Effect of heat transfer on oscillatory flow of a fluid through a
porous medium in a channel with an inclined magnetic field
Mrs. B. Swaroopa
Prof. K. Ramakrishna Prasad
Lecturer in Mathematics
Govt. Degree College (W)
Srikalahasti , Chittoor(Dist)
Prof. in Mathematics
S.V. University
Tirupati, Chittoor(Dist)
Andhrapradesh-517644
Andhrapradesh-517502
swaroopa1983@yahoo.com
krp@yahoo.com
ABSTRACT
In this paper, the effect of heat transfer on
oscillatory flow of a fluid trough a porous medium between in
channel under the effect of inclined magnetic field is
investigated. The expressions for the velocity and temperature
fields are obtained analytically. The effects various emerging
parameters on the velocity field, temperature field, skin
friction and Nusselt number are discussed in detail with the
aid of graphs and tables.
Keywords
Darcy number, Inclined magnetic field, Oscillatory flow, Heat
transfer.
INTRODUCTION
The Problem of convective flow in fluid saturated
porous medium has been the subject of several recent papers.
Interest in understanding the convective transport processes in
porous material is increasing owing to the development of
geothermal energy technology, high performance insulation
for building and cold storage, renewed interest in the energy
efficient drying processes and many other areas. It is also
interest in the nuclear industry, particularly in the evaluation
of heat removal from a hypothetical accident in a nuclear
reactor and to provide effective insulation. Compressive
literature surveys concerning the subject of porous media can
be found in the most recent books by Ingham and Pop (1998),
Nield and Bejan (1999), Vafai (2002), Pop and Ingham (2001)
and Bejan and Kraus (2003). Many studies related to nonNewtonian fluids saturated in a porous medium have been
carried out. Dharmadhikari and Kale (1985) studied
experimentally the effect of non-Newtonian fluids in a porous
medium. Rees (1996) analyzed the effect of inertia on free
convection over a horizontal surface embedded in a porous
medium. Oscillatory viscous flow in a porous channel with
arbitrary wall suction was studied by Jankowski and
Majdalani (2002).
The flow of an electrically conducting fluid has
significant applications in many branches of engineering
science such as magnetohydrodynamics (MHD) generators,
plasma studies, nuclear reactor, geothermal energy extraction,
electromagnetic propulsion, the boundary layer control in the
field of aerodynamics and so on. Heat transfer effect on
laminar flow between parallel plates under the action of
transverse magnetic field was studied by Nigam and Singh
(1960). Soundalgekar and Bhat (1971) have investigated the
MHD oscillatory flow of a Newtonian fluid in a channel with
heat transfer. MHD flow of viscous fluid between two parallel
plates with heat transfer was discussed by Attia, and Kotb
(1996). Raptis et al. (1982) have analyzed the hydromagnetic
free convection flow through a porous medium between two
parallel plates. Aldoss et al. (1995) have studied mixed
convection flow from a vertical plate embedded in a porous
medium in the presence of a magnetic field. Makinde and
Mhone (2005) have considered heat transfer to MHD
oscillatory flow in a channel filled with porous medium.
Mostafa (2009) have studied thermal radiation effect on
unsteady MHD free convection flow past a vertical plate with
temperature dependent viscosity. Unsteady heat transfer to
MHD oscillatory flow through a porous medium under slip
condition was investigated by Hamza et al. (2011).
Manyonge et al. (2012) have studied the steady MHD
Poiseuille flow between two infinite parallel porous plates in
an inclined magnetic field. The effect of an inclined magnetic
field on unsteady free convection flow of a dusty viscous fluid
between two infinite flat plates filled by a porous medium was
investigated by Sandeep and Sugunamma (2013). Recently,
Joseph et al. (2014) have analyzed the unsteady MHD couette
flow between two infinite parallel porous plates in an inclined
magnetic field with Heat Transfer.
In view of these, we modeled the effect of heat
transfer on oscillatory flow of a fluid trough a porous medium
between in channel under the effect of inclined magnetic field.
The expressions for the velocity and temperature fields are
obtained analytically. The effects various emerging
parameters on the velocity field, temperature field, skin
friction and Nusselt number are discussed in detail with the
aid of graphs and tables.
MATHEMATICAL FORMULATION
We consider the oscillatory flow of a Newtonian
fluid through a porous medium in a channel of width h under
the influence of inclined magnetic field and radiative heat
transfer as depicted in Fig.1. It is assumed that the fluid has
small electrical conductivity and the electromagnetic force
produced is very small. We choose the Cartesian coordinate
system  x, y  , where x - is taken along center of the channel
and the y - axis is taken normal to the flow direction.
the Hartmann number, N is the radiation parameter, Pe is
the Peclet number and Gr is the Grashof number.
The corresponding non-dimensional boundary
conditions are
y0
at
(2.8)
u0,  0
y 1
at
(2.9)
u  0 ,  1
SOLUTION
In order to solve equations (2.6) – (2.9) for purely
oscillatory flow, let
p

