Oscillatory MHD Convective Flow of Second Order Fluid through Porous Medium in a Vertical Rotating Channel in Slip-Flow Regime with Heat Radiation

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1 International Jornal of Scientific and Innovative Mathematical Research (IJSIMR) Volme 2, Isse 5, May 214, PP ISSN X (Print) & ISSN (Online) Oscillatory MHD Convective Flow of Second Order Flid throgh Poros Medim in a Vertical Rotating Channel in Slip-Flow Regime with Heat Radiation B. P. Garg Research Spervisor, Pnjab Technical University, Jalandhar, India bkgarg27@gmail.com Abstract: An analysis of an oscillatory magnetohydrodynamic (MHD) convective flow of a second order (viscoelastic), incompressible, and electrically condcting flid throgh a poros medim bonded by two infinite vertical parallel poros plates is presented.the two poros plates with slip-flow condition and the no-slip condition are sbjected respectively to a constant injection and sction velocity. The pressre gradient in the channel varies periodically with time. A magnetic field of niform strength is applied in the direction perpendiclar to the planes of the plates. The indced magnetic field is neglected de to the assmption of small magnetic Reynolds nmber. The temperatre of the plate with no-slip condition is nonniform and oscillates periodically with time and temperatre difference of the two plates is assmed high enogh to indce heat radiation.the entire system rotates in nison abot the axis perpendiclar to planes of the plates. Adopting complex variable notations, a closed form soltion of the problem is obtained. The analytical reslts are evalated nmerically and then presented graphically to discss indetail the effects of different parameters entering into the problem. The velocity, temperatre and the skin-friction in terms of its amplitde and phase angle have been shown graphically to observe the effects of viscoelastic parameter γ, rotation parameter Ω, sction parameter, Grashof nmber Gr, Hartmann nmber M, the pressre A, Prandtl nmber Pr, Radiation parameter N and the freqency of oscillation. Keywords: Slip-flow, Convective, Poros medim, Rotating, Oscillatory. 1. INTRODUCTION The wall slip flow is a very important phenomenon that is widely encontered in this era of indstrialization. It has nmeros applications, for example in lbrication of mechanical devices where a thin film of lbricant is attached to the srface slipping over one another or when the srfaces are coated with special coatings to minimize the friction between them. Marqes et al [1] have considered the effect of the flid slippage at the plate for Coette flow. Hayat et al. [2] analyzed slip flow and heat transfer of a second grade flid past a stretching sheet throgh a poros space in view of the importance of the slip-flow regime in this era of modern science, technology and its vast ranging indstrialization applications. The magnetohydrodynamic (MHD) flow of a viscoelastic flid as a lbricant is also of primary interest in many indstrial applications. The MHD lbrication is an externally pressrized thrst that has been investigated both theoretically and experimentally by Maki et al [3]. Hghes and Elco [4] have investigated the effects of magnetic field in lbrication.hamza [5] considered the sqeezing flow between two discs in the presence of a magnetic field. The problem of sqeezing flow between rotating discs has been stdied by Hamza[6] and Bhattacharya and Pal[7].Sweet et al.[8] solved analytically the problem of nsteady MHD flow of a viscos flid between moving parallel plates. Hayat et al. [9] have presented analytical soltion to the nsteady rotating MHD flow of an incompressible second grade flid in a poros half space. There isa variety of indstrial applications wherein de to high temperatre radiative heat is also accompanied along with condction and convection heat. Algoa et al. [1] stdied the radiative and free convective effects on a MHD flow throgh a poros medim between infinite parallel plates with time-dependent sction. Mebine and Gms[11] investigated steady-state soltions to MHD thermally radiating and reacting thermosoltal flow throgh a channel with poros medim. ARC Page 5

