MHD Transient Flow with Hall Current Past an Accelerated Horizontal Porous Plate in a Rotating System ()
1. Introduction
Many engineering problems are susceptible to MHD analysis. The study of MHD flow problems has achieved remarkable interest due to its application in MHD generators, MHD pumps and MHD flow meters etc. The study of effects of magnetic field on free convection flow is important in liquid metals, electrolytes and ionized gases. Geophysics encounters MHD phenomena in interactions of conducting fluids and magnetic fields. The rotating flow of an electrically conducting fluid in presence of magnetic field has got its importance in Geophysical problems. The study of rotating flow problems are also important in the solar physics dealing with the sunspot development, the solar cycle and the structure of rotating magnetic stars. It is well known that a number of astronomical bodies possess fluid interiors and magnetic fields. Changes that take place in the rate of rotation, suggest the possible importance of hydro magnetic spinup. The general theory of rotating fluids has received growing interest during last decade because of its application in Cosmic and Geophysical science. In this regard, we may cite the works done by Raptis [1], Singh [2,3], Alam et al. [4] and Debnath [5].
MHD in the present form is due to the pioneer contri bution of several notable authors like Alfven [6], Cowling [7], Ferraro and Pulmpton [8] etc. It was emphasized by Cowling (1975) that when the strength of the applied magnetic field is sufficiently large, Ohm’s law needs to be modified to include Hall current. The Hall effect is due merely to the sideways magnetic force on the drifting free charges. The electric field has to have a component transverse to the direction of the current density to balance this force. In many works of Plasma physics, it is not paid much attention to the effect caused due to Hall current. However, the Hall effect can not be completely ignored if the strength of the magnetic field is high and the number density of electrons is small as it is responsible for the change of the flow pattern of an ionized gas. Hall effect results in a development of an additional potential difference between opposite surfaces of a conductor for which a current is induced perpendicular to both the electric and magnetic field. This current is termed as Hall current. It was discovered in 1879 by Edwin Herbert Hall while working on his doctoral degree at the Johns Hopkins University in Baltimore, Maryland, USA. Pop [9], Kinyanjui et al. [10], Archrya et al. [11] and Ahmed and Kalita [12] etc. have presented some model studies on the effect of Hall current on MHD convection flow because of its possible application in the problems of MHD generators and Hall accelerators. An unsteady MHD free convective flow past a vertical porous plate immersed in a porous medium with Hall current, thermal diffusion and heat transfer have been studied by Ahmed et al. [13]. Recently, Ahmed and Sarmah [14] have carried out an investigation of MHD transient flow past an impulsively started infinite horizontal porous plate in a rotating system with Hall current.
Due to the importance of studying MHD flow problems in rotating fluid with Hall current, we have proposed in the present paper to investigate an unsteady MHD transient flow with Hall currents past a uniformly accelerated porous plate in a rotating system.
2. Basic Equations
The equations governing the motion of an incompressible viscous electrically conducting fluid in a rotating system in presence of a magnetic field are as under:
Equation of continuity:
(1)
Momentum equation:
(2)
Kirchhoff’s first law:
(3)
General Ohm’s law:
(4)
Gauss’s law of magnetism:
(5)
where 
is the velocity vector, 
the angular velocity of the fluid, 
the position vector of the fluid particle P considered, 
the fluid density, p the pressure, 
the current density, 
the magnetic induction vector, 
the co-efficient of viscosity, 
the electrical conductivity, 
the time, 
the strength of the applied magnetic field, 
the electron frequency, 
the electron collision time, 
the electron charge, 
the number density of electron, 
the electron pressure, 
the electric field, 
is the Coriolis acceleration, 
is the centripetal acceleration and the other
Flow configuration
symbols have their usual meanings and the other symbols have their usual meanings.
We now consider an unsteady flow of an incompressible viscous electrically conducting fluid past a suddenly started infinite horizontal porous plate relative to a rotating system with constant suction in presence of a uniform transverse magnetic field taking into account the effect of Hall current. Our investigation is restricted to the following assumptions:
• All the fluid properties are constants and the buoyancy force has no effect on the flow.
• The plate is electrically non-conducting.