  eit
(3.1)
x
i t
(3.2)
u  y , t   u0  y  e
Fig. 1 Physical model of the problem
The basic equations of momentum (Joseph, 2014)
and energy governing such a flow, subject to the Boussinesq
approximation, are:
u
p
 2u


    2   B02 sin 2   u  u   g  T  T0 
t
x
y
k
(2.1)

T K  2T 1 q


t c p y 2 c p y
u0,
(2.2)
T  T1
at
yh
(3.3)
where  is a real constant and  is the frequency of the
oscillation.
Substituting the equations (3.1) - (3.3) in to the
equations (2.6) – (2.9), we get
d 2 u0
 b 2u0    Gr 0
(3.4)
dy 2
d 2 0
 a 2 0  0
dy 2
with the boundary conditions
y0
u0  0 , 0  0 at
The boundary conditions are given by
(2.3)
T  T0 at y  0
u0,
  y, t   0  y  eit
(2.4)
u0  0 ,
0  1
at
(3.5)
(3.6)
y 1
(3.7)
where   0      is the angle between velocity and
in which a  N  i Pe and b  M  i Re .
magnetic field strength, u is the axial velocity, T is the fluid
temperature, p is the pressure,  is the fluid density, B0 is
Solving equations (3.4) and (3.5) using the
boundary conditions (3.6) and (3.7), we obtain
the magnetic field strength,  is the conductivity of the
fluid, k is the permeability of the porous medium, g is the
acceleration due to gravity,  is the coefficient of volume
expansion due to temperature, c p is the specific heat at
constant pressure, K is the thermal conductivity and q is the
radiative heat flux. Following Cogley et al. (1968), it is
assumed that the fluid is optically thin with a relatively low
density and the radiative heat flux is given by
q
 412 T0  T 
(2.5)
y
here 1 is the mean radiation absorption coefficient.
Introducing
the
following
non-dimensional
variables
pa
T  T0
x
y
u
tU
x , y , u ,  
, , t
,
p
h
h
U
h
T1  T0
U
 g  T1  T0 
 hU
 h 2 B02
k
, Gr 
, Re 
, Da  2 ,
h

U

 hUc p
4 2h2
Pe 
, N2  1
K
K
M2 
here U is the mean flow velocity, into the equations (2.1) and
(2.2), we get (after dropping bars)
u
p  2u
Re
   2  M 12u  Gr
(2.6)
t
x y
Pe
  2

 N 2
t y 2
where
Re
(2.7)
is
the
Reynolds
number,
1 

M1   M 2 sin 2  
 , Da is the Darcy number, M is
Da 

2
2
1
 Gr  sin ay sinh by    cosh b  1



sinh by 
 2
2

2 
 a  b   sin a sinh b  b sinh b

u0  y   


   1  cosh by 

2
 b

(3.8)
sin ay
0  y  
and
(3.9)
sin a
Therefore, the fluid velocity and temperature are
given as
 Gr  sin ay sinh by  

