Heat and Mass transfer on Unsteady MHD Free
Convection Rotating flow through a porous
medium over an infinite vertical plate with Hall effects
D. Dastagiri Babu1,a), S.Venkateswarlu2,b) and E.Keshava Reddy3,c)
1, 2
Dept. Of Mathematics, RGM College of Engineering and Technology, Nandyal, Kurnool, AP, India
3
Dept. Of Mathematics, JNTUA College of Engg., Anantapuramu, Andhra Pradesh, India
a)
Corresponding author: [email protected]
b)
[email protected]
c)
[email protected]
Abstract: In this paper, we have considered the unsteady MHD free convection flow of an incompressible electrically
conducting fluid through porous medium bounded by an infinite vertical porous surface in the presence of heat source and
chemical reaction in a rotating system taking hall current into account. The flow through porous medium is governed by
Brinkman’s model for the momentum equation. In the undisturbed state, both the plate and fluid in porous medium are in
solid body rotation with the same angular velocity about normal to the infinite vertical plane surface. The vertical surface is
subjected to the uniform constant suction perpendicular to it and the temperature on the surface varies with time about a nonzero constant mean while the temperature of free stream is taken to be constant. The exact solutions for the velocity,
temperature and concentration are obtained making use of perturbation technique. The velocity expression consists steady
state and oscillatory state. It reveals that, the steady part of the velocity field has three layer characters while the oscillatory
part of the fluid field exhibits a multi-layer character. The influence of various flow parameters on the velocity, temperature
and concentration is analysed graphically, and computational results for the skin friction, Nusselt number and Sherwood
number are also obtained in the tabular forms.
Keywords: Hall effects, Heat and mass transfer, MHD flows, infinite vertical plates, porous medium, rotating channels,
second grade fluids.
INTRODUCTION
Magneto hydrodynamic (MHD) free convection flow induced by thermal and solutal buoyancy forces
acting over bodies with different geometries in a fluid saturated porous medium is prevalent in many physical
phenomena and has varied and wide range of industrial applications. For example, in atmospheric flows, the
presence of pure air or water is impossible because some foreign mass may be present either naturally or mixed
with air or water due to industrial emissions. Comprehensive reviews of natural convection boundary layer flow
over various geometrical bodies with heat and mass transfer in porous and non-porous media are well
documented by Incropera et al. [1]. Makinde [2] studied MHD mixed convection flow and mass transfer past a
vertical porous plate embedded in a porous medium with constant heat flux. Eldabe et al. [3] discussed unsteady
MHD flow of a viscous and incompressible fluid with heat and mass transfer in a porous medium near a moving
vertical plate with time-dependent velocity. Mention may be made of research studies of Singh et al. [4]. The
text books by Sparrow and Howell et al. [5] present the most essential features and state of the art applications
of radiative heat transfer. Mahmoud [6] investigated the effects of thermal radiation on unsteady MHD free
convection flow past an infinite vertical porous plate taking into account the effects of viscous dissipation. Patra
et al. [7] considered the effects of radiation on natural convection flow of a viscous and incompressible fluid
near a stationary vertical flat plate with ramped temperature. Narahari [8] discussed unsteady free convection
flow between two vertical plates with ramped temperature within one of the plates in the presence of thermal
radiation and mass diffusion. Seth et al. [9] studied unsteady MHD natural convection flow with radiative heat
transfer past an impulsively moving vertical plate with ramped temperature embedded in a fluid saturated
porous medium with thermal diffusion. Recently, Seth et al. [10] considered the effects of rotation on unsteady
hydro magnetic free convection flow of a viscous, incompressible and optically thick radiating fluid past an
impulsively moving vertical plate with ramped temperature in a fluid saturated porous medium. It is noticed that
when the density of an electrically conducting fluid is low and/or applied magnetic field is strong, Hall current is
produced in the flow-field which plays an important role in determining flow features of the problems because it
induces secondary flow in the flow-field. Keeping in view this fact, significant investigations on hydromagnetic
free convection flow past a flat plate with Hall effects under different thermal conditions are carried out by
several researchers in the past. Mention may be made of the research studies of Pop and Watanabe [11], AboEldahab and Elbarbary [12], Takhar et al. [13] and Saha et al. It is worthy to note that Hall current induces
secondary flow in the flow-field which is also the characteristics of Coriolis force. Therefore, it becomes very
important to compare and contrast the effects of these two agencies and also to study their combined effects on
such fluid flow problems. Satya Narayana et al. studied the effects of Hall current and radiation–absorption on
MHD convection heat and mass transfer flow of a micropolar fluid in a rotating frame of reference. Seth et al.
investigated effects of Hall current and rotation on unsteady hydromagnetic convection flow of a viscous,
incompressible, electrically conducting and heat absorbing fluid past an impulsively moving vertical plate with
ramped temperature in a porous medium taking into account the effects of thermal diffusion.Motivated the
above studies, the aim of the present study was to analyze the effects on the unsteady MHD free convection
flow of an incompressible electrically conducting second grade fluid through porous medium bounded by an
infinite vertical porous surface in the presence of heat source and chemical reaction in a rotating system taking
hall current into account.
FORMULATION AND SOLUTION OF THE PROBLEM
We consider the unsteady MHD free convection flow of an electrically conducting viscous
incompressible fluid bounded by a vertical porous surface in a rotating system in the presence of heat source and
chemical reaction subjected to a uniform transverse magnetic field of strength B 0 normal to plate and taking hall
current into account. The temperature on the surface varies with the time about a non-zero constant mean while
the temperature of free stream is taken to be constant. We consider that the vertical infinite porous plate rotates
with the constant angular velocity about an axis is perpendicular to the vertical plane surface. The physical
configuration of the problem is as shown in Fig. 1.
FIGURE 1: Physical configuration of the problem
The unsteady hydro magnetic flow in a rotating system is governed by the equation of motion for
momentum, the conservation of mass, energy and the equation of mass transfer, under usual Boussinesq
approximation, are given by
w
0
z
(1)
u
u
 2u σB02