2 B. P. Garg The flow of viscoelastic flids throgh poros media has attracted the attention of a large nmber of scholars owing to their application in the fields of extraction of energy from geothermal regions and in the flow of oil throgh poros rocks. Many common liqids sch as certain paints, polymer soltions, some organic liqids and many new materials of indstrial importance exhibit both viscos and elastic properties. The flids with sch characteristics are called viscoelastic flids. Flows throgh the poros medim are also very sefl in chemical engineering for the processes of prification and filtration. The scientific treatment of the problem of irrigation, soil erosion etc. are present developments of poros media. Singh [12] analyzed viscoelastic mixed convection MHD oscillatory flow throgh a poros medim filled in a vertical channel. Rahman and Sarkar[13] investigated the nsteady MHD flow of a viscoelastic Oldroyd flid nder time varying body forces throgh a rectanglar channel. Singh and Singh[14] stdied MHD flow of a dsty viscoelastic (Oldroyd B-liqid)throgh a poros medim between two parallel plates inclined to horizon. Rajgopal and Gpta [15] obtained an exact soltion for the flow of a non- Newtonian flid past an infinite poros plate.applying qasilinearization to the problem Verma et al [16] analyzed steady laminar flow of a second grade flid between two rotating poros disks.rhodes and Rolea[17] stdied the hydrodynamic lbrication of partial poros metal bearings. Mehmood and Ali [18] extended the problem of oscillatory MHD flow in a channel filled with poros medim stdied by Makinde and Mhone[19] to slip-flow regime. Frther by applying the pertrbation techniqe Kmar et al [2] investigated the same problem of slip-flow regime for the nsteady MHD periodic flow of viscos flid throgh a planer channel. Very recently, Chodhry and Das [21] stdied the combined effects of magnetic field and heat radiation on a viscoelastic flow in a channel filled with poros medim. This stdy is also an extension of the paper by Makinde and Mhone[19].Singh and Devi [22] stdied the effect of slip velocity on MHD oscillatory flow throgh poros medim in a channel. Sivaraj and Kmar [23] analyzed nsteady MHD oscillatory chemically reacting slip-flow in a planer channel with varying concentration. Hamza et al. [24] investigated the effects of slip condition on an nsteady heat transfer to MHD oscillatory flow throgh poros medim. Singh and Kmar [26] analytically stdy the flctating heat and mass transfer MHD free convection flow of radiating and reacting flid past a vertical poros plate in slip-flow regime. The objective of the present analysis is to stdy an oscillatory convection flow of an incompressible, electrically condctingsecond order flid in a vertical poros channel. Constant injection and sction is applied at the left and the right infinite poros plates respectively. The entire system rotates abot an axis perpendiclar to the planes of the plates and a niform magnetic field is also applied along this axis of rotation. A general exact soltion of the partial differential eqations governing the flow problem is obtained and the effects of varios flow parameters on the velocity field and the skin friction are discssed in the last section of the paper with the help of figres. 2. BASIC EQUATIONS Consider the flow of a second order (viscoelastic), incompressible and electrically condcting flid in a rotating vertical channel. In order to derive the basic eqations for the problem nder consideration following assmptions are made: (i) The two infinite vertical parallel plates of the channel are permeable and electrically non-condcting. (ii) The vertical channel is filled with a poros medim. (iii) The flow considered is flly developed, laminar and oscillatory. (iv) The flid is second order, incompressible and finitely condcting. (v) All flid properties are assmed to be constant except that of the inflence of density variation with temperatre is considered only in the body force term. (vi) The pressre gradient in the channel oscillates periodically with time. (vii) A magnetic field of niform strength B is applied perpendiclar to the plates of the channel. International Jornal of Scientific and Innovative Mathematical Research (IJSIMR) Page 51