• The entire system is rotating with angular velocity 
about the normal to the plate and 
is so small that 
can be neglected.
• The magnetic Reynolds number is so small that the induced magnetic field can be neglected.
• 
is constant.
• 
.
Initially the plate and the fluid were rotating in unison with a constant angular velocity 
about the normal to the plate. At time
, the plate is moved in its own plane relative to the rotating system with acceleration
.We introduce a coordinate system 
with X-axis horizontally in the direction of the plate velocity, Y-axis horizontally perpendicular to the direction of the plate velocity and Z-axis along the normal to the plate which is the axis of rotation. Let 
be the fluid velocity, 
be the current density at the point 
and 
be the applied magnetic field, 
being the unit vectors along X-axis, Y-axis and Z-axis respectively. As the plate is infinite in X-direction and Y-direction, therefore all the quantities except possibly the pressure are independent of 
and
.
The Equation (1) gives
(6)
which is trivially satisfied by
(7)
Therefore the velocity vector 
is given by
(8)
The Equation (5) is satisfied by
(9)
The Equation (3) reduces to
which shows that
(10)
(as the plate is electrically non-conducting).
Hence the current density is given by
(11)
Under the above assumptions, the Equation (4) takes the form:
(12)
where 
is the Hall parameter.
The Equations (8)-(12) yield,
(13)
(14)
With the foregoing assumptions and under the usual boundary layer approximation the Equation (2) reduces to
(15)
(16)
with
.
where 
is the constant suction velocity and 
is the kinematic viscosity The relevant initial and boundary conditions are
(17)
(18)
We introduce the following non-dimensional variables and parameters:
The non-dimensional form of the Equations (15) and (16) are
(19)
(20)
Subject to the initial and boundary conditions:
(21)
(22)
3. Method of Solution
Let us introduce the complex variable 
defined by 
where
.
The non-dimensional forms of the equation governing the flow can be rewritten as follows:
(23)
where 
Subject to the boundary conditions:
(24)
(25)
On applying Laplace Transform to the Equation (23), the following ordinary differential equation is derived
(26)
with relevant boundary conditions :
(27)
(28)
The solution of the Equation (26) under the conditions (27) and (28) is
(29)
Taking inverse Laplace transforms of the Equation (29) we derive the following:
(30)
where, 
, 
, 
and 
4. Skin Friction
The non-dimensional skin-friction at the plate is given by
where 
and 
are the skin-frictions at the plate due to the primary and the secondary velocity fields. The expressions for 
and 
are obtained but not presented here for the sake of brevity.
5. Results and Discussion
In order to get the physical insight into the problem we have carried out numerical calculations for the representative velocity field and skin-friction at the plate for different values of the physical parameters involved and these values have been demonstrated in different graphs. Our investigation is restricted to t equal to 1 and the other parameters namely, rotational parameter
, Hartmann number
, Hall parameter 
and accelerating parameter 
has been considered arbitrarily.
Figure 1 depicts the variation in skin-friction 
due to the primary velocity field versus Hall parameter 
for different values of rotational parameters
. It is noticed that 
decreases with increasing values of Hall parameter 
whereas a rise in the values rotational parameter 
results a growth in
.
The influence of Hartmann number 
on skin-friction 
against Hall parameter 
is presented in Figure 2. It is observed that a growth in the Hartmann number 
leads to an increase of
. Moreover, it is seen
Figure 1. The behavior of 
versus m under the effect of
.
Figure 2. The behavior of 
versus m under the effect of
.
that 
decreases very slowly and steadily as the Hall parameter 
rises. In other words, an increase in the Hall parameter or a decrease in Hartmann number results a decrease in drag force at the plate due to the primary velocity.
Figure 3 demonstrates the nature of 
against Hall parameter 
under the effect of accelerating parameter
. Figure 3 exhibits a substantial growth in 
with increasing values of accelerating parameter
. It is also seen that 
is unalterable for small values accelerating parameter or in absence of plate acceleration. That is, 
(drag force) due to the primary velocity is undisturbed whenever the plate is at rest as well for the slow movement of the plate. However, for higher values of the accelerating parameter 
skin-friction 
decreases gradually with increasing values of Hall parameter
.
