Er25885892

E. Omokhuale, I. J. Uwanta, A. A. Momoh, A. Tahir. / International Journal of Engineering
Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue5, September- October 2012, pp.885-892
The Effects of Concentration and Hall Current on Unsteady Flow
of a Viscoelastic Fluid in a Fixed Plate
1
E. Omokhuale, 1I. J. Uwanta,2A. A.Momoh, 2 A. Tahir.
1
Department of Mathematics, UsmanuDanfodiyo University, P. M. B. 2346, Sokoto, Nigeria
.2
Department of Mathematics, Modibbo Adama University of Technology, P.M.B. 2076, Yola. Adamawa State.
Nigeria

ABSTRACT
The present Study deals with the effects numerically. Kai- Long (2010) studied Heat and
of concentration and Hall current on mass transfer for a viscous flow with radiation effect
unsteadyflow of a viscoelastic fluid in a fixed past a non-linear stretching sheet. Hall Effect on
plate. The resultant equations have been solved MHD mixed convective flow of a viscous
analytically. The velocity, temperature and incompressible fluid past a vertical porous plate
concentration distributions are derived, and their immersed in a porous medium with Heat
profiles for various physical parameters are Source/Sink was studied by Sharma et al. (2007).
shown through graphs. The coefficient of Skin Salehet al. (2010) examined Heat and mass transfer
friction, Nusselt number and Sherwood number in MHD viscoclastic fluid flow through a porous
at the plate are derived and their numerical medium over a stretching sheet with chemical
values for various physical parameters are reaction. Das and Jana (2010) examined Heat and
presented through tables. The influence of Mass transfer effects on unsteady MHD free
various parameters such as the thermal Grashof convection flow near a moving vertical plate in a
number, mass Grashof number, Schmidt porous medium. The effects of chemical reaction,
number, Prandtl number, Hall Parameter, Hall current and Ion – Slip currents on MHD
Hartmann number, Chemical Reaction micropolar fluid flow with thermal Diffusivity using
parameter and the frequency of oscillation on the Novel Numerical Technique was studied by Motsa
flow field are discussed. It is seen that, the and Shateyi (2012). Aboeldahab and Elbarby (2001)
velocity increases with the increase in Gc and examined Hall current effect on
Gr,and it decreases with increase in Sc, M, m, Kc, Magnetohydrodynamics free convection flow past a
and Pr, temperature decreases with increase in Semi – infinite vertical plate with mass transfer. Hall
Pr, Also, the concentration decreases with the current effect with simultaneous thermal and mass
increase in Sc and Kc. diffusion on unsteady hydromagnetic flow near an
accelerated vertical plate was investigated Acharyaet
KEY WORDS: Hall Current, Viscoelastic, Porous al. (2001). Takhar (2006) studied Unsteady flow free
medium, Magnetohydrodynamics. convective flow over an infinite vertical porous
platedue to the combined effects of thermal and
1.0 INTRODUCTION mass diffusion, magnetic field and Hall current.
There exist flows which are caused not only Exact Solution of MHD free convection flow and
by temperature differences but also by concentration Mass Transfer near a moving vertical porous plate in
differences. There are several engineering situations the presence of thermal radiation was investigated
wherein combined heat and mass transport arise by Das (2010). Anjali Devi and Ganga examined
such as dehumidifiers, humidifiers, desert coolers, effects of viscous and Joules dissipation and MHD
and chemical reactors etc. Magnetohydrodynamics is flow, heat and mass transfer past a Stretching porous
currently undergoing a period of great enlargement surface embedded in a porous medium. Sonthet al.
and differentiation of subject matter. It is important (2012) studied Heat and Mass transfer in a
in the design of MHD generators and accelerators in viscoelastic fluid over an accelerated surface with
geophysics, underground water storage system, soil heat source/sink and viscous dissipation. Shateyiet
sciences, astrophysics, nuclear power reactor, solar al. (2010) investigated The effects of thermal
structures, and so on. Radiation, Hall currents, Soret and Dufour on MHD
Moreso, the interest in these new problems flow by mixed convection over vertical surface in
generates from their importance in liquid metals, porous medium. Effect of Hall currents and
electrolytes and ionized gases. On account of their Chemical reaction and Hydromagnetic flow of a
varied importance, these flows have been studied by stretching vertical surface with internal heat
several authors – notable amongst them are Singh generation / absorption was examined by Salem and
and Singh (2012) they investigated MHD flow of El-Aziz (2008).
Viscous Dissipation and Chemical Reaction over a
Stretching porous plate in a porous medium 2.0 MATHEMATICAL FORMULATION