3 (viii) The magnetic Reynolds nmber is taken to be small enogh so that the indced magnetic field is neglected. (ix) Hall effect, electrical and polarization effects are also neglected. (x) The temperatre of the platewith no-slip condition is non-niform and oscillates periodically with time. (xi) The temperatre difference of the two plates is also assmed to be high enogh to indce heat transfer de to radiation. (xii) The flid is assmed to be optically thin with relatively low density. (xiii) The entire system (consisting of channel plates and the flid) rotates abot an axis perpendiclar to the plates. Under these assmptions we write hydromagnetic governing eqations of motion and continity in a rotating frame of reference as:. (3) In eqation (2) the last term on the left hand side is the Coriolis force. On the right hand side of (2) the last term acconts for the force de to boyancy. The second last term is the Lorentz Force de to magnetic field B and is given by and the modified pressre International Jornal of Scientific and Innovative Mathematical Research (IJSIMR) Page 52 (1) (2) (4), where R denotes the positionvector from the axis of rotation, denotes the flid pressre, J is the crrent density, and all other qantities have their sal meaning and have been defined from time to time in the text. In the term third from last of eqation (2), is the Cachy stress tensor and the constittive eqation derived by Coleman and Noll [26] for an incompressible homogeneos flid of second order is. (5) Here is the interdeterminate part of the stress de to constraint of incompressibility,, and are the material constants describing viscosity, elasticity and cross-viscosity respectively. The kinematic and are the Rivelen Ericson constants defined as,, where denotes the gradient operator and d/dt the material time derivative. According to Markovitz and Coleman [27] the material constants, are taken as positive and as negative. 3. FORMULATION OF THE PROBLEM In the present analysis we consider an nsteady flow of a second order (viscoelastic), incompressible and electrically condcting flid bonded by two infinite vertical poros plates distance d apart. A coordinate system is chosen sch that the X * -axis is oriented pward along the centerline of the channel and Z * -axis taken perpendiclar to the planes of the plates lying in planes. The flid is injected throgh the poros plate at with constant velocity w and simltaneos scked throgh the other poros plate at with the same velocity w. The non-niform temperatre of the plate at is assmed to be varying periodically with time. The temperatre difference between the plates is high enogh to indce the heat de to radiation. The Z * - axis is considered to be the axis of rotation abot which the flid and the plates are

4 B. P. Garg assmed to be rotating as a solid body with a constant anglar velocity Ω *. A transverse magnetic field of niform strength B (,, B ) is also applied along the axis of rotation. All physical qantities depend on z * and t * only for this problem of flly developed laminar flow. A schematic diagram of the flow problem is shown in figre 1. X * Poros Medim W W B O W W d Fig.1. Physical configration of the flow. On integration of the eqation of continity we get.then the velocity may reasonably be assmed with its components along x *, y *, z * directions as V ( *, v *, w ). Under the sal Bossinesq approximation and following Attia and Ewis[28] the MHD flow in the rotating channel is governed by the following Cartesian eqations:, (5), (6). (8) where is the density, is the kinematic viscosity, is viscoelasticity, p * is the modified pressre, t * is the time, is the electric condctivity, B is the component of the applied magnetic field along the z * -axis, g is the acceleration de to gravity, k is the thermal condctivity, c p is the specific heat at constant pressre and the last term in eqation (8) is the radiative heat flx. Following Cogley et al [29] it is assmed that the flid is optically thin with a relatively low density and the heat flx de to radiation in eqation (8) is given by. (9) where is the mean radiation absorption coefficient. After the sbstittion of eqation (9), eqation (8) becomes. (1) Eqation (7) shows the constancy of the hydrodynamic pressre along the axis of rotation. We shall assme now that the flid flows nder the inflence of pressre gradient varying periodically with time in the X * -axis is of the form (7) International Jornal of Scientific and Innovative Mathematical Research (IJSIMR) Page 53

5 and, where A is a constant. (11) The bondary conditions for the problem are, (12), (13) where is the freqency of oscillations,, L being mean free path and r 1 is the Maxwell s reflexion coefficient. Introdcing the following non-dimensional qantities: (14) into eqations (5), (6) and (1), we get, (15), (16), (17) where * represents the dimensional physical qantities, is the injection/sction parameter, is the visco-elastic parameter, is the rotation parameter, is the Hartmann nmber, is the permeability of the poros medim, is the Grashof nmber, is the Prandtl nmber, is the radiation parameter, is the freqency of oscillations. The bondary conditions in the dimensionless form become, (18) For the oscillatory internal flow we shall assme that the flid flows nder the inflence of a nondimension pressre gradient varying periodically with time in the direction of X-axis only which implies that (19), and. (2) International Jornal of Scientific and Innovative Mathematical Research (IJSIMR) Page 54