(mv  u )  u  g (T  T )  g  (C  C )
 w  2 v   2 
2
K1
t
z
z 1  m
σB02
 2 v   3v
v
v

(v  mu)  v
 w  2 u   2  1 2 
2
K1
z
z
t
 z t 1  m
0
1 p 
 w
 z k
(2)
(3)
(4)
k  2T
T
T
w

 S1 (T  T )
t
z C p z 2
(5)
C
C
 2C
w
 D 2  KC (C  C )
t
z
z
(6)
Where all the physical quantities are their usual meaning
The corresponding boundary conditions are
u  v  0, T  Tw   (Tw  T ) eit , C  Cw   (Cw  C ) eit at z  0
Where   1 and

u  v  0, T  T , C  C
at z  
(7)
(8)
is the frequency of oscillation. There will be always some fluctuation in the
temperature, the plate temperature is assumed to vary harmonically with time. It varies from
Tw   (Tw  T )
as t varies from 0 to  / 2 . Now there may also occur some variation in suction at the plate due to the
variation of the temperature, here we assume that, the frequency of suction and temperature variation are same.
Integrating the equation (1), we get
w(t )  w0 (1  A eit )
Where A is the suction parameter,
number such that
the absence of
w0
is the constant suction velocity and
(9)

is the small positive
A  1. The equation (4) determines the pressure distribution along the axis of rotation and
p
in the equation (4) implies that there is a net cross flow in the y  direction. We choose,
y
q  u  iv and taking into consideration (9), the momentum equation (2) and (3) can be written as
 B02

q
q
2q

q  q  g (T  T )  g (C  C )
 w0 ( 1   Aeit )  2i q   2 

(
1
im
)
K
t
z

z
1
(10)
Introducing the following non-dimensional quantities:
q
T  T
C  C
tw02

z* 
, q*  , T * 
, C* 
, *  2 , t* 
Cw  C
w0
w0
Tw  T


w0 z
Making use of non-dimensional quantities (dropping asterisks), the equation (10), (5) and (6) can be
written as
q
q
2q  M 2
1
 ( 1   Aeit )  2 iRq  2  
  q  GrT  GmC
t
z
z  1  im K 
T
T 1  2T
 (1  A eit )

