Abstract: An exact solution of hydromagnetic, oscillatory flow of an incompressible, electrically conducting and viscous fluid in horizontal porous channel embedded with porous medium of time dependent permeability is obtained in the presence of hall current. A magnetic field of uniform strength is applied along an axis perpendicular to the plane of the plate about which the entire system rotates. The effects of various parameters on the exact solution so obtained are discussed with the help of graphs and tables in detail.

Mathematical Subject Classification: 76A05

Keywords: Hydromagnetic, oscillatory, rotating, porous medium, hall current.

Introduction:

Channels are frequently used in various applications in designing ventilating and heating of buildings, cooling electronic components, drying several types of agriculture products grain and food, and packed bed thermal storage. Convective flows in channels driven by temperature differences of boundary walls have been studied and reported, extensively in literature. On the other hand flows of fluid through porous medium are very important particularly in the fields of agriculture engineering for irrigation processes; in petroleum technology to study petroleum transport; in chemical engineering for filtration and purification processes. Abdussattar [1] and Ahmed [3] have studied free convection and mass transfer flow of a viscous fluid through porous medium. In most of these studied the permeability of the porous medium is assumed to be constant while the porosity of medium may not necessarily be constant because porous material containing the fluid is a non homogenous medium and there can be numerous in homogeneities present in a porous medium. Acharya et al [2] have analyzed free convection and mass transfer in steady flow through porous medium with constant suction in the presences of magnetic field. A number of investigation have been made by Raptis et al [9-11] into steady two –dimensional flow past a vertical walls of constant –permeability of porous medium. Notable among them are Attia and Kotb [4], Crammer and Pai[ 5], Ferraro and Plumpton [6], Shercliff [12] investigated a two-dimensional MHD flow between two porous parallel, insulated horizontal plates and the heat transfer through when the lower plate is kept stationary and upper plate is moving with uniform velocity. Singh et al [14] investigated a three –dimensional fluctuating flow through a porous medium when the permeability various both in time and space. Singh et al [13] have discussed hydromagnetic free convective and mass transfer flow of a viscous stratified fluid considering variation in permeability with direction. Further the flow of electrically conducting fluids in channel and pipes under the effect of transverse magnetic field occur in magnetohydrodynamic (MHD) generators, accelerators, pump and flow meters. In view of these and many others important applications of these types of flows a number of scholars have shown their interest.

There are other industrial applications of flows of electrically conducting fluids in the fields of geothermal system, nuclear reactor, filtration etc, where the conducting fluid flows through porous medium which rotates about an axis. In view of the importance of rotating flows a number of studies appeared in literatures. Mazumder [8] studied an oscillatory Ekman boundary layer flow bounded by horizontal plates one of which oscillate and other is at rest. Singh and Kumar [15] studied an oscillatory MHD flow through a porous medium bounded by rotating porous channel in the presences of hall current by taking into consideration the constant suction and permeability.

In this paper we present an investigation of the effect of magnetic field, hall current and rotation on a viscous incompressible and electrically conducting fluid in a porous horizontal channel filled with porous material by considering periodic suction and permeability.

Mathematical Analysis:

Consider an unsteady flow of an electrically conducting, viscous, incompressible fluid through a porous medium bounded between two insulated infinite parallel plates’ distances d apart. Choose the origin at the lower plate lying inplane and -axis parallel to the direction of motion of the upper plate. A strong magnetic field of uniform strength is applied along -axis taken perpendicular to the planes of the plates. The entire system rotates with angular velocity about -axis. The magnetic Reynolds number is considered to be small so that the induced magnetic field is neglected. All physical quantities depend onand for this problem of fully developed laminar flow.

The equation of continuity gives on integration at whereand solenoidal relation for the magnetic field gives (constant) everywhere in the flow field. The physical configuration of the problem is shown in Figure 1.

Figure 1: The physical configuration of the problem

Basic Equation:

The equation of conservation of electric charge gives= constant. This constant is zero i.e. at the plates which are electrically non-conducting. Taking Hall current into account the generalized Ohm’s law is of the form

(1)

Where is the velocity vector , is the magnetic field,is the current density, E is the electric field, is the electric conductivity, is the magnetic permeability, is the cyclotron frequency and is the electron collision time.

For very large magnetic field, the and components of Ohm’s law (1), which include Hall current, are

(2)

(3)

Since the external electric field arising due to polarization of charges is negligible.

Hence Therefore, solving for and , we get

and (4)

Thus within the frame work of these assumptions and making use of equation(4),the flow in the presence of Hall current by the following equations:

(5)

(6)

where is the fluid density is time is the permeability of the porous medium. , is the Hall parameter. The permeability is taken in the form .