 2

2 
sinh b  
  a  b   sin a


  cosh b  1
 eit
u  y, t    
sinh by
(3.10)
2


b sinh b
 

  1  cosh by 

 b2



sin ay it
  y, t  
e
and
(3.11)
sin a
The skin friction at the upper plate y  1 of the
channel is given by
 Gr
 a cot a  b coth b 
 2
2
a

b


u
 eit
 
 
  cosh b  1

y y 1



 
coth b  sinh b 
b
b


(3.12)
The rate of heat transfer coefficient in terms of
Nusselt number Nu at the plate y  1 of the channel is given
by

(3.13)
Nu  
   a cot a  eit
y y 1
RESULTS AND DISCUSSIONS
Fig. 2 shows the influence of Hartmann number M
on the velocity u for N  1, Gr  1 , Re  1 ,   1 ,

Da  0.1 ,   ,   1 , Pe  0.71 and t  0 . It is observed
4
that, the velocity u decreases with an increase in Hartmann
number M .
The influence of Darcy number Da on the velocity

u for N  1, Gr  1 , Re  1 ,   1 ,   , M  1 ,   1 ,
4
Pe  0.71 and t  0 is shown in Fig.3. It is noticed that, the
velocity u increases with an increase in Darcy number Da .
Fig. 4 depicts the influence of inclination angle 
on the velocity u for N  1, Gr  1 , Re  1 ,   1 ,
Da  0.1 , M  1 ,   1 , Pe  0.71 and t  0 . It is found
that, the velocity u decreases with increasing inclination
angle  .
The influence of Peclet number Pe on the velocity

u for N  1, Gr  1 , Re  1 ,   1 ,   , M  1 ,   1 ,
4
Da  0.1 and t  0 is depicted in Fig. 5. It is observed that,
the velocity u decreases with increasing Peclet number Pe .
Fig. 6 illustrates the influence of Grashof number
Gr on the velocity u for N  1, Da  0.1 , Re  1 ,   1 ,

  , M  1 ,   1 , Pe  0.71 and t  0 . It is noted that
4
the velocity u increases with an increase in Grashof number
Gr .
The influence of Reynolds number Re on the

velocity u for N  1, Gr  1 , Da  0.1 ,   1 ,   ,
4
M  1 ,   1 , Pe  0.71 and t  0 is presented in Fig. 7. It
is observed that, the velocity u decreases with increasing
Reynolds number Re .
Fig. 8 shows the influence of radiation parameter
N on the velocity u for Da  0.1 , Gr  1 , Re  1 ,   1 ,

  , M  1 ,   1 , Pe  0.71 and t  0 . It is seen that
4
the velocity u increases with increasing radiation parameter
N.
The influence of oscillation parameter  on the

velocity u for Da  0.1 , Gr  1 , Re  1 , N  1 ,   ,
4
M  1 ,   1 , Pe  0.71 and t  0 is shown in Fig. 9. It is
noted that the velocity u decreases with an decrease in
frequency of oscillations  .
Fig. 10 depicts the influence of pressure constant  on the

velocity u for Da  0.1 , Gr  1 , Re  1 ,   1 ,   ,
4
M  1 , N  1 , Pe  0.71 and t  0 . It is found that the
velocity u increases with increasing pressure constant  .
Fig. 11 illustrates the influence of radiation
parameter N on the temperature  for   1 , Pe  0.71
and t  0 . It is found that, the temperature  increases with
increasing radiation parameter N .
The influence of Peclet number Pe on the
temperature  for   1 , Pe  0.71 and t  0 is depicted in
Fig. 12. It is observed that the temperature  decreases with
an increase in Peclet number Pe .
Fig. 13 shows the influence of oscillation parameter
 on the temperature  for N  1 , Pe  0.71 and t  0 . It
is noted that the temperature  decreases with increasing  .
Table-1 shows the effects of Da , N , Gr , Re ,  ,
M ,  , Pe and  on the skin friction  . From Table-1, it is
observed that the skin friction  decreases with increasing
 , M , Pe, Re and  , whereas it increases with increasing
N , Gr , Da and  .
Table-2 shows the effects of N ,  and Pe on the
Nusselt number Nu at the upper plate. From Table-2, it is
found that the Nusselt number Nu increase with increasing
N , while it decreases with increasing Pe and  .
CONCLUSIONS
In this paper, we studied the effect of heat transfer on an
oscillatory flow of a fluid through a porous medium between
two infinite parallel plates under the influence of inclined
magnetic field. The expressions for the velocity and
temperature fields are obtained analytically. It is found that,
the velocity u and skin friction  increases with increasing
Da , N , Gr , Pe and  , whereas they decreases with increasing
 , M , Re and  . The temperature  and the Nusselt number
Nu increases with increasing N , while they decreases with
increasing Pe and  .
Fig 2: The influence of Hartmann number M on the velocity