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We consider the unsteady flow of a viscous volume expansion, k 0 is the kinematic
incompressible and electrically conducting
viscoelastic fluid over afixed porous plate with viscoelasticity,  is the density,  is the viscosity,
oscillating temperature. The x- axis is assumed to be is the kinematic viscosity, KT is the thermal
oriented vertically upwards along the plate and the conductivity, Cp is the specific heat in the fluid at
y-axis is taken normal to the plane of the plate. It is constant pressure,  is the electrical conductivity of
assumed that the plate is electrically non – the fluid,  e is the magnetic permeability, D is the
conducting and a uniform magnetic field of straight
molecular diffusivity, T0 is the temperature of the
B0 is applied normal to the plate. The induced
magnetic field is assumed constant. So that plane and Td is the temperature of the fluid far away

B   0, B0 ,0 ,. The plate is subjected to a constant from plane. C 0 is the concentration of the plane and
suction velocity. Cd is the concentration of the fluid far away from
1.
The equation of conservation of charge J  0, gives constant. the plane.
And v  v0 , the negative sign indicate that the
      Pe  suction is towards the plane.
2.
J  e e ( J  E )   V  B   (1) Introducing the following non-dimensionless
 ene 
parameters
Equation (1) reduces to v0 y v 2t u * T  T C  C 
 B0   , t  0 , u  ,   ,  ,C  
2 
J x*  mu*   *    4 v0 v0 T  T C  C 
(1  m ) 
 g  (T  T ) g  c (C  C )  B0 2 Cp 

 B0 Gr  , Gc  ,M  , Pr  
J y* 
(1  m2 )
 u  m 
* *  (2) v0 2
v0 2
 v0 2
KT 
 
 k1v0 2 k *v0 2
Where Sc  , K  2 , k  2 
D 4  

m  e e
is the Hall parameter. (8)
The governing equations for the momentum, energy Substituting the dimensionless variables in (8) into
and concentration are as follows; (3) to (6), we get (dropping the bars)

u u u  3u (u  m ) u 1 u u  2u K  3u M (u  m ) u
 v0    k1 2   B02  g  (T  Td )  g  * (C  Cd )  *       Gr  GcC
t y y y t  (1  m )
2
k 4 t   2 4 2 t (1  m 2 ) k

(3) (9)
   2  3 (  mu ) u 1    2 K  3 M (  mu ) u
 v0   2  k1 2   B0 2      
t y y y t  (1  m 2 ) k * 4 t   2 4 2 t (1  m 2 ) k
(4)
(10)
T T K T 2
 v0  T 1   1  2
t y  Cp y 2  
4 t  Pr  2
(5) (11)
C C C 2
1 C C 1  C 2
 v0  D 2  KrC  
t y y 4 t  Sc  2
(6) (12)
The boundary conditions of the problem are: The corresponding boundary conditions are
u  0,   0, T  Td  (T0  Td )eit , C  Cd  (C0  Cd )eit aty  0 u(0, t )  0,  (0, t )  0, (0, t )  eit , C (0, t )eit aty  0
u  0,   0, T  0, C  0aty  1 u(1, t )   (1, t )   (1, t )  C (1, t )  0aty  1
(7) (13)
Where u and v are the components of velocity in the
x and y direction respectively, g is the acceleration Equations (9) and (10) can be combined into a single
due to gravity,  and  are the coefficient of
*
equation by introducing the complex velocity.