The behaviour of 
(drag force per unit area) due to the secondary velocity field versus Hall parameter m under the effects of Hartmann number
, rotational parameter 
and acceleration parameter 
respectively are depicted in Figures 4-6. It is noticed that a rise in
, 
and accelerating parameter 
results a growth in
. Moreover 
is undisturbed by
Figure 3. The behavior of 
versus m under the effect of a.
Figure 4. The behavior of 
under the effect of 
versus m.
Figure 5. The behavior of 
versus m under the effect of
.
Hall effect in absence of magnetic field and acceleration of the plate. The same figures also indicate the growth of 
for small and moderate values of Hall parameter 
and afterwards a reversal trend on 
is noticed.
The effect of Hartmann number 
on primary velocity field is presented in Figure 7. This figure clearly
Figure 6. The behavior of 
versus m under the effect of a.
Figure 7. The variation of 
versus 
under the effect of M.
shows that the primary velocity sharply decreases in a thin layer of the fluid adjacent to the plate surface and then it decreases slowly and steadily to its minimum value 
as
. The same figure also indicates that an increase in the Hartmann number 
has an inhibiting effect on the primary velocity
. The primary fluid velocity 
gets continuously reduced with increasing
. That is the application of the transverse magnetic field retards the primary motion. This phenomenon has an excellent agreement with the physical fact that the Lorentz force generated in present flow model due to interaction of the transverse magnetic field and the fluid velocity acts as a resistive force to the fluid flow which serves to decelerate the flow. As such the magnetic field is an effective regulatory mechanism for the flow regime.
Further, it is worthwhile to mention that the effect of Hartmann number 
on primary velocity 
is negligible for large
. In other words the fluid motion far away from the plate is undisturbed due to imposition of the magnetic field.
Figure 8 indicates that an increase in the plate acceleration causes the primary flow to retard comprehendsively near the plate. That is the role of accelerating parameter 
on primary velocity field is almost similar to the role of Hartmann number
. However, in absence of acceleration a very small growth of primary velocity is noticed in a thin fluid region adjacent to the plate surface and afterwards it falls slowly and steadily to its free stream value as
.
Figures 9-12 demonstrate the behaviour of the secondary velocity field 
under the effects of rotational parameter
, Hartmann number
, Hall parameter 
and accelerating parameter 
respectively. From these figures, it is interesting to observe that the magnitude of 
increases from its zero value at the plate surface into a fluid region adjacent to
Figure 8. The variation of 
versus 
under the effect of a.
Figure 9. The behavior of ν under the effect of
, versus
.
Figure 10. The behavior of ν under the effect of M, versus
.
Figure 11. The behavior of ν under the effect of
, versus
.
Figure 12. The behavior of ν under the effect of
, versus
.
the plate and then it decreases to zero value (attainable at possibly large distances from the plate surface). This clearly agrees with the boundary conditions of the present flow problem. Figures 9 and 12 indicate that a rise in the values of rotational parameter 
and accelerating parameter 
causes a growth in the magnitude of secondary velocity. However, this effect seems to be negligible in the fluid region far away from the plate. Further, it is worthwhile to mention that for small and moderate values of Hartmann number 
and Hall parameter
, the behaviour of secondary velocity field is identical to its behaviour under the effects of 
and accelerating parameter 
whereas, for higher values of Hartmann number 
and Hall parameter 
the behaviour of secondary velocity 
takes a reverse trend.
It is inferred from Tables 1 and 2 that a rise in rotational parameter
, and Hall parameter 
results in a growth in the primary velocity
.
• 6. Conclusions
• The main flow velocity 
decreases with an imposition of magnetic field.
• An increase in accelerating parameter 
results in a remarkable growth in the main flow velocity 
in a thin layer adjacent to the plate.
• The drag force due to primary velocity 
rises due to rotation or imposition of the transverse magnetic field or the acceleration of the plate.
• The skin friction 
(drag force due to secondary velocity) shows a growth for increasing each of Hartmann number
, rotational parameter 
as well as accelerating parameter
.
7. Acknowledgements
The Authors are grateful to the CSIR-HRDG (India) for sponsoring this work through their Research-Grant-InAid. No. 25 (0209)/12/EMR-I.
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