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IIUM Engineering Journal, Vol. 15, No. 2, 2014 Azram and Zaman
101
COUETTE FLOW PROBLEM FOR AN UNSTEADY MHD
THIRD-GRADE FLUID WITH HALL CURRENTS
MOHAMMAD AZRAM
1
AND HAIDER ZAMAN
2
1
Department of Science in Engineering,
Faculty of Engineering, International Islamic University Malaysia,
PO Box 10, 50728, Kuala Lumpur, Malaysia.
2
Faculty of Numerical Sciences, Islamia College University 25120,
Peshawar 25000, Pakistan.
azram50@hotmail.com
ABSTRACT: In this work, we analyze Couette flow problem for an unsteady mangneto-
hydrodynamic (MHD) third-grade fluid in the presence of a pressure gradient and Hall
currents. Existing literature on the topic shows that the effecs of Hall current on Couette
flow of an unsteady MHD third-grade fluid with a pressure gradient has not yet been
investigated. The arising non-linear problem is solved by the homotopy analysis method
(HAM) and the convergence of the obtained complex series solution is carefully
analyzed. The effects of pressure number, Hartmann number and Hall parameter on
unsteady velocity are discussed via analysis of plots.
ABSTRAK: Kajian dijalan untuk menganalisa masalah aliran Couette bagi bendalir
MHD gred ketiga dan arus Hall. Bagi topik ini kesan arus Hall terhadap aliran Couette
dalam bendalir MHD gred ketiga tak mantap dengan kecerunan tekanan, belum pernah
dikaji selidik. Masalah tak linear berbangkit diselesaikan dengan kaedah analisis
homotopi (HAM) dan ketumpuan solusi rangkaian kompleks dianalisa dengan teliti.
Kesan nilai tekanan, nombor Hartmann dan parameter Hall terhadap halaju tak mantap
diperbincangkan melalui plot yang dianalisis.
KEYWORDS: Couette flow; hall currents; unsteady flow; third-grade fluid; HAM
1. INTRODUCTION
In fluid mechanics, everyone is familiar with the Couette flow problem, the flow
between two parallel plates in which bottom plate is fixed and upper plate is initially at
rest and is suddenly set into motion in its own plane with a constant velocity, is termed as
Couette flow [1,2]. Bhaskara and Bathaiah [3] have analyzed the Couette flow problem
with Hall effects for flow through a porous straight channel. Ganapathy [4] has written a
note on the oscillatory Couette flow in a rotating system. Erdogan [5] solved unsteady
Couette flow for viscous fluid by the Laplace transform method. Stokes and Couette flows
due to an oscillating wall have been discussed by Khaled and Vafai [6]. Hayat et al. [7]
used the Laplace transform method to determine the analytic solutions of Couette flows of
a second grade fluid. Oscillatory Couette flow has been studied by Singh [8]. Guria [9]
discussed the Couette flow problem for rotating and oscillatory flow. Couette flow of an
unsteady third-grade fluid with variable magnetic field has been investigated by Hayat and
Kara [10], where the fluid is in an annular region between two coaxial cylinders. The axial
Couette flow problem of an electrically conducting fluid in an annulus has been examined
by Hayat et al. [11]. Das et al. [12] studied unsteady Couette flow problem in a rotating
system. Recently, Zaman et al. [13] presented a solution for unsteady Couette flow
IIUM Engineering Journal, Vol. 15, No. 2, 2014 Azram and Zaman
102
problem for the Eyring-Powell model. When a strong magnetic field is applied in an
ionized gas of low density, the conductivity normal to the magnetic field is decreased by
free spiraling of electrons and ions about the magnetic lines of force before suffering
collisions. This phenomenon is known as the Hall effect and a current induced in a
direction normal to the electric and magnetic fields is called the Hall current [14]. The
study of the effects of Hall currents on flow of non-Newtonian fluids [15-23] is important
because of its applications in power generators and pumps, Hall accelerators, refrigeration
coils, electric transformers, in flight MHD, electronic system cooling, cool combustors,
fiber and granular insulation, oil extraction, thermal energy storage, and flow through
filtering devices.
In order to understand the interaction of electric, magnetic, and hydrodynamic forces
in the unsteady third-grade fluid, we considered a simple flow problem, known as the
Couette flow. The effects of a pressure gradient and Hall current on the flow are also taken
in to account. The complex analytical solution for the non-linear problem is found by
using the homotopy analysis method (HAM) [24- 31].