6 B. P. Garg 4. SOLUTION OF THE PROBLEM Now combining eqations (15) and (16) into single eqation by introdcing a complex fnction of the form F = + iv and with the help of eqation (2), we get with corresponding bondary conditions as, (21), (22) In order to solve eqation (21) and (17) nder bondary conditions (22) and (23) it is convenient to adopt complex notations for the velocity, temperatre and the pressre as nder: The soltions will be obtained in terms of complex notations, the real part of which will have physical significance. The bondary conditions (22) and (23) in complex notations can also be written as (23) (24), (25) Sbstitting expressions (24) in eqations (21) and (17), we get The transformed bondary conditions redce to (26), (27). (28), (29) The soltion of the ordinary differential eqation (27) nder the bondary conditions (29) and (3) gives the velocity field as (3), (31) where,,,,,,. International Jornal of Scientific and Innovative Mathematical Research (IJSIMR) Page 55

7 Similarly, the soltion of eqation (28) for the temperatre field can be obtained nder the bondary conditions (29) and (3) as. (32) From the velocity field obtained in eqation (31) we can get the skin-friction at the left plate (η = -.5) in terms of its amplitde and phase angle as, with (33) (34) The amplitde is and the phase angle is (35) Similarly the Nsselt nmber N in terms of its amplitde and the phase angle can be obtained from eqation (32) for the temperatre field as, (36) with, (37) where the amplitde and the phase angle ψ of the rate of heat transfer are given as The temperatre field, amplitde and phase of the Nsselt nmber need no frther discssion becase these have already been discssed in detail by Singh [3]. 5. RESULTS AND DISCUSSIONS The problem of oscillatory magnetohydrodynamic (MHD) convective and radiative flow in a vertical poros channel is analyzed in slip-flow regime. The viscoelastic, incompressible and electrically condcting flid is injected throgh one of the poros plates and simltaneosly removed throgh the other poros plate with the same velocity. The entire system (consisting of Poros channel plates and the flid) rotates abot an axis perpendiclar to the plates. The closed form soltions for the velocity and temperatre fields are obtained analytically and then evalated nmerically for different vales of parameters appeared in the eqations. To have better insight of the physical problem the variations of the velocity, the skin-friction in terms of its amplitde and phase angle are evalated nmerically for different sets of the vales of injection/sction parameter, viscoelastic parameter γ, rotation parameter Ω, Hartmann nmber M, the permeability parameter K, Grashof nmber Gr, Prandtl nmber Pr, radiation parameter N, pressre gradient Aand the freqency of oscillations. These nmerical vales are then shown graphically to assess the effect of each parameter for the two cases of rotation Ω = 5 (small) and Ω = 15 (large). The velocity variations are presented graphically in Figs. 2 to 11 for γ=.2, =.5, Gr=5, M=2, K=.2, Pr=.7, N=1, A=5, =5 and t= except the ones shown in these figres. (38) International Jornal of Scientific and Innovative Mathematical Research (IJSIMR) Page 56

