Magnetohydrodynamic Oscillatory Flow in a Planer Porous Channel with Suction and Injection

Magnetohydrodynamic Oscillatory Flow in a Planer Porous Channel with Suction and Injection

Sahin Ahmed1 and Hamida Khatun2

1Heat Transfer and Fluid Mechanics Research, Department of Mathematics and Computing, Rajiv Gandhi University, Rono Hills, Itanagar 791112, Arunachal Pradesh, India.

2Department of Mathematics, South Salmara College, Dhubri 783127, Assam.

u(y, t)

B0 y = a , u = 0

y

y

v 2

a

X

y = a , u = 0 2

Figure 1: Physical Model

Abstract

The theoretical analysis of an Oscillatory MHD flow through porous medium bounded by the horizontal parallel porous plates is carried out. Both the stationary plates are subjected to the same constant injection / suction velocities. A uniform magnetic field is applied normal to the planes of the plates. A closed form analytical solution is obtained and the affects of different flow parameters on velocity field and skin-friction are discussed with the help of graphs in detail. It is found that, when the Darcy number (Da) or

suction/injection parameter (A) increased, the fluid

velocity profiles were decreased. An increase in Da or A is found to escalate the shear stress (r). Possible applications of the present study include laminar aerodynamics, materials processing and thermo-fluid dynamics.

1. Introduction

   Magnetohydrodynamics deals with dynamics of an electrically conducting fluid, which interacts with a magnetic field. The study of MHD flow, through and across porous media, is of great theoretical interest because it has been applied to a variety of geophysical and astrophysical phenomena. Practical interest of such study includes applications in electromagnetic lubrication, boundary cooling, bio-physical systems and in many branches of engineering and science. In fact, flows of fluids through porous media have attracted the attention of a number of scholars because of their possible applications in many branches of science and technology. In fact a porous material containing the fluid is a non-homogeneous medium but it may be possible to treat it as a homogeneous one, for the sake of analysis, by taking its dynamical properties to be equal to the averages of the original non-homogeneous continuum. Thus a complicated problem of the flow through a porous medium gets reduced to the flow problem of a homogeneous fluid with some additional resistance. The hydrodynamic channel flow is a classical problem for which exact solution can be obtained Schllicting [1]. Eckert [2] obtained the exact solution of Navier-Stokes equations for the flow between two parallel porous plates with constant injection/suction.

   In view of numerous important engineering and geophysical applications of the channel flows through porous medium, for example in the fields of chemical engineering for filtration and purification processes, in the fields of agriculture engineering for channel irrigation and to study the underground water resources, in petroleum technology to study the movement of natural gas,

   oil and water through the oil channel/reservoirs. A series of investigations have been made by different scholars like Ahmadi and Manvi [3]; Raptis [4]; Raptis and Perdikis [5]; Singh and Garg

   [6] and Singh and Sharma [7] where the porous medium is either bounded by a channel or by a plane surface. On the other hand, in view of the increasing technical applications using magnetohydrodynamic (MHD) effect, it is desirable to extend many of the available hydrodynamic solution to include the effects of magnetic field for those cases where the viscous fluid is electrically conducting. The effect of a transverse magnetic field on free convective flows of an electrically conducting viscous fluid has been discussed in recent and past years by several authors, notably by Gupta [8], Soundalgekar [9], Mishra and Mudili [10], Mahendra [11], Sarojamma and Krishna [12] and Singh and Garg [13]. Such types of flows have wide range of applications in aeronautics, fluid fuel nuclear reactors and chemical engineering. The various applications of MHD flows in technological fields have been complied by Moreau [14]. Recently Makinde and Mhone [15] investigated the effects of radiative heat and magnetic field on the unsteady flow of a fluid through a channel filled with saturated porous media. This problem is further extended by Mehmood and Ali [16] by considering the fluid slip conditions at the stationary plate. Major mistakes found in both the above [15], [16] studies had been marked by Singh and Garg [17]. Kuznetsov [18] presented an analytical solution to the flow and heat transfer in Couette flow through a rigid saturated porous medium where the fluid flow occurs due to a moving wall and it is described by the Brinkman-Forchheimer-extended Darcy equation. The problem of free convection heat transfer flow through a porous medium bounded by a wavy wall and a vertical wall is studied by Ahmed [19]. A three-dimensional Couette flow through a porous medium with heat transfer has also been investigated by Ahmed [20]. Ahmed and Zueco [21] investigate the effects of Hall current, magnetic field, rotation of the channel, and suction- injection on the oscillatory free convective MHD flow in a rotating vertical porous channel when the entire system rotates about an axis normal to the channel plates and a strong magnetic field of uniform strength is applied along the axis of rotation. Ahmed [22] investigated the effect of periodic heat transfer on unsteady MHD mixed convection flow past a vertical porous flat plate with constant suction and heat sink when the free stream velocity oscillates in about a non-zero

   D

   D

   constant mean. Very recently, Ahmed and Kalita

   au au ap 1 a2u 1 2 1

   [23] investigated the effects of thermal radiation and magnetohydrodynamic forces on transient flow

   at + ay = ax + A ay2 A M

   + u (5)

   a

   over a hot vertical plate in a Darcian regime.