 ST
t
z Pr z 2
C 1  2C
C
 KC C

 (1  A eit )
z Sc z 2
t
(11)
(12)
(13)
Where,
B02
M 
is the
w02

R  2 is the
w0
2
Gm 
Hartmann number (Magnetic field parameter), K 
Rotation
g * (Cw  C )
w03
Source parameter,
KC 
parameter,
Gr 
K1w02
is the Porosity parameter,
ν2
g (Tw  T )
is the thermal Grashof number,
w03
is the mass Grashof number, Pr 
 C p
is Prandtl parameter,
k
S
S1
is the
w0
K C

chemical reaction parameter, m   ee is the hall parameter and Sc 
is
2
w0
D
the Schmidt number.
The corresponding non-dimensional boundary conditions
q  0, T  1   eit , C  1   eit
q T C  0
at
z 0
at z  
(14)
(15)
In order to reduce the system of partial differential equations (11) – (13) under their boundary
conditions (14) and (15), to a system of ordinary differential equations in the non-dimensional form, In view of
the equation (9) and oscillating plate temperature T , The solution form of the equations (11), (12) and (13) are,
q( z, t )  q0 ( z)   q1 ( z) eit
(16)
T ( z, t )  T0 ( z)   T1 ( z ) eit
(17)
C ( z, t )  C0 ( z)   C1 ( z) eit
(18)
These equations (16) – (18) are valid for small amplitude of oscillation. Substituting from (16) to (18)
into the system of equations (11) – (13) respectively, and we get equating the harmonic and non-harmonic
terms, solving these equations with corresponding boundary conditions, we obtained the velocity, temperature
and concentration distribution.
The non-dimensional skin friction at the plate z  0 in term of amplitude and phase angle is given by
 dq 
 dq 
 dq 
i t
   0    1  e
 dz  z  0  dz  z  0
 dz  z  0
 
The  xz and
 yz
(19)
components of skin friction at the plate are given by
 du0 
 dv 
 dv 
 du 
and  yz   0 
  1 
   1 
 dz  z  0
 dz  z  0
 dz  z  0
 dz  z  0
 xz  
The rate of heat transfer co-efficient at the plate z  0 in term of amplitude and phase angle is given
by
 dT 
 dT 
 dT 
Nu      0     1  ei  t
 dz  z 0  dz  z 0
 dz  z 0
(20)
The rate of mass transfer co-efficient at the plate z  0 in term of amplitude and phase angle is given
by
 dC0 
 dC 
 dC 
Sh  
   1  ei  t
 

 dz  z 0  dz  z 0
 dz  z 0
(21)
RESULTS AND DISCUSSION
The Figures (2-4) shown the velocity profiles, the figure (5) exibit the temperature distribution and the
figure (6) depicts the concentration profiles. For computational purpose we are fixing the values A  0.05 ;
  5 / 2 ;   0.001 , t  0.2 .
From figures (2) depicts the magnitudes of the velocity components reduces with increasing Hartmenn
number M. The application of magnetic field defines the Larentz force and then reduces the velocity. Similarly
the behaviour are observed for Pr, Sc or R. From the figures (3), the primary velocity enhances whereas the
secondary velocity shown the reverse effect with increasing permeability of the porous medium. The similar
effect is shown for Kc. From the figures (4), we notice that both the velocity components increases with
increasing Hall parameter m. Likewise, we observed that same nature for Gr, Gm., t or S. It is observed that, from
figures (5), Prandtl number Pr leads to decrease the temperature uniformly in all layers being the heat source
parameter fixed. It is found that the temperature decreases in all layers with increase in the heat source
parameter S. It is concluded that the heat source parameter S and Prandtl number Pr reduces the temperature in
all layers. The temperature increases with increasing the frequency of oscillation