8 Figre2 illstrates the variation of the velocity with the increasing slip-flow parameter δ. It is qite obvios from this figre that velocity goes on increasing with increasing slip-flow parameter δ. The velocity variations with the rotation parameter Ω are presented in figre 3.The velocity decreases with increasing rotation Ω of the entire system. The velocity profiles initially remain parabolic for small vales of rotation parameter Ω with maximm at the centre of the channel and then as rotation increases the velocity profiles flatten. The variations of the velocity profiles with the viscoelastic parameter γ are presented in Fig.4. It is very clear from this figre that the velocity decreases with the increase of viscoelastic parameter for both the cases of small (Ω = 5) and large (Ω = 15) rotations of the channel. However, for both these cases of rotation, the velocity increases significantly with increasing injection/sction parameter as is revealed in Fig.5.The variations of the velocity profiles with the Grashof nmber Gr isshown in Fig. 6.Althogh the velocity for large rotation () is mch less than the velocity for small rotation () bt in both these cases velocity increases with the increasing Grashof nmber Gr. The variation of the velocity profiles with Hartmann nmber M is presented in Fig.7. For both cases of small Ω (=5) andlarge Ω (=15) rotationsthe velocity goes on increasing with increasing M. This means that the backward flow cased by the rotation of the system and the viscoelasticity of the flid is resisted by the increasing Lorentz force de to increasing magnetic field strength. Fig.8 depicts the variations of the velocity with the permeability of the poros medim K. It is observed from the figre that the velocity decreases with the increasingpermeability K of the poros medim for both small () and large () rotations of this system of the viscoelastic flid flow.we also find from Fig.9 and Fig.1 respectively that with the increase of Prandtl nmber Pr and the radiation parameter N the velocity decreases for small () and large () rotations both. From Fig.11 it is evident that the velocity goes on increasing with the increasing favorable pressre gradient P ( ). The effects of the freqency of oscillations on the velocity are exhibited in Fig.12. It is noticed that velocity decreases with increasing freqency for either case of channel rotation large or small. The skin-friction in terms of its amplitde and phase angle has been shown in Figs. 13 and 14 respectively. The effect of each of the parameter on and is assessed by comparing each crve with dotted crves in these figres. In Fig.13 the comparison of the crves V,VI, VIII,and XI with dotted crve I (---)indicates that the amplitde increases with the increase of, injection/sction parameter, Grashof nmber Gr, permeability of the poros medim K and the pressre gradient parameter A.Similarly the comparison of other crvesii, III, IV,VII, IX, X and XIwith the dotted crve I (---) reveals that the skin-friction amplitde decreases with the increase of slip-flow parameter δ, rotation parameter Ω, viscoelastic parameter γ, Hartmann nmber M, Prandtl nmber Pr, the radiation parameter N.It is obvios that goes on decreasing with increasing freqency of oscillations. Fig. 14 showing the variations of the phase angle of the skin-friction it is clear that there is always a phase lag becase the vales of remains negative throghot. Here again the comparison of crves III, IV, V, VI and VIII with the dotted crve I (-- -) indicates that the phase lag increases with the increase rotation parameter Ω, viscoelastic parameter γ, injection/sction parameter, Grashof nmber Gr and permeability of the poros medim K. The comparison of other crves like VII, IX, X and XIwith the dotted crve (---) I shows that the phase lag decreases with the increase of Hartmann nmber M,Prandtl nmber Pr, radiation parameter N and the pressre gradient parameter A. For increasing slip-flow parameter δ (crves I & II) the phase lag decreasessmall vales of the freqency of oscillations bt then increases for large vales of. 6. CONCLUSIONS An oscillatory hydromagnetic convective flow of viscos incompressible and electrically condcting flid in a vertical poros channel is investigated when the entire system consisting of channel plates and the flid rotates abot an axis perpendiclar to the plates. A closed form soltion of the problem is obtained. The velocity increases with the increase of the slip-flow parameter δ, injection/sction parameter, Grashof nmbergr, the Hartmann nmber M and the pressre gradient parameter A. However, the velocity decreases with the increase of rotation parameter Ω, viscoelastic parameter γ, permeability of the poros medim K, Prandtl nmber Pr, radiation parameter N, and the freqency of oscillation. The increase of Hartmann nmber i.e. International Jornal of Scientific and Innovative Mathematical Research (IJSIMR) Page 57

9 increasing magnetic field strength resists the deceleration of the flow de to rotation and the viscoelasticity of the flid. The amplitde increases with the increase of injection/sction parameter, the Grashof nmber Gr, and the pressre gradient parameter A. There is always a phase lag of the skin friction. NOMENCLATURE A a constant B magnetic field applied c p specific heat at constant pressre g gravitational force Gr Grashof nmber k thermal condctivity M Hartmann nmber N heat radiation parameter p pressre Pr Prandtl nmber t time variable T flid temperatre T 1, T 2 constant temperatres, v, w velocity components along X, Y, Z- directions injection/sction velocity w.14 δ=,.5, 1.12 x, y, z variables along X, Y, Z-directions Greek symbols mean radiation absorption coefficient coefficient of volme expansion injection/sction parameter freqency of oscillations kinematic viscosity flid density Ω rotation parameter skin-friction at the left wall phase angle of the skin-friction mean non-dimensional temperatre * sperscript representing dimensional qantities η Fig.2. Velocity variations with slip-flow parameter δ , 1, η International Jornal of Scientific and Innovative Mathematical Research (IJSIMR) Page 58

10 Fig. 3.Velocity variations with rotation parameter Ω. γ=.2,.4, η Fig. 4.Velocity variations with viscoelastic parameter γ =.2,.5, η Fig. 5. Velocity variations with injection/sction parameter η Gr=2, 5, 1 International Jornal of Scientific and Innovative Mathematical Research (IJSIMR) Page 59