This solution is valid for all values of time in the whole spatial domain 10 <≤ η . The
convergence of the analytic solution is ensured with the help of the ħ-curve. The effect of
pressure number, Hartmann number, Hall parameter, second-grade parameter and third-
grade parameters on the unsteady velocity are illustrated through plots. Also, the effects of
the pertinent parameters on the local skin friction co-efficient at the surface of the wall are
presented numerically in tabular form.
2. FORMULATION OF THE PROBLEM AND ITS ANALYTIC
SOLUTION
Consider the unsteady flow of an electrically conducting incompressible third-grade
fluid between two parallel flat plates, subjected to a uniform transverse magnetic field. We
assume that the bottom plate is fixed and the top plate is stationary when t < 0 and at t = 0,
the top plate starts moving impulsively in its own plane with a constant velocity U and a
pressure gradient is also applied. The flow here is maintained by the motion of the top
plate. The Cauchy stress tensor T for a third-grade fluid is given as [32]
1)
2
1
(3)1221(231
2
12211
AtrAAAAAAAAApIT βββααµ +++++++−=
(1)
where p is the scalar pressure, I is the identity tensor, µ is the co-efficient of viscosity
iα , jβ , (i = 1, 2 and j = 1, 2, 3) are the material parameters of third-grade fluid, and Ai (i
= 1, 2, 3) are the first three Rivlin-Ericksen tensors defined by [32]
A1 = (grad V) + (grad V)
T
(2)
.2,1,)()(1 =++=+ nnA
T
VgradVgradnA
dt
ndA
nA (3)
The equations governing the magneto-hydrodynamic flow with Hall effects are:
Velocity Field: V = [u (y; t) , 0, 0] (4)
IIUM Engineering Journal, Vol. 15, No. 2, 2014 Azram and Zaman
103
Continuity equation: div V = 0 (5)
Equation of motion: )()( BJTdiv
dt
dV
×+=ρ (6)
Maxwell equations:
t
B
EcurlJmBcurlBdiv
∂
∂
−=== )(,)(,0)( µ (7)
Generalized Ohm’s law: )( BVEBJ
oB
eewJ ×+=×+ σ
τ
(8)
Where u( y; t) is the velocity component in the x-direction, t is time, B (= B0 + b) is
the total magnetic field, B0 is the applied magnetic field, b is the induced magnetic field, J
is the current density, σ is the electrical conductivity of the fluid, E is the electric field, µ m
is the magnetic permeability, ρ is the fluid density, we and τe are the cyclotron frequency
and collision time of the electrons respectively. We assume that, the quantities ρ, µ m and σ
are constants throughout the flow field, the magnetic field B is normal to the velocity
vector V and the induced magnetic field is neglected compared with the imposed magnetic
field, so that the magnetic Reynolds number is small [33]. We also suppose that weτe ~ O
(1) and wiτi << 1 (where wi and τi are the cyclotron frequency and collision time for ions
respectively). Continuity Eqn. (5) is identically satisfied by the velocity taken in Eqn. (4).
Under the aforementioned assumptions the unsteady Couette flow problem with Hall
currents become;
���� � � �� � �
� � �
���
� ���
�����������
� � � ��� �����
��
� ��� �����
��
� ���
����� ������ �
���
(9)
0 � � 1# $%$&,
(10)
The boundary and initial conditions are
��&, (� � 0, )( & � 0, *+, ( - 0,
(11)
��&, (� � �. � /, )( & � 0, *+, ( 1 0, ��&, (� � 0, )( ( � 0, *+, 0 2 & 3 0
where ν is the kinematic viscosity, � �� 45 65 � is the Hall parameter, U is the velocity
of the upper plate and h is the distance between two parallel plates and it will be
considered as a length scale of the flow, eqn. (10) shows that p is independent of y. In
order to non-dimensionalize the problem let us introduce the similarity transformations,
� � /*�7, 8�, 7 � �9 , 8 � :�9
, (12)
where *�7, 8� is the dimensionless velocity function, η is the dimensionless distance
from the bottom wall and ξ is the dimensionless time. Eqn. (9) and eqn. (11) become
*" � < �="�> � ? �
="�>
� @����������
� * � A*B *" � �=�> � C � 0 (13)
*�0, 8� � 0, *�1, 8� � 1, *+, 8 - 0, (14)