u for N  1, Gr  1 , Re  1 ,   1 , Da  0.1 ,   ,
4
  1 , Pe  0.71 and t  0 .
Fig 3: The influence of Darcy number Da on the velocity u

for N  1, Gr  1 , Re  1 ,   1 ,   , M  1 ,   1 ,
4
Pe  0.71 and t  0 .
Fig 4: The influence of inclination angle  on the velocity u
for N  1, Gr  1 , Re  1 ,   1 , Da  0.1 , M  1 ,   1 ,
Pe  0.71 and t  0 .
Fig 5: The influence of Peclet number Pe on the velocity u

for N  1, Gr  1 , Re  1 ,   1 ,   , M  1 ,   1 ,
4
Da  0.1 and t  0 .
Fig 6: The influence of Grashof number Gr on the velocity

u for N  1, Da  0.1 , Re  1 ,   1 ,   , M  1 ,
4
  1 , Pe  0.71 and t  0 .
Fig 7: The influence of Reynolds number Re on the velocity

u for N  1, Gr  1 , Da  0.1 ,   1 ,   , M  1 ,
4
  1 , Pe  0.71 and t  0 .
Fig 8: The influence of radiation parameter N on the
velocity u for Da  0.1 , Gr  1 , Re  1 ,
 1,  

4
, M  1 ,   1 , Pe  0.71 and t  0 .
Fig. 11 The influence of radiation parameter N on the
temperature  for   1 , Pe  0.71 and t  0 .
Fig 9: The influence of oscillation parameter  on the
velocity u for Da  0.1 , Gr  1 , Re  1 , N  1 ,  

4
,
M  1 ,   1 , Pe  0.71 and t  0 .
Fig 12: The influence of Peclet number Pe on the
temperature  for   1 , Pe  0.71 and t  0 .
Fig 10: The influence of pressure constant  on the velocity

u for Da  0.1 , Gr  1 , Re  1 ,   1 ,   , M  1 ,
4
N  1 , Pe  0.71 and t  0 .
Fig 13: The influence of oscillation parameter  on the
temperature  for N  1 , Pe  0.71 and t  0 .
Table-1 Skin friction  for t  0 .

Da
0.1
0.1
0.1
1
0.1
0.1
0.1
0.1
0.1
0.1
M
1
1
2
1
1
1
1
1
1
1
0
 /4
 /4
 /4
 /4
 /4
 /4
 /4
 /4
 /4

N
1
1
1
1
2
1
1
1
1
1
1
1
1
1
1
5
1
1
1
1
Pe
0.71
0.71
0.71
0.71
0.71
0.71
7
0.71
0.71
0.71
Gr
1
1
1
1
1
1
1
5
1
1
Table - 2 Nusselt number for t  0
N