and time t. It is observed
that from figures (6), for be heavier diffusing foreign species, i.e., the velocity reduces with increasing Schmidt
number Sc in both magnitude and extent and thinning of thermal boundary layer occurs. Likewise, the
concentration profiles decrease with increase in chemical reaction parameter Kc. It is concluded that the
Schmidt number and the chemical reaction parameter reduces the concentration in all layers. The concentration
increases with increasing the frequency of oscillation

and time t.
It is noted from the table 1 that the magnitudes of both the skin friction components
 xz
and
 yz
increase with increase in permeability parameter K, hall parameter m, thermal Grashof number Gr and mass
Grashof number Gm, and where as it reduces with increase in Hartmann number M, heat source parameter S,
Schmidt number Sc, chemical reaction parameter Kc and Prandtl number Pr. Likewise the rotation parameter R
enhances skin friction component
 xz
and reduces skin friction component  yz . From the table 2 that the
magnitude of the Nusselt number Nu increases for the parameters heat source parameter S and Prandtl number
Pr or time t, and it reduces with the frequency of oscillation  . Also from the table 4, the similar behaviour is
observed. The magnitude of the Sherwood number Sh increases for increasing the parameters Schmidt number
Sc and chemical reaction parameter Kc or time t and reduce with increasing the frequency of oscillation  .
FIGURE 2: The velocity profile for the component u against M
K  1, R  1.2, m  1, S  2, Gr  5, Gm  10, Sc  0.22, Kc  2,Pr  0.71, t  0.2
FIGURE 2: The velocity profile for the component v against M
K  1, R  1.2, m  1, S  2, Gr  5, Gm  10, Sc  0.22, Kc  2,Pr  0.71, t  0.2
FIGURE 3: The velocity profile for the component u against K
M  2, R  1.2, m  1, S  2, Gr  5, Gm  10, Sc  0.22, Kc  2,Pr  0.71, t  0.2
FIGURE 4: The velocity profile for the component u against m
M  2, R  1.2, S  2, Gr  5, Gm  10, Sc  0.22, Kc  2,Pr  0.71, t  0.2, K  1
FIGURE 5: The temperature profile for
Pr=0.71, t=0.2

against

with S=2,
FIGURE 3: The velocity profile for the component v against K
M  2, R  1.2, m  1, S  2, Gr  5, Gm  10, Sc  0.22, Kc  2,Pr  0.71, t  0.2
FIGURE 4: The velocity profile for the component v against m
M  2, R  1.2, S  2, Gr  5, Gm  10, Sc  0.22, Kc  2,Pr  0.71, t  0.2, K  1
FIGURE 5: The temperature profile for  against t with S=2,
  5 / 2 , Pr=0.71
FIGURE 6: The Concentration profile for C against Sc with
Kc=2,   5 / 2 , t=0.2
FIGURE 7: The Concentration profile for C against
Kc=2, Sc=0.22, t=0.2
M
K
m
R
S
Gr
2
3
4
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
4
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
1
1
1
1
2
3
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1.2
1.2
1.2
1.2
1.2
1.2
1.2
1.4
1.8
1.2
1.2
1.2
1.2
1.2
1.2
1.2
1.2
1.2
1.2
1.2
1.2
2
2
2
2
2
2
2
2
2
3
4
2
2
2
2
2
2
2
2
2
2
5
5
5
5
5
5
5
5
5
5
5
6
7
5
5
5
5
5
5
5
5