   The aim of the present paper is to study the combined effects of injection/suction and magnetic field on the oscillatory flow through saturated porous medium bounded by two parallel porous plates.

2. Mathematical Analysis

   Consider a two-dimensional flow of an electrically conducting, viscous incompressible, electrically conducting and Newtonian fluid through saturated

   where p is the density, t is time, u is the axial velocity, v is the transverse velocity, v is the kinematic viscosity, p is the pressure, K is constant of permeability of the porous medium, B0 uniform

   magnetic field, a is electrical conductivity, A is the

   injection/suction parameter, Da is the Darcy number and M is the Hartmann number.

   The transformed boundary conditions become

   u = 0 , v = V , at y = 1

   2

   porous medium filled in an infinite horizontal

   channel. The plates of the channel are distance a apart. A coordinate system is chosen with x-axis lies along the centerline of the channel and y- axis is normal to the planes of the plates. Both the lower

   u = 0 , v = V , at y = 1

   2

3. Method of solution

   (6)

   and the upper stationary porous plates of the channel are subjected to the same constant injection and suction velocity. A homogeneous magnetic field B0 is applied normal to the planes of the plates as shown in Figure-1. The flow becomes oscillatory due to the time dependence of the

   pressure. All the physical quantities are independent of x for the problem of fully developed laminar flow. Under all these assumptions the flow is depicted mathematically as:

   Conservation ofMass:

   In order to solve equation (5) under the boundary conditions (6), let us assume the solution of the following form

   0

   0

   u(y, t) = u (y)eiwt , ap = eit (7)

   ax

   where is constant and w is the frequency of oscillations.

   Substituting expressions (7) into equations (5), we obtain

   u Au m2u0 = A (8)

   0 0

   av = 0 (1)

   ay

   Conservation of Momentum:

   au au 1 ap a2u aB2 v

   where m = M2 + J + iwA .

   Da

   The corresponding transformed boundary conditions become

   0

   0

   u = 0 , at y = 1

   + v = + v 0 u (2) 2

   at ay

   p ax

   ay2 p K

   u = 0 , at y = 1

   0

   0

   2

   (9)

   The boundary conditions of the problem are

   u = 0 , v = V , at y = a

   2 a

   u = 0 , v = V , at y = 2 (3)

   On introducing the following non-dimensional quantities,

   Equation (8) is solved under boundary conditions

   1. and the solution for the fluid velocity is obtained as under:

      1)

      1)

      ( 2

      ( 2

      u(y, t) = A 1 cos (A1y) eiwt (10) m2 cos A

      x y u tv p

      where A

      = 1 A + A2 + 4m2

      x = a , y = a , u = a , t = a , p = pV2 ,

      1 2 2

      A = Va , D = KV , M = aB a (4)

      a 0

      a 0

      v va µ

      Knowing the fluid velocity, the shear stress at the

      lower plate is given by:

      into equations (1), (2), we get

      T = µ au = ;\ tan (11)

      ay y=- /2 m2 2 0.7

4. Results and Discussion

   The analytical solutions obtained in equation

   0.6

   0.5

   1.0 5

   0.5 5

   0.1 5

   1. and (11) for the velocity and skin friction have

   0.1 10

   u

   u

   been calculated numerically to have a physical

   insight into the problem. The effects of variations of different parameters like the Darcy number(Da), injection / suction parameter(;\), Hartmann number(M), and frequency of oscillations (w) on the velocity field (u) and skin friction (T) variations are presented graphically in Figures 2 to 6 below.

   Figure 2 shows the collective effects of

   0.4

   0.3

   0.2

   0.1

   0

   0.1 15

   Hartmann number (M) and Darcy number (Da) on the flow velocity (u) for different values of w = 10 , ;\ = 0.2 , = 10. The influence of M and Da on u profiles is therefore expected to be strong. This is indeed the case as seen in fig. 2; for constant Da(= 0.3), with a rise in M, from 1, 5 to 10 there is a strong reduction in velocity across the region y [0.5, 0.5]. The flow is therefore decelerated with increasing Hartmann number owing to the corresponding increase in the Lorentizian hydromagnetic drag force. Moreover, with constant M = 1.0 value, as Da increases from

   1. through 0.5 to 1.0, there is a distinct escalation in velocity across the region y [0.5, 0.5]. A velocity peak arises in the middle of the channel for all profiles. No back flow is sustained throughout the channel.