IIUM Engineering Journal, Vol. 15, No. 2, 2014 Azram and Zaman
104
*�7, 0� � 0, *+, 0 2 7 3 1,
where prime denotes differentiation with respect to 7, <�� <�/#0 � is dimensionless
second-grade parameter, ?�� ?��/#0E�, A �� ���
����F
�:9
� are dimensionless third-grade
fluid material parameters, C �� 9
�FG H H
� is the dimensionless pressure number and I��JKL 0 /#�� is the dimensionless modified Hartmann number [33]. The local skin friction
coefficient or fractional drag coefficient on the surface of the moving wall is
M= � N OPQRQST��U
, (15)
Now using equations (1), (2), (3), (4) and (12) the eqn. (15) can be written in
dimensionless variables as
V5 W M= � 2Y* Z N�1, 8� � < ��> * Z�1, 8� � ? �
=[��,>��>
� �\ A�* Z�1, 8��\] (16)
where V5 �� /0/^� is the Reynolds number. The boundary conditions (14) lead us to
take base functions for velocity *�7, 8� as
Y7_ N8` /, a 1 0, b 1 0] (17)
The velocity *�7, 8� can be expressed in terms of base functions as
*�7, 8� � ∑ ∑ Ωefg̀hLg_hL 7_ 8` (18)
To start with the homotopy analysis methods, due to the boundary conditions (14) it is
reasonable to choose the initial guess approximation
*L�7, 8� � 7�1 � 8 � 78�, 8 - 0 (19)
and the auxiliary linear operator
i�*� � ��=�j,>��j� , (20)
Following the HAM and trying higher iterations with the unique and proper
assignment of the results converge to the exact solution:
*�7, 8� k *L�7, 8� � *��7, 8� � *��7, 8� � * �7, 8� � l � *m �7, 8� (21)
At �� 0.1, N=0.1, <=0.1, ?=0.1, A=0.1, P= 0.1 using the symbolic computation we
obtained
*��7, 8� � 7�1 � 8 � 78� � 3p7810 � 3p7
810
� p78 15 � p7
8 10 � p7
\8 6 � 13p7815
\ � 11p7\10
� 3p7\8\10 � p7
E8\15 � s 2271515 � u606v p78E � p7
8E20
IIUM Engineering Journal, Vol. 15, No. 2, 2014 Azram and Zaman
105
� s 5303 � u606v p7\8E � p7
E8E12 � � 5606 � u1212�p78w
� � w\L\ � ��L�� p7\8w � � w�L� � �� � � p7E8w, (22)
Similarly * �7, 8�, *\�7, 8�, *E�7, 8� and so on are calculated. The obtained values of *L, *�, * , ----- lead us to take
*m �7, 8� � x x Ωm,_` 7_
\m�\y_
`hL
m�
_hL
8` . (23)
The total complex analytic solution in compact form is as
*�7, 8� � x *m �7, 8� � lim}~g � x x Ωm,L` 7_
\m�\
`hL
}
mhL
8` �g
mhL
(24)
where from initial guess in eqn. (19) we obtain
ΩL,�L � 1, ΩL,�� � 1, ΩL, � � �1
(25) ΩL,_` � 0, �b � 0,1,2,3�, �a � 0,1,2�
all other unknown constants can be determined by utilizing eqn. (25) and using the
recurrence relations, which we have calculated but it is not possible to write here due to
their length. We know that the auxiliary parameter p gives the convergence region and
rate of approximation for the homotopy analysis method. From p-curve in Fig.1 we note
that the range for the admissible value for p is -1.5 < p < 0. Our calculations depict that
the series of the dimensionless velocity in eqn. (25) converges in the whole region of η and
ξ for p = -0.5.
3. GRAPHS, TABLES AND DISCUSSION
The discussion of emerging parameters on the dimensionless velocity f ( η, ξ ) is as
follows:
Figures 2 to 7 are plotted in the absence of Hall currents and in Fig. 8 the Hall current
is taken into account.
Figure 2 presents the velocity profile f for various values of ξ. This figure shows that
with the passage of time the velocity of the fluid increases due to continuous motion of the
upper plate. The boundary layer thickness decreases with the passage of time and shear
thinning is seen. Figure 3 elucidates the variation of Hartmann number on the velocity. It
is found that the velocity increases with an increase in N and the boundary layer thickness
decreases. This means that the magnetic force provides a mechanism to the control of
boundary layer thickness. Figure 4 illustrates the influence of a second-grade parameter
on the velocity profile f. It is evident from the figure that an increase in α results in the
decrease of the velocity; here boundary layer thickness increases and shear thickening is
observed.
