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An Oscillatory MHD Visco-Elastic Fluid Flow through a Porous Medium Bounded by Rotating Porous Channel in the presence of Hall Current

Utpal Jyoti Das

Department of Mathematics, Rajiv Gandhi University, Doimukh-791112, Arunachal Pradesh, INDIA.

Corresponding Address:

Research Article

Abstract: A theoretical solution for an oscillatory flow of an electrically conducting, viscous, incompressible visco-elastic fluid is obtained without neglecting Hall current. The magnetohydrodynamic (MHD) flow is bounded by two infinite horizontal plates filled with porous medium. The fluid is injected with constant velocity through the lower stationary plate and the upper plate is subjected to the same constant suction velocity. The effects of Hall current and Hartmann numbers on velocity profile and shear stress for different values of the visco-elastic parameter with the combination of the other flow parameters are illustrated graphically and physical aspects of the problem are discussed.

2000 Mathematics Subject Classification: 76A05, 76A10.

Keywords: visco-elastic, oscillatory, porous channel, porous medium, MHD, Hall current.

1. Introduction

The study of flows through porous medium have stimulated considerable interest due its applications in the fields of agricultural engineering for irrigation processes; in petroleum technology to study petroleum transport; in chemical engineering for filtration and purification processes. Raptis [1] investigated the unsteady two-dimensional flow through a porous medium bounded by an infinite porous plate subjected to a constant suction and variable temperature. Further, Raptis and Perdikis [2] studied the problem of free convective flow through a porous medium bounded by a vertical porous plate with constant suction when the free stream velocity oscillates in time about a constant mean value. Singh et al. [3] investigated the problem of heat transfer in three dimensional flow through a porous medium with periodic permeability. Singh and Verma [4] studied further the three dimensional oscillatory flow through a porous medium where the free stream velocity oscillates in time about a non-zero constant mean. Singh et al. [5] studied the effect of permeability variation on the heat transfer and three dimensional flow through a highly porous medium bounded by an infinite porous plate with constant suction. Attia [6] studied the Hall current effect on the velocity and temperature fields of an unsteady Hartmann number. Singh and Mathew [7] studied the injection/suction effect on a hydromagnetic oscillatory flow in a horizontal porous channel in a rotating system. Choudhury and Das [8] extended the problem studied by Singh and Mathew [7] to the case of visco-elastic fluid. Singh and Kumar [9] investigated the problem of on oscillatory MHD flow through a porous medium bounded by rotating porous channel in the presence of Hall current. The aim of the present investigation is to study the effect of the Hall current on the visco-elastic fluid flow when the porous horizontal channel filled with a porous medium is rotating about an axis normal to the planes of the plates.

2.  Mathematical Analysis

Consider an unsteady oscillatory flow of an electrically conducting visco-elastic, incompressible second order fluid through a porous medium bounded between two insulated infinite parallel porous plates distance d apart. The fluid is injected with constant velocitythrough the lower stationary plate and is being sucked with the same velocity through the upper plate which is oscillating in its own plane with a velocity  about a non-zero constant mean velocity   . A coordinate system is taken with the origin at the lower stationary plate lying in plane and -axis parallel to the direction of motion of the upper plate. The  -axis taken perpendicular to the planes of the plates, is the axis of the rotation about which the entire system is rotating with constant angular velocity. A strong magnetic field of uniform strength  is applied along - axis. The magnetic Reynolds number is considered to be small so that the induced magnetic field is neglected. Since the plates are infinite in extent, all the physical quantities except the pressure depend only on and for this fully developed laminar flow.

The constitutive equation for the incompressible second order fluid is of the form

(1)

where   is the stress tensor,    (n=1, 2) are the kinematic Rivlin-Ericksen tensors;  are the material coefficients describing the viscosity, elasticity and cross-viscosity respectively. The material coefficients  are taken constants with  and    as positive and as negative (Coleman and Markovitz [10]). The equation (1) was derived by Coleman and Noll [11] from that of the simple fluids by assuming that stress is more sensitive to the recent deformation than to the deformation that occurred in the distant past.

Denoting the velocity components in the directions, respectively, the continuity equation  gives on integration  and solenoidal relation for the magnetic field  gives (constant) everywhere in the flow field. The physical configuration of the problem is shown in Figure 1.

Figure1: Physical configuration of the problem.

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 (Cowling [12]) is of the form

(2)

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

For the  and  components of Ohm’s law (2), for large magnetic field, which include Hall current, are

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

Hence  Therefore, solving for  and  we get

and

Under the above assumptions, the governing equations for the flow, in presence of Hall current, are as follows:                                                                                             (3)

(4)

where  are the kinematic viscosity,  is the time, is the density, is the pressure and is the permeability of the porous medium,  is the Hall parameter.

The boundary conditions for the problem are

at  ,

at                                                                            (5)

where is the frequency of oscillations, is the mean velocity and  is a very small positive constant.

Eliminating the pressure gradient, under the usual boundary layer approximations, equations (3) and (4) reduces to

(6)

(7)

Introducing the following non-dimensional quantities

the rotation parameter, , the frequency parameter,  is the injection/suction parameter,  is the permeability parameter,  is the Hartmann number, and  into the equations (6) and (7), we get

(8)

(9)

subject to boundary conditions

(10)

Equations (8) and (9) can now be combined into a single equation, by introducing the complex function ,  as

(11)

subject to the boundary conditions

at

at                                (12)

where , the visco-elastic parameter and

3. Solution of the problem

In order to solve equation (11) subject to the boundary conditions (12), we look for a solution of the form

(13)

Substituting (13) into the equations (11) and (12) and comparing the harmonic   and non-harmonic terms, we get

(14)

(15)

(16)

where primes denote differentiation with respect to.