Pe
Nu
1
2
1
1
1
1
5
1
0.71
0.71
0.71
7
-0.6572
0.8657
-6.5753
-1.5824
References
Aldoss, T. K., Al-Nimr, M. A., Jorah, M. A. and Al-Shaer, B.
1995. Magnetohydrodynamic mixed convection from a
vertical plate embedded in a porous medium, Numer. Heat
Transfer, 28A(1995), 635-645.
Attia, H. A. and Kotb, N. A. 1996. MHD flow between two
parallel plates with heat transfer, Acta Mechanica, 117 (1996),
215-220.
Bejan, A. and Kraus, A.D. 2003. Heat transfer Handbook,
wiley, New York.
Cogley, A.C.L., Vinvent, W.G. and Giles, E.S. 1968.
Differential approximation for radiative heat transfer in nonlinear equations-grey gas near equilibrium, American Institute
of Aeronautics and Astronautics, 6(1968), 551-553.
Dharmadhikari, R.V. and Kale, D.D. 1985. Flow of nonNewtonian fluids through porous media, Chem. Eng. Sci.,
40(1985), 527-529.
Hamza, M. M., Isah, B.Y. and Usman, H. 2011. Unsteady
heat transfer to MHD oscillatory flow through a porous
medium under slip condition, International Journal of
Computer Applications, 33(4) (2011), 12-17.
Ingham, D.E. and Pop, I. 1998. Transport Phenomena in
Porous Media, Pergamon, Oxford, Vol. II (1998), 200.
Jankowski, T.A. and Majdalani, J. 2002. Oscillatory Viscous
Flow in a Porous Channel with Arbitrary Wall Suction, AIAA
2002-2855 (2002), 1-11.
Re
1
1
1
1
1
1
1
1
5
1


1
1
1
1
1
1
1
1
1
2
-6.9182
-7.3380
-8.6981
-1.8714
-7.2799
-6.5753
-7.3829
-6.4365
-6.5543
-14.9013
Joseph, K.M., Daniel, S. and Joseph, G.M. Unsteady MHD
couette flow between two infinite parallel porous plates in an
inclined magnetic field with heat transfer, International
Journal of Mathematics and Statistics Invention, 2(3)(2014),
103-110.
Makinde, O. D. and Mhone, P. Y. 2005. Heat transfer to
MHD oscillatory flow in a channel filled with porous
medium, Rom. J. Phys., 50 (2005), 931-938.
Manyonge, W.A., Kiema, D.W. and Iyaya, C.C.W. 2012.
Steady MHD Poiseuille Flow between two infinite parallel
porous plates in an inclined magnetic field, Int. Journal of
Pure and Applied Maths., 76(5) (2012), 661-668.
Mostafa, A.A. 2009. Thermal radiation effect on unsteady
MHD free convection flow past a vertical plate with
temperature dependent viscosity, The Canadian Journal of
Chemical Engineering, 87(1) (2009), 171- 181.
Nield, D.A. and Bejan, A. 1999. Convection in porous
Media, 2nd Ed. Springer, New York.
Nigam, S. D. and Singh, S. N. 1960. Heat transfer by laminar
flow between parallel plates under the action of transverse
magnetic field, Q. J. Mech. Appl. Math., 13 (1960), 85-87.
Pop, I. and Ingham, D.B. 2001. Convective heat transfer,
Computational and Mathematical modeling of viscous fluids
and porous media, Pergamon, Oxford.
Raptis, A., Massias, C. and Tzivanidis, G. 1982. MHD flow
between two parallel plates with heat transfer, Phys. Lett.,
90A(1982), 288-289.
Ress, D.A.S. 1996. The Effect of Inertia on Free Convection
from a Horizontal Surface Embedded ina Porous Medium",
Int. J. Heat Mass Transfer, 39 (1996), 3425-3430.
Sandeep, N. and Sugunamma, V. 2013. Effect of an inclined
magnetic field on unsteady free convection flow of a dusty
viscous fluid between two infinite flat plates filled by a porous
medium. Int. Journal of Appl. Maths. Modelling, 1 (1) (2013),
16 – 33.
Soundalgekar, V. M. and Bhat, J. P. 1971. Oscillatory MHD
channel flow and heat transfer, J. Pure Appl. Math., 15
(1971), 819-828.
Vafai, K. 2004. Handbook of Porous media, Marcel Dekker,
New York, Vol. II (2002).