The boundary conditions of the problem are

(7)

The flow in presences of Hall current is governed by the following equations

(8)

(9)

Introducing the following non dimensional quantities

is the frequcency of oscillations is the injection and suction parameter, is the Hartmann number, is the rotation parameter , is the permeability of the porous medium.

Eliminating the modified pressure gradient under the usual boundary layer approximation, equation (8) and (9) become

(10)

(11)

Using the non dimensional quantities the equation (10) and (11) become

(12)

(13)

The corresponding transformed boundary condition becomes

(14)

Equation (12) and (13) can be combined into a single equation by using the complex velocity of the form

(15)

(16)

The boundary conditions in complex notation is given by

(17)

Method of Solution:

(18)

Substituting (18) into (16) and comparing harmonic and non harmonic part we have

(19)

(20)

With corresponding transformed boundary condition

(21)

The solution of the equation (19) and (20) under the boundary condition (21) are obtained as:

(22)

(23)

Results and Discussion:

Now for the resultant velocities and the shear stresses of the steady and unsteady flow, we write

(24)

(25)

The solution corresponds to the steady part which as the primary and as the secondary velocity components. The amplitude and the phase differences due to these primary and secondary velocities for the steady flow are given by

(26)

The resultant velocity or amplitude and the phase differences of the unsteady flow are given by

(27)

The resultant velocities and and the phase angle, for the steady and unsteady part of the flow are respectively shown graphically in figures 2 to 5. It is observed from figures 2 and 4 that and both increase rapidly from zero near the stationary plate and then approach unity in the form damped oscillations. Figure 2 and 4 for the steady and unsteady resultant velocities and respectively shown. The resultant velocities decrease slightly with the increase value of hall parameter and it increase with the increase value of suction parameter , permeability and the rotation parameter, the Hartman number. Unsteady resultant velocities decrease with the increase value of frequency of oscillation and increase with the increase value of hall parameter , suction parameter , permeability , the rotation parameter and the Hartmann number .

It is also found from figure 3 and 5 that the phase differences increase quickly from zero at the stationary plate and then approach zero again at the moving plate in the form of damped oscillations. Particularly for large values of rotations a phase lag is also observed for both steady and unsteady phase angles and . The phase differences increase with the increasing value of hall parameter and permeability and the phase differences decreases with the increasing value of Hartmann number , suction parameter and rotation parameter . The phase differences is increase in the boundary and decrease at the middle of the channel and then increase at the boundary of the channel. The phase differences increase with the increasing of hall parameter , rotation parameter , suction parameter , frequcency of oscillation and permeability in the beginning and in the middle it decrease and than again a phase lag occur and than at the boundary it again increase and with the decreasing value of A and B the phase differences decrease in the starting and in the middle of the channel it increase and then it again decrease and than a phase lag occur and then it again increase at the boundary of the channel.

Curve

2 1 10 0.2 0.2 I

4 1 10 0.2 0.2 II

2 3 10 0.2 0.2 III

2 1 20 0.2 0.2 IV

2 1 10 1 0.2 V

2 1 10 0.2 1 VI

�

�

�

Figure 2: The Resultant velocity due to and .

Curve

2 1 10 0.2 0.2 I

4 1 10 0.2 0.2 II

2 3 10 0.2 0.2 III

2 1 20 0.2 0.2 IV

2 1 10 1 0.2 V

2 1 10 0.2 1 VI

�

�

�

Curve

2 1 10 0.2 0.2 5 0.5 0.5 I

4 1 10 0.2 0.2 5 0.5 0.5 II

2 3 10 0.2 0.2 5 0.5 0.5 III

2 1 20 0.2 0.2 5 0.5 0.5 IV

2 1 10 1 0.2 5 0.5 0.5 V

2 1 10 0.2 1 5 0.5 0.5 VI

2 1 10 0.2 0.2 15 0.5 0.5 VII

2 1 10 0.2 0.2 5 0 0.5 VIII

2 1 10 0.2 0.2 5 0.5 0 IX

� Figure 3: Phase angle due to and .

�

�

Curve

2 1 10 0.2 0.2 5 0.5 0.5 I

4 1 10 0.2 0.2 5 0.5 0.5 II

2 3 10 0.2 0.2 5 0.5 0.5 III

2 1 20 0.2 0.2 5 0.5 0.5 IV

2 1 10 1 0.2 5 0.5 0.5 V

2 1 10 0.2 1 5 0.5 0.5 VI

2 1 10 0.2 0.2 15 0.5 0.5 VII

2 1 10 0.2 0.2 5 0 0.5 VIII

2 1 10 0.2 0.2 5 0.5 0 IX

�Figure 4: The Resultant velocity due to and .