IIUM Engineering Journal, Vol. 15, No. 2, 2014 Azram and Zaman
106
Fig.1: ħ-curve for f (η, ξ). Fig. 2: Influence of ξ on f (η, ξ).
Fig. 3: Influence of N on f (η, ξ). Fig. 4: Influence of α on f (η, ξ).
In Fig. 5 and Fig. 6 the velocity distribution is presented for the various values of
third-grade parameters β and ζ. It is observed that the velocity increases by increasing the
influence of β and decreases by increasing the influence of ζ.
Fig. 5: Influence of β on f (η, ξ). Fig. 6: Influence of ζ on f (η, ξ).
IIUM Engineering Journal, Vol. 15, No. 2, 2014 Azram and Zaman
107
Fig. 7: Influence of P on f (η, ζ). Fig. 8: Influence of ε on Abs {f (η, ξ)}.
Figure 7 depicts the variation of the pressure number on the velocity. It is observed
that the velocity increases with an increase in P, which is consistent with what is expected.
Due to the Hall current velocity field becoming complex, the absolute value of the velocity
profile is plotted in Fig. 8. It is observed that with an increase in the Hall parameter ε, the
absolute value of the velocity increases and boundary layer thickness decreases.
It is observed from Table 1 that for fixed values of α, β, ζ, ħ, P and ε with increase in
Hartmann number N, the absolute value of the skin friction co-efficient Re × Cf given in
equation (16) increases for all values of the time ξ. Also with an increase in the Hall
current ε, the absolute value of the skin friction co-efficient decreases for all values of the
time ξ. For fixed values of N, ħ, α, β, ζ, P and ε with increase in dimensionless time ξ the
absolute value of the skin friction co-efficient decreases at all points. Table 2 displays the
variation of the skin friction co-efficient with the pressure number P. It is observed from
Table 2 that increase in the pressure leads to an increase in the shear stress at the wall at all
points which is consistent with our expectation.
Table 1: Absolute values of the skin friction co-efficient Re × Cf with
α = 0.1, β = 0.1, ζ = 0.1, P = 0.1, ħ = 0.5.
ξ
ε = 0.1 ε = 0.1 ε = 0.1 N = 0.1 N = 0.1 N = 0.1
N = 0.1 N = 0.3 N = 0.5 ε = 0.2 ε = 0.3 ε = 0.5
0.2 1.98649 1.99305 1.99961 1.98639 1.98625 1.98586
0.4 1.89823 1.92579 1.95333 1.89783 1.89722 1.89559
0.6 1.80866 1.87047 1.93679 1.80777 1.80641 1.80276
0.8 1.74278 1.84279 1.93615 1.74147 1.73924 1.73350
1 1.74219 1.84016 1.93205 1.74139 1.73920 1.73325
Table 2: Absolute values of the skin friction co-efficient Re × Cf with
α = 0.1, β = 0.1, ζ = 0.1, ξ = 0.1, ħ = 0.5.
P
ε = 0.1 ε = 0.1 ε = 0.1 N = 0.1 N = 0.1 N = 0.1
N = 0.1 N = 0.3 N = 0.5 ε = 0.2 ε = 0.3 ε = 0.5
0.2 2.02849 2.03009 2.03168 2.02847 2.02844 2.02834
0.4 2.03078 2.03237 2.03396 2.03076 2.03072 2.03063
0.6 2.03306 2.03465 2.03624 2.03304 2.03300 2.03291
0.8 2.03535 2.03694 2.03853 2.03532 2.03529 2.03519
1.0 2.03763 2.03922 2.04081 2.03761 2.03757 2.03748
IIUM Engineering Journal, Vol. 15, No. 2, 2014 Azram and Zaman
108
4. CONCLUSION
The Couette flow between two parallel plates filled with MHD unsteady third-grade
fluid is studied analytically. The effects of the pressure and Hall current are also
incorporated. A non-linear third-grade model for the fluid is used. The model is invoked
into the governing equations and the resulting one dimensional equation for unsteady
MHD flow is derived. This equation is solved by HAM in general to study the sensitivity
of the flow to the parameters that are used in the third-grade model. The various
dimensionless parameters seem to affect the velocity a lot. The velocity profile and local
skin friction co-efficient are greatly influenced by the Hall parameter, pressure and
Hartmann numbers.
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