with
FIGURE 6: The Concentration profile for C against Kc with
Sc=0.22,   5 / 2 , t=0.2
FIGURE 7: The Concentration profile for C against t with Kc=2,
  5 / 2 , Sc=0.22
TABLE 1: Skin Friction
Gm
Sc
Kc
10
10
10
10
10
10
10
10
10
10
10
10
10
5
8
10
10
10
10
10
10
0.22
0.22
0.22
0.22
0.22
0.22
0.22
0.22
0.22
0.22
0.22
0.22
0.22
0.22
0.22
0.3
0.6
0.22
0.22
0.22
0.22
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
4
7
2
2
Pr
0.71
0.71
0.71
0.71
0.71
0.71
0.71
0.71
0.71
0.71
0.71
0.71
0.71
0.71
0.71
0.71
0.71
0.71
0.71
3
7
 xz
5.620268
5.280022
4.994062
5.630642
5.633692
5.781484
5.936172
5.368144
4.939473
5.513113
5.431000
5.938664
6.257066
3.606121
4.814612
5.438912
4.923213
5.310184
4.999061
4.900980
4.533414
 yz
-2.685635
-2.431979
-2.238832
-2.798579
-2.856412
-3.117423
-3.295582
-2.707081
-2.707612
-2.599534
-2.539932
-2.802592
-2.919556
-1.635212
-2.265465
-2.441874
-1.885492
-2.286522
-1.957933
-2.261319
-2.153403
It is noted from the table 1 that the magnitudes of both the skin friction components
 xz
and
 yz
increase with increase in permeability parameter K, hall parameter m, thermal Grashof number Gr and mass
Grashof number Gm, and where as it reduces with increase in Hartmann number M, heat source parameter S,
Schmidt number Sc, chemical reaction parameter Kc and Prandtl number Pr. Likewise the rotation parameter R
enhances skin friction component
 xz
and reduces skin friction component  yz . From the table 2 that The
magnitude of the Nusselt number Nu increases for the parameters heat source parameter S and Prandtl number
Pr or time t, and it reduces with the frequency of oscillation  . Also from the table 4, the similar behaviour is
observed. The magnitude of the Sherwood number Sh increases for increasing the parameters Schmidt number
Sc and chemical reaction parameter Kc or time t and reduce with increasing the frequency of oscillation  .
TABLE 2: Nusselt Number
S
Pr
2
3
4
2
2
2
2
2
2
0.71
0.71
0.71
3
7
0.71
0.71
0.71
0.71

5 / 2
5 / 2
5 / 2
5 / 2
5 / 2
7 / 2
9 / 2
5 / 2
5 / 2
t
Nu
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.4
0.6
-1.59653
-1.85503
-2.07512
-4.36861
-8.61827
-1.59538
-1.59431
-1.59854
-1.60026
TABLE 3: Sherwood Number
Sc
Kc
2
3
4
2
2
2
2
2
2
0.22
0.22
0.22
0.3
0.6
0.22
0.22
0.22
0.22

5 / 2
5 / 2
5 / 2
5 / 2
5 / 2
7 / 2
9 / 2
5 / 2
5 / 2
t
Sh
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.4
0.6
-0.781334
-0.928700
-1.053333
-0.937762
-1.434060
-0.780754
-0.778487
-0.782446
-0.783434
Conclusions
1.
2.
3.
4.
5.
6.
The resultant velocity enhances with increasing K, m, R, Gr, Gm, Pr and time t; and reduces with
increasing M, S, Kc and Sc.
Lower the permeability of porous medium lesser the fluid speed in the entire fluid region.
The parameters S and Pr reduce the temperature in all layers. The temperature increases with
increasing  and time.
The Schmidt number and Kc reduce the concentration in all layers. The concentration increases with
increasing  and time.
The heat transfer coefficient increases with increasing S and Pr or time span, and it reduces with  .
The Sherwood number enhances for increasing the parameters Schmidt number and chemical reaction
parameter or time span and reduces with increasing  .
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