The corresponding transformed boundary conditions are

at         ,

at                                         (17)

To solve the equations (14) to (16) under boundary conditions (17), we consider very small value of non-Newtonian parameter, and substituting

(18)

into equations (14) to (16) and boundary conditions (18) up to first order of  and  equating the coefficients of like powers of , we obtain the following sets of ordinary differential equations and corresponding boundary conditions:

(19)

with

(20)

(21)

with

(22)

(23)

with

(24)

Solving the equations (19), (21), (23) under the boundary conditions (20), (22), (24) respectively and substituting these values in (18), we get the solutions for velocity. The solutions and constants of the differential equations are obtained but not presented here for the sake of brevity.

4. Results and discussion

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

(25)

and

(26)

The steady part consists of as the primary and as the secondary velocity components. The amplitude and phase difference due to these primary and secondary velocities for the steady flow are given by

(27)

Figures 2 and 3 depict the resultant velocity  for the steady and unsteady part of the flow against  to observe the visco-elastic effects for the various values of Hall current  and Hartmann number. It is observed from figures 2 and 3 that both  and increase rapidly

Figure 2: Variation of resultant velocity  against  with

Figure 3: Variation of resultant velocity  against  with

from zero near the stationary plate for Newtonian as well as non-Newtonian  cases. It is evident from the Figures 2 and 3 that both  and increase with the increase of Hartmann number for both Newtonian and non-Newtonian cases. But these results decrease with the increase with the increase of Hall current

The amplitude and the phase difference of the shear stress at the stationary plate for the steady flow can be obtained as,

where

,      (28)

,                            (29)

Here are, respectively, the shear stresses at the stationary plate due to the primary and the secondary velocity components.

Table 1: Values of  andfor various values of  with .

 Case α I 1 2 0 2.7482 0.7864 -0.05 3.4562 0.6842 -0.10 4.2684 0.5678 II 2 1 0 2.5684 0.8997 -0.05 3.2772 0.8224 -0.10 4.1282 0.7826 III 4 1 0 4.1226 0.6422 -0.05 6.2254 0.5864 -0.10 8.7265 0.5216

Table 1 exhibits the effects of the visco-elastic parameter on the amplitude and the phase difference of the shear stress at the stationary plate  for the steady part of the flow with the combination of the other flow parameters   and  It is observed from the table 1 that the values of increase but decrease with the increasing values of the non-Newtonian  parameter  This table shows that increase with the increase of Hartmann number  but decrease with the increase of Hall current  for both Newtonian as well as non-Newtonian cases.

The solutions of  and  together give the unsteady part of the flow. The unsteady primary and secondary velocity components and , respectively, for the fluctuating flow can be obtained as

(30)

(31)

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

(32)

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

(33)

which gives

(34)

Table 2: Values of  andfor various values of  with .

 Case α I 1 2 0 1.6412 0.5262 -0.05 3.8926 0.4721 -0.10 6.2542 0.3946 II 2 1 0 1.5864 0.5987 -0.05 3.1225 0.5678 -0.10 5.8462 0.4996 III 4 1 0 2.5778 0.4884 -0.05 7.8892 0.4263 -0.10 10.7884 0.3692

The amplitude  and phase difference  of the unsteady shear stresses at the stationary plate have been listed in Table 2. It is observed from the Table 2 that the values of  decrease with the increase of Hall current  But   increase with the increase of Hartmann number for both Newtonian as well as non-Newtonian cases. The phase difference increase with the increase of , but decrease with the increase of  for both Newtonian and non-Newtonian cases.

References

1. A.A. Raptis, Unsteady free convection through  porous medium, Int. J.Engg. Sci., 21, 345-348, 1983.
2. A.A. Raptis and C.P. Perkidis, Oscillatory flow through a porous medium by the presence of free convective flow, Int. J. Engg. Sci., 23, 51-55, 1985.
3. K.D. Singh, K. Chand and G.N. Verma, Heat transfer in three dimensional flow through a porous medium with periodic permeability, Z. Angew. Math. Mech., 75(12), 950-952, 1995.
4. K.D. Singh and G.N. Verma, Three dimensional oscillatory flow through a  porous medium, Z.Angew. Math. Mech., 75(8), 599-604, 1995.
5. K.D. Singh, R. Sharma and K. Chand, Three dimensional fluctuating flow and heat transfer through porous medium with variable permeability, Z.Angew Math. Mech., 80(7), 473-480, 2000.
6. H.A. Attia, Time varying hydromagnetic Couette flow with heat transfer of a dusty fluid in the presence of uniform suction and injection considering the Hall effect. Turkish J. Eng. Env. Sci., 30, 285-297, 2006.
7. K.D. Singh and A. Mathew, An exact solution of an oscillatory flow through porous  medium in a rotating porous channel with injection/suction, J. Rajasthan Acad. Phy. Sci, 7(1), 1-14,  2008.
8. R. Choudhury and U.J. Das, An oscillatory visco-elastic flow through porous medium in a rotating porous channel with injection/suction, Far East Journal of Applied Mathematics, 38(1), 1-18, 2010.
9. K.D. Singh and R. Kumar,  An exact solution of an oscillatory mhd flow through a porous medium bounded by rotating porous channel in the presence of hall current, Int. J. of Appl. Math and Mech. 6 (13): 28-40, 2010.
10. B.D. Coleman and H. Markovitz, Incompressible second order fluids, Adv. Appl. Mech., 8, 69-101, 1964.
11. B.D. Coleman and W. Noll, An approximation theorem for functionals with applications in continuum mechanics, Archs Ration, Mech. Analysis 6, 355, 1960.
12. T.C. Cowling, Magnetohydrodynamics. Interscience, New York, 1957.

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