�

�

Figure 5: Phase angle due to and

For the steady flow the amplitude and the phase differences of shear stress at the stationary plate (can be obtained as

(28)

(29)

Here and are respectively, the shear stress at the stationary plate due to the primary and the secondary velocities components. The numerical values for the resultant shear stress and the phase angle are listed in table-1. This table shows that and phase angle goes on increasing with the increasing rotations of the channel.

Table1: Amplitude and the phase angle due to and

m

K

2

1

10

0.2

0.2

4.8896

0.78540

4

1

10

0.2

0.2

5.6420

0.55842

2

3

10

0.2

0.2

4.7618

0.64822

2

1

20

0.2

0.2

6.6017

0.69284

2

1

10

1

0.2

5.2293

0.57062

2

1

10

0.2

1

4.7958

0.70385

For the unsteady part of the flow the amplitude and the phase differences of the shear stresses at the stationary plate ( can be obtained as, for as

(30)

(31)

From figure 6 it is conclude that decrease slightly with increase value of hall current and permeability of the porous medium , and it increase with increase of Hartman number ,rotation parameter ,suction parameter and . From figure 7 it is conclude that the phase differences increases with the increasing value of permeability of the porous medium , hall parameter and rotation parameter . The phase differences decreases with the increasing value of suction parameter and Hartman number.

Curve

2 1 10 0.2 0.2 0.5 0.5 I

4 1 10 0.2 0.2 0.5 0.5 II

2 3 10 0.2 0.2 0.5 0.5 III

2 1 20 0.2 0.2 0.5 0.5 IV

2 1 10 1 0.2 0.5 0.5 V

2 1 10 0.2 1 0.5 0.5 VI

2 1 10 0.2 0.2 0 0.5 VII

2 1 10 0.2 0.2 0.5 0 VIII

�

�

�

Figure 6: The amplitude of unsteady shear stress for .

�

Curve

2 1 10 0.2 0.2 0.5 0.5 I

4 1 10 0.2 0.2 0.5 0.5 II

2 3 10 0.2 0.2 0.5 0.5 III

2 1 20 0.2 0.2 0.5 0.5 IV

2 1 10 1 0.2 0.5 0.5 V

2 1 10 0.2 1 0.5 0.5 VI

2 1 10 0.2 0.2 0 0.5 VII

2 1 10 0.2 0.2 0.5 0 VIII

�

�

Figure 7: Phase angle of unsteady shear stress at .

Concluding Remarks:

The resultant velocities and increases with the increasing value of Hartmann number, permeability parameter , and rotation parameter. The resultant velocities decrease with the increasing value of frequency of oscillation. The resultant velocities and both increases rapidly from zero near the stationary plate and then approach unity in the form of damped oscillation.

References

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M. Acharya, G.C. Dabh and L.P. Singh, Magnetic field effects on the free convection and mass transfer flow through porous medium with constant suction and constant heat flux, Ind. J. pure appl. Math.,31,1-18, 2000.

N. Ahmed and D. Sarma,Three dimensional free convective flow and heat transfer through a porous medium, Ind. J. pure appl. Math., 28 (10) ,1345-1353, 1997.

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K.P. Crammer and S.I. Pai, Magneto fluid dynamical for engineer and applied Physicist, Mc. Graw Hill book Co. New York, 1973.

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T.A. Shercliff, A text-book of Magnetohydrodynamics Pergamon Press London, 1965.

N.P. Singh , Ajay Kumar Singh, M.K. Yadav and Atul Kumar Singh ,Hydromagnetic free convection and mass transfer flow of a viscous stratified fluid, J. Energy Heat Mass transfer, 21 11-115, 1999.

K.D. Singh, Rakesh Sharma and Khem Chand ,Three –dimensional fluctuating flow and heat transfer through a porous medium with variable permeability, ZAMM, 80, 473-480, 2000.

K. D. Singh and Rakesh Kumar ,An exact solution of an oscillatory MHD flow through a porous medium bounded by rotating channel in the presences of hall current, In .J. of Appl. and Mech, 6 (13), 28-40, 2009.

V.M. Soundalgekar, S.N. Ray and U.N.Das, MHD flow past an infinite vertical oscillating plate with mass transfer and heat flux, Proc. Math. Soci. 11, 95-98,1995.

Appendix

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