11 Fig. 6. Velocity variations with Grashof nmber Gr M=1, 2, η Fig. 7. Velocity variations with Hartmann nmber M..14 K=.2,.5, η Fig. 8. Velocity variations with permeability parameter K..14 Pr=.7, η International Jornal of Scientific and Innovative Mathematical Research (IJSIMR) Page 51

12 Fig. 9. Velocity variations with Prandtl nmber Pr..14 N=1, η Fig. 1. Velocity variations with radiation parameter N η A=3, 5, 7 Fig. 11. Velocity variations with pressre gradient A η =5, 6, 7 Fig. 12. Velocity variations with freqency. International Jornal of Scientific and Innovative Mathematical Research (IJSIMR) Page 511

13 II XI V VIII.3 I.2.1 VII III X IV VI IX Fig.13. Amplitde of the skin friction II VII VI IX X XI I VIII III IV V Fig.14. Phase of the skin friction. Vales of parameters plotted in Figs.13 & 14. δ Ω γ Gr M K Pr N A Crve I(---) II III IV V VI VII VIII IX X XI International Jornal of Scientific and Innovative Mathematical Research (IJSIMR) Page 512

14 ACKNOWLEDGEMENTS The athor is gratefl to the learned referee for his valable sggestions and technical comments which led a definite improvement in the research paper. REFERENCES [1] Marqes WJr., Kremer M. and Shapiro F.M., Coette Flow with Slip and Jmp Bondary Conditions, Continm Mech. Thermodynam., 12, pp , (2). [2] Hayat T., Javed T. and Abbas Z., Slip flow and heat transfer of a second grade flid past a stretching sheet throgh a poros space, Int. J. Heat Mass transfer, 51, pp , (28). [3] Maki E. R., Kzma D., Donnelly, R. L. and Kim B., Magnetohydrodynamic lbrication flow between parallel plates, J. Flid Mech., 26, pp , (1966). [4] Hghes W. F. and Elco R. A., Magnetohydrodynamic lbrication flow between parallel rotating discs, J. Flid Mech., 13, pp , (1962). [5] Hamza E. A., The magnetohydrodynamic sqeeze film, J. Flid Mech., 19, pp , (1964). [6] Hamza E. A., The magnetohydrodynamic effects on a flid film sqeezed between two rotating srfaces, J. Phys. D: Appl. Phys., 24, pp , (1991). [7] Bhattacharyya S. and Pal A., Unsteady MHD sqeezing flow between two parallel rotating discs, Mech. Res. Commn., 24, pp , (1997). [8] Sweet Erik, Vajravel K., Robert A. Gorder Van and Pop, I., Analytical soltion for the nsteady MHD flow of a viscos flid between moving parallel plates, Commn. Nonlinear Sci. Nmer. Simlat., 16, pp , (211). [9] Hayat T., Fetecaa C. and Sajid M., Analytic soltion for MHD transient rotating flow of a second grade flid in a poros space, Nonlinear Analysis: Real World Applications, 9, pp , (28). [1] Alagoa K. D., Tay G. and Abbey T.M., Radiative and free convective effects of a MHD flow throgh a poros medim between infinite parallel plates with time-dependent sction, Astrophysics and Space Science, 26, pp , (1999). [11] Mebine Promise and Gms Rhoda H., On steady MHD thermally radiating and reacting thermosoltal viscos flow throgh a channel with poros medim, Int. J. of Mathematics and Mathematical sciences, Article ID287435, 12 pages, (21). [12] Singh, K. D., Viscoelastic mixed convection MHD oscillatory flow throgh a poros medim filled in a vertical channel, Int. J. Physical and Mathematical Sciences, 3, pp , (212). [13] Rahmann M. M. and Sarkar M. S. A., Unsteady MHD flow of viscoelastic Oldroyd flid nder time varying body forces throgh a rectanglar channel, Blletin of Cactta Mathematical Society, 96, pp , (24). [14] Singh A. K. and Singh N. P., MHD flow of a dsty viscoelastic liqid throgh a poros medim between two inclined parallel plates, Proc. of National Academy of Sciences India, 66A, pp , (1966). [15] Rajgopal K. R. and Gpta A. S., An exact soltion for the flow of a non-newtonian flid past an infinite poros plate, Meccanica, 19, pp , (1984). [16] Verma P. D. Sharma P. R. and Ariel P. D., Applying qasilinearization to the problem of steady laminar flow of a second grade flid between two rotating poros disks, J. Tribol. Trans. ASME, 16, pp , (1984). [17] Rhodes C. A. and Rolea W.T., Hydromagnetic lbrication of partial metal bearings, J.Basic Eng.-T. ASME, 88, pp. 53-6, (1966). [18] Mehmood A. and Ali A., The effect of slip condition on nsteady MHD oscillatory flow of a viscos flid in a planer channel, Rom. Jorn. Phys., 52, pp , (27). International Jornal of Scientific and Innovative Mathematical Research (IJSIMR) Page 513