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U  u , t   i , t  Solving (20) to (22) under the boundary
conditions (23) and (25).And substituting the
(14) obtained solutions into (17) to (19) respectively.
Then
Where the velocity distribution can be expressed as
i  1 U (, t )   A5em5  A6em6  A8em3  A9em4  A10em1  A11em2   eint
 
Thus, (24)
1 U U  2U K  3U M (1  im)U U And, the temperature field is given by
      Gr  GcC
4 t   2 4 2 t (1  m 2 ) k  ( , t )   A3em   A4e m    ei t
3 4

(15) (25)
With boundary conditions: Similarly, the concentration distribution gives
t  0 : U (0, t )  0, (0, t )  eit , C (0, t )eit at  0
*
C ( , t )   A1em1  A2e m2   ei t
U (1, t )   (1, t )  C (1, t )  0, at  1 (26)
The Skin friction, Nusselt number and
(16) Sherwood number is obtained by differentiating (24)
where Gr is the thermal Grashof number,
to (26) and at
evaluated at   0 respectively. .
Gc is the mass Grashof number, Sc is the Schmidt
number, Pr is the Prandtl number, Rm is the Reynold U ( , t )
number, K is the viscoelastic Parameter and S is the    m5 A5  m6 A6  m3 A8  m4 A9  m1 A10  m2 A11   ei t
pearmeability.   0
(27)
3.0 METHOD OF SOLUTIONS  ( , t )
To solve (11), (12) and (15) subject to the boundary    m3 A3  m4 A4   ei  t
conditions (16), we assume solutions of the form   0
U (, t )  U1    eit (28)
C ( , t )
(17)
   m1 A1  m2 A2 
 ( , t )  1 ( ) e i t
  0
(18) (29)
C(, t )  C1    e it
(19)
4.0 RESULTS AND DISCUSSION
where U1   1   and C1   are to be The effects of concentration and Hall
determined. current on unsteady flow of a viscoelastic fluid in a
Substituting (17) to (19) into (11), (12) and (15), fixed plate has been formulated and solved
Comparing harmonic and non harmonic terms, we analytically. In order to understand the flow of the
obtain fluid, computations are performed for different
parameters such as Gr, Gc, Sc, Pr,  , Kc, M, and
U Gr GcC
U    L3U   it
 m.
L1 4 L1e 4 L1eit
4.1 Velocity profiles
(20) Figures 1 to 7connote the velocity profiles;
1
1  Pr 1  i1  0
figure 8 depicts the temperature profiles and
figures9& 10display the concentration profiles with
4
varying parameters respectively.
(21) The effect of velocity for different values of
1 (Pr = 0.71, 1, 3) is displayed in figure 1, the graph
C1 ScC1  iC1  KcScC1  0 show that velocity decreases with increase in Pr.
4 (22) The effect of velocity for different values of (Sc =
and boundary conditions give 0.18, 0.78, 2.01, 2.65) is given in figure 2, the graph
  0 : U1 (0)   (0)  C (0)  1  show that velocity decreases with the increase in Sc.
Figure 3 denotes the effect of velocity for different

  1: U1 (1)  0, (1)  0, C (1)  0 values of (m = 2, 4, 6, 10), it is seen that velocity
decreases with the increase in m.
(23)
The effect of velocity for different values of (M = 1,
where the primes connotes differentiation with
respect to  .
3, 6, 10) is shown in figure 4, it depict that velocity
decreases with increase in M.

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Figure 5 depicts the effect of velocity for (Gr = 1, 2, The effect of velocity for different values of (Kc = 1,
3, 5), the graph connote that velocity increases with 2, 3, 4) is displayed in figure 7, it is clear that
the increase in Gr. velocity decreases with the increase in Kc.
The effect of velocity for different values of
(Gc = 2, 3, 4, 8) is presented in figure 6, it is seen
that velocity increases with the increase in Gc.

Figure 1. Velocity profiles for different values of Pr.