      -0.5 -0.3 -0.1 0.1 0.3 0.5

      y

      Fig 3: Velocity distributions for ;\ and w at t = 0

      Figure 3 presents the flow velocity profiles for various injection/suction parameter (;\) and frequency of oscillations w for different values of M = 5 , Da = 0.5 , = 10. With constant w = 5, an increase in ;\ from 0.1 through 0.5 to 1.0, the flow velocity is accelerated throughout the channel and attains its maximum velocity in the middle of the channel. Moreover with constant ;\ = 0.1, the flow velocity is decelerated throughout the channel when the frequency of oscillations rises from 5.0 through 10 to 15. No back flow is sustained throughout the channel.

      0.6

      0.5

      M Da

      1 1.0

      1 0.5

      1 0.3

      0.5

      0.4

      0.3

      M

      0.2 1.0

      1.0 1.0

      0.2 5.0

      0.4 5 0.3 0.5 1.0

      u 0.3

      0.2

      10 0.3

      0.2

      0.1

      0.2 10.0

      0.1

      0

      -0.5 -0.3 -0.1 0.1 0.3 0.5

      y

      0

      -0.1

      -0.2

      0 5 10 15 20

      Fig 2: Velocity distributions for M and Da at t = 0

      Fig 4: Variations of Skin friction for M and ;\

      versus w

      Figure 4 depicts the influence of the applied magnetic field (M) and injection/suction parameter (;\) on the non-dimensional coefficient of skin friction r at the lower plate for different values of Da = 0.5 , = 10. The imposition of the magnetic field causes to decrease r. It is noticed that, the influence of injection/suction on r is significantly elevated. Velocities are decreased and the shear stress at the wall will therefore be depressed with a rise in Hartmann number(M). For the higher values of w (> 4.8), significant flow reversal is sustained with maximum magnetohydrodynamic forces, M = 5 and 10 i.e. shear stresses become negative. No flow reversal however arises for small frequency of oscillations. The back flow effect is still present for M = 5 (magnetic body force is five times the viscous hydrodynamic force), but is stifled somewhat and the inception of backflow is further delayed. However for M = 1 and ;\ = 0.2, 0.5 and 1.0, all backflow is eliminated entirely from the regime for all frequency of oscillations and only positive shear stresses arise at the plate. Generally with frequency of oscillations, shear stresses are found to reduce

      i.e. the flow is retarded.

      0.4

      with rising Darcy number from 0.1 through 0.5, 0.7 to 1.0. Significantly, shear stress at Da = 1.0 is more fluctuates than the others.

5. Conclusions

   A mathematical model has been presented for the hydromagnetic unsteady boundary layer flow from a horizontal channel bounded by two parallel plates filled with saturated porous medium with transverse magnetic field effects, subject to a constant suction/injection velocity. Analytical solution for the non-dimensional momentum equation subject to transformed boundary conditions has been obtained. The flow has been shown to be accelerated with increasing suction/injection parameter, but reduced with Magnetic field. Increasing Hartmann number also decreases the shear stresses. A positive increase in Da strongly accelerates the flow. The study has important applications in materials processing and nuclear heat transfer control, as well as MHD energy generators. Currently the authors are exploring extensions of this work to study rheological (non-Newtonian) working fluid effects, the results of which will be reported in the near future.

6. References

0.3

0.2

0.1

0

-0.1

Da = 0.7

Da = 0.5

Da = 1.0

Da = 0.1

0 5 10 15 20

1. H. Schllicting, Boundary Layer Theory, McGraw- Hill, New York, 1979.

2. E. R. G., Eckert, Heat and Mass Transfer, McGraw Hill New-York, 1958.

3. G. Ahmadi, and R. Manvi, Equation of Motion for Viscous Flow through a Rigid Porous Medium, Indian J. Tech. 9, pp. 441-444, 1971.

4. A. Raptis, Unsteady Free Convection Flow through Porous Medium, Int. J. Enging. Sci., 21, pp. 345-348, 1983.

5. A. Raptis, and C. P. Perdikis, Oscillatory Flow through a Porous Medium by the Presence of Free Convective Flow, Int. J. Enging. Sci., 23, pp. 51-55, 1985.

6. K. D. Singh, and B. P. Garg, Exact Solution of an Oscillatory Free Convective MHD Flow in a Rotating Porous Channel with Radiative Heat, Proc. Nat. Acad. Sci., 80(A), pp. 81-89, 2010.