15 [19] Makinde O.D. and Mhone P.Y., Heat Transfer to MHD Oscillatory Flow in a Channel Filled with Poros Medim, Rom. Jorn. Phys., 5, pp , (25). [2] Kmar Anil, Varshney C.L. and SajjanLal, Pertrbation Techniqe to Unsteady MHD Periodic Flow of Viscos Flid throgh a Planer Channel, J. of Engineering and Tech. Res., 2, pp , (21). [21] Chodhary Rita and Das UtpalJyoti, Heat transfer to MHD oscillatory viscoelastic flow in a channel filled with poros medim, Physics Research International, doi:11155/212/879537, (212). [22] Singh K.D. and Devi R., Effect of slip velocity on MHD oscillatory flow throgh poros medim in a channel, International Jornal of Physics, 3, pp.75-83, (21). [23] Sivaraj R. and Kmar B. Rshi, Chemically reacting nsteady MHD oscillating slip flow in a planer channel with varying concentration, Int. J. of Mathematics and Scientific Comptation, 1, pp , (211). [24] Hamaza M. M., Isah B. Y. and Usman H., Unsteady heat transfer to MHD oscillatory flow throgh poros medim nder slip condition, Int. J. of Compter Application, 33, pp , (211). [25] Singh K. D. and Kma, R., Flctating heat and mass transfer on nsteady MHD free convection flow of radiating and reacting flid past a vertical poros plate in slip-flow regime, J. Appl. Flid Mech., 4, pp , (211). [26] Coleman B. D. and Noll W., An approximation theorem for fnctional, with applications in continm mechanics, Archive for Rational Mechanics and Analysis, 6, pp , (196). [27] Markovitz H. and Coleman B. D., Incompressible second order flids, Advances in Applied Mechanics, 8, pp , (1964). [28] AttiaHazem Ali and EwisKarem Mahmod, Unsteady MHD Coette flow withheat transfer of a viscoelastic flid nder exponentially decaying pressre gradient, Tamkang J. of Science and Engineering, 13, pp , (21). [29] Cogley A. C. L., Vinvent W. G. and Giles E. S., Differential approximation for radiative transfer in a non-gray near eqilibrim, American Institte of Aeronatics and Astronatics, vol. 6, pp , (1968). [3] Singh K. D., Effect of injection/sctionon convective oscillatory flow throgh poros medim bonded by two vertical poros plates, Int. J. of Physical and Mathematical Sciences, 2(1), pp , (211). AUTHOR S BIOGRAPHY BP Garg, (b. 1969) is Director-Principal of one of the repted engineering colleges (Adesh Institte of Engineering & Technology, Faridkot) affiliated to PTU Jallandhar, India. He attained his Post gradate and Doctorate degree from H.P. University, Shimla. His field of specialization is Flid Mechanics. He has twenty seven international and national research papers in his credit. He is life member of varios academic and professional bodies. He is serving as Referee of many prestigios and internationally repted jornals. He has been awarded Best Edcationist Award for otstanding achievements in the field of edcation by International Institte of Edcation & Management and RashtriyaVidyaGarav Gold Medal Award for otstanding achievements in the field of edcation by Indian Solidarity Concil, New Delhi. International Jornal of Scientific and Innovative Mathematical Research (IJSIMR) Page 514