Figure 2. Velocity profiles for different values of Sc.

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Figure 3. Velocity profiles for different values of m.

Figure 4. Velocity profiles for different values of M
.

Figure 5. Velocity profiles for different values of Gr.

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Figure 6. Velocity profiles for different values of Gc.

Figure 7. Velocity profiles for different values of Kc.

4.2 Temperature profiles
In figure 8, the effect of temperature for different
values of (Pr = 0.71, 1, 3, 7) is given. The graph
show that temperature increases with increasing Pr.

Figure 8. Temperature profiles for different values of Pr.

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4.3 Concentration profiles concentration for (Kc= 0, 0.5, 1, 3) is given in figure
Figure 9 depicts the effect of concentration for (Sc = 10, it is seen that concentration decreases with the
0.3, 0.78, 1, 2.01), it is seen that concentration decrease in Kc.
increases with the increase in Sc. The effect of

Figure 9. Concentration profiles for different values of Sc.

Figure 10. Concentration profiles for different values of Kc.

Table 1 to 3 represent the Skin friction, Nusselt Table 2: Nusselt number
number and Sherwood number respectively.  Pr Nu
Table 1 depicts that the Skin friction increases with 1 0.71 0.8181
increase in Gc and Grand decreases with increase in 4 0.71 0.7400
Sc, m, M, Kc, and Pr. 1 3 0.7830
Table 2 represents that the Nusselt number decreases
with increase in Pr, and decreases with increase in  Table 3: Sherwood number
Table 3 shows that the Sherwood number decreases  Sc Sh
with increase in Sc and  . 1 0.3 0.4015
5 0.3 0.3890
Table 1: Skin friction  4 0.6 0.3977
 Gc Sc Kc Gr M Pr M 
1 1 0.3 0.1 1 1 0.71 0.5 6.1082 5.0 SUMMARY AND CONCLUSION
3 1 0.3 0.1 1 1 0.71 0.5 5.5219 We have examined and solved the
1 1 0.6 0.1 1 1 0.71 0.5 5.7548 governing equations for the effects of concentration
1 4 0.3 0.1 1 1 0.71 0.5 6.5671 and Hall current on unsteady flow of a viscoelastic
1 1 0.3 1 1 1 0.71 0.5 6.3654 fluid in a fixed plate analytically. In order to point
1 1 0.3 0.1 4 1 0.71 0.5 6.8760 out the effect of physical parameters namely;
1 1 0.3 0.1 1 3 3 0.5 5.9006 thermal Grashof number, mass Grashof number,
1 1 0.3 0.1 1 1 0.71 2 5.0126 Schmidt number, Prandtl number, Chemical reaction
1 1 0.3 0.1 1 1 0.71 0.5 6.8741 parameter, Hartmann number, Hall parameter and

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the frequency of oscillation on the flow field and the 8. Salem, A. M. and El-Aziz, M. Abd. (2008).
following conclusions are drawn: Effect of Hall currents and Chemical
*The velocity increases with the increase in Gc and reaction and Hydromagnetic
Gr. flow of a stretching vertical surface with
*The velocity falls due to increasing in Sc,  , M, m, internal heat generation / absorption.
Kc, and Pr. Applied Mathematical Modelling: 32, 1236
*The temperature decreases with increase in Pr, – 1254.
*The concentration reduces with the increase in Sc 9. Saleh, M.A.,Mohammed ,A.A.B. and
and Kc. Mahmoud S.E. (2010). Heat and mass
*The Skin friction increases with increase in Gc and transfer in MHD viscoclastic fluid flow
Gr and decreases with increase in Sc, M, m,Kc, and through a porous medium over a stretching
Pr. sheet with chemical reaction. Journal of
*The rate of heat transfer decreases with increase in Applied Mathematics, 1, 446-455.
Pr and  ,
*The Sherwood number decreases with the increase 10. Shateyi, S., Motsa, S. S. and Sibanda, P.
in Sc and  . (2010). The effects of thermal Radiation,
Hall currents, Soret and
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