a g e

with chemical reaction, The Bulletin of
0.2
GUMA, Vol.5, pp.13-20, 1999.
0.18
Pr=0.71;Sc=0.60;M=4;R=5;
[10]. R. Muthucumaraswamy, G. Senthil kumar,
0.16 Gm=5; =1;a=0.2;Gr=20;t=0.2; Heat and Mass transfer effects on moving
0.14 vertical plate in the presence of thermal
0.12
radiation, Theoret. Appl. Mech., Vol.31,
Concentration

No.1, pp.35-46, 2004.
0.1
[11]. R. Muthucumaraswamy, K.E. Sathappan, R.
0.08 Kr =1,2, 3, 4
Natarajan, Mass transfer effects on
0.06 exponentially accelerated isothermal vertical
0.04 plate, Int. J. of Appl. Math. and Mech., 4(6),
0.02
19-25, 2008.
0
0 0.5 1 1.5 2 2.5
y

Fig.15. Effects of chemical reaction parameter on
concentration profiles.

REFERENCES
[1]. P. L. Chambre, J. D. Young, On the
diffusion of a chemically reactive species in
a laminar boundary layer flow, The Physics
of Fluids, Vol.1, pp.48-54, 1958.
[2]. A. S. Gupta, I. Pop, V. M. Soundalgekar,
Free convection effects on the flow past an
accelerated vertical plate in an
incompressible dissipative fluid, Rev. Roum.
Sci. Techn. -Mec. Apl., 24, pp.561-568,
1979.
[3]. N. G. Kafousias, A.A. Raptis, Mass transfer
and free convection effects on the flow past
an accelerated vertical infinite plate with
variable suction or injection, Rev. Roum.
Sci. Techn.- Mec. Apl., 26, pp.11-22, 1981.
[4]. A. K. Singh, N. Kumar, Free convection
flow past an exponentially accelerated
vertical plate, Astrophysics and Space
science, 98, pp. 245-258, 1984.
[5]. M. A. Hossain, L. K. Shayo, The Skin
friction in the unsteady free convection flow
past an accelerated plate, Astrophysics and
Space Science, 125, pp. 315-324, 1986.
[6]. B.K. Jha, MHD free-convection and mass
transform flow through a porous medium,
Astrophysics and Space science, 175, 283-
289, 1991.
[7]. B.K. Jha, R. Prasad, S. RAI, Mass transfer
effects on the flow past an exponentially
accelerated vertical plate with constant heat
flux, Astrophysics and Space Science, 181,
pp.125-134, 1991.
[8]. U. N. Das, R. K. Deka, V. M. Soundalgekar,
Effects of mass transfer on flow past an
impulsively started infinite vertical plate
with constant heat flux and chemical
reaction, Forschung in Ingenieurwsen,
Vol.60, pp.284-287, 1994.
[9]. U. N. Das, R. K. Deka, V. M. Soundalgekar,
Effects of mass transfer on flow past an
impulsively started infinite vertical plate

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