   Fig 5: Variations of Skin friction for Da versus w

   Figure 5 shows the distribution of shear stress at the lower plate for various Darcy numbers over frequency of oscillations for different values of M = 3 , ;\ = 0.5 , = 10 = 10. Again it is seen that the shear stresses are reduced substantially at the lower plate throughout the channel. For all w > 4.8, flow reversal is observed for small Darcy numbers (Da = 0.1 and 0.5), and therefore, back flow is sustained throughou the regime. Shear stresses are significantly boosted

7. K. D. Singh, and R. Sharma, Three Dimensional Couette Flow through a Porous Medium with Heat Transfer, Indian J. Pure and Appl. Math., 32(12), pp. 1819-1829, 2001.

8. A. S. Gupta, Combined Free and Forced Convection Effects on the Magnetohydrodynamic Flow through a Channel, ZAMP, 20, pp. 506-513, 1969.

9. V. M. Soundalgekar, Free Convection Effects on Steady MHD Flow Past a Vertical Porous Plate, J. Fluid Mechanics, 66, pp. 541-551, 1974.

10. S. P. Mishra, and J. C. Mudili, Combined Free and Forced Convection Effects on the

    Magnetohydrodynamic Flow through a Porous Channel,

    Proc. Ind. Acad. Sci., 84A, pp. 257-272, 1976.

11. M. Mahendra, Combined Effects of Free and Forced Convection on Magnetohydrodynamic Flow in a Rotating Channel, Proc. Indian Acad. Sci., 84, pp. 383- 401, 1977.

12. G. Sarojamma, and D. V. Krishna, Transient Hydromagnetic Convective Flow in a Rotating Channel with Porous Boundaries, Acta Mech., 40, pp. 277-288, 1981.

13. K. D. Singh, and B. P. Garg, Radiation Effects on Unsteady MHD Free Convective Flow through Porous Medium Past a Vertical Porous Plate, Proc. Indian Natn. Sci. Acad., 75(1), pp. 41-48, 2009.

14. R. Moreau, Magnetohydrodynamics, Kluwer Academic Publishers, Dordrecht, 1990.

15. O. D. Makinde, and P. Y. Mhone, Heat Transfer to MHD Oscillatory Flow in a Channel Filled with Porous Medium, Rom. Journ. Phys., 50, pp. 931-938, 2005.

16. A. Mehmood, and A. Ali, The Effect of Slip Condition on Unsteady MHD Oscillatory Flow of a Viscous Fluid in a Planer Channel, Rom. Journ. Phys., 52(1-2), pp. 85-91, 2007.

17. K. D. Singh, and B. P. Garg, Radiative Heat Transfer in MHD Oscillatory Flow through Porous Medium Bounded by Two Vertical Porous Plates, Bull. Cal. Math. Soc., 102 (2), pp. 129-138, 2010.

18. A. V. Kuznetsov, Analytical investigation of heat transfer in Couette flow through a porous medium utilizing the Brinkman-Forchheimer-extended Darcy model, Acta Mechanica, 129, pp. 13-24, 1998.

19. S. Ahmed, Free convective flow in a vertical channel through a porous medium with heat transfer, International Journal of Applied Mathematics, 21 (4), pp. 671-684, 2008.

20. S. Ahmed, Three-dimensional Channel flow through a porous medium, Bulletin of Calcutta Mathematical Society, 101 (5), pp. 503-514, 2009.

21. S. Ahmed, and J. Zueco, Modeling of Heat and Mass Transfer in a Rotating Vertical Porous Channel with Hall Current, Chemical Engineering Communications, 198, pp. 12941308, 2011.

22. S. Ahmed, Free and Forced Convective MHD Oscillatory Flows over an Infinite Porous Surface in an Oscillating Free Stream, Latin American Applied Research, 40, pp. 167-173, 2010.

23. S. Ahmed and K. Kalita, Magnetohydrodynamic transient flow through a porous medium bounded by a hot vertical plate in presence of radiation: A theoretical analysis, Journal of Engineering Physics and Thermophysics (Springer), 86 (1), pp. 31-39, 2013.

Authors Bio Statement:

Sahin Ahmed, Ph. D., is an Associate Professor, working in the Department of Mathematics and Computing; Rajiv Gandhi University, Itanagar, Arunachal Pradesh, India. He has been doing his research work in the field of Thermo-fluid Magneto Hydrodynamics since 1999 and presently he is working with the Mathematics Education

Research. More than 60 research papers have been published in internationally reputed peer reviewed journals to his credit. Thirteen (13) Research Scholars have obtained their M. Phil degrees and some Scholars doing their Ph. D. works under his supervision. At present, he is involved as a Principal Investigator in several Research projects under UGC of India and he has attended many learned National and International Conferences and presented papers.

Hamida Khatun, M. Phil., is an Assistant Professor in the Department of Mathematicas; Mankachar College, Dhubri, affiliated to Gauhati University, Assam, India. His area of research is Fluid Dynamics. She is carried out her research work under the supervision of Dr. Sahin Ahmed since 2012. She has attended many learned National and International Conferences and presented papers.