Heat and Mass Transfer on Unsteady M.H.D Oscillatory Flow of Non-Newtonian Fluid through Porous Medium in Parallel Plate Channel

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1 Heat and Mass Transfer on Unstead M.H.D Oscillator Flow of Non-Newtonian Fluid through Porous Medium in Parallel Plate Channel Dr. G. Kathaani, Dr. P. Sambasivudu, D. M. Praveen Babu Assistant Professor, Department of Applied Mathematics, Yogi Vemana Universit, Kadapa, A.P., India Assistant Professor, Department of Applied Mathematics, C.B.I.T, Proddatur,A.P., India Ph.DScholar, Department of Applied Mathematics, Yogi Vemana Universit, Kadapa, A.P., India Abstract: We have considered the unstead two dimensional MHD oscillator flow of non-newtonian fluid through a porous medium under the influence of uniform transverse magnetic field. A mathematical model is developed and analzed b using appropriate mathematical techniques. The mathematical epressions for the velocit profile, wall shear stress and rates of heat and mass transfer have been obtained and computationall discussed with respect to the different parameters in detail. Kewords: Heat and mass transfer, MHD Oscillator flows, porous medium. Introduction The effect of heat and mass transfer on unstead MHD oscillator flow of fluid in horizontal media are encountered in a wide range of engineering and industrial applications such as molten iron flow, recover etraction of crude oil, geothermal sstems. Man chemical engineering processes like metallurgical and polmer etrusion processes involve cooling of a molten liquid being stretched in a cooling sstem; the fluid mechanical properties of penultimate product depend mainl on the cooling liquid used and the rate of stretching. Some polmers fluids like polethlene oide and polsobutlene solutions in a cetane, having better electromagnetic properties are normall used as cooling liquid as their flow can be regulated b eternal magnetic fields in order to improve the qualit of the final product. Also, the radiative heat transfers is an important factor of thermodnamics of ver high temperature sstems such as electric furnaces, solar collectors, storage of nuclear wastes packed bed cataltic reactors, satellites, steel rolling, crogenic engineering etc. The stud of such flow under the influence of magnetic field and heat transfer has attracted the interest of man investigators and researchers. Asadullah et al.,[] consider the MHD flow of a Jeffre fluid in converging and diverging channels. The flows between non parallel walls have a ver significant role in phsical and biological sciences. Kavita et al.,[] investigated the influence of heat transfer on MHD oscillator flow of Jeffer fluid in a Channel. An analsis of first order homogeneous chemical reaction and heat source on MHD oscillator flow of viscous elastic fluid through a channel filled with saturated porous medium are reported b Devika et al.,[8]. An oscillator flow of a Jeffre Fluid in an elastic tube of variable cross section has been investigated at low Renolds number b Badari et al.,[]. Their main concentration is on the ecess pressure of the tube. The equation has been solved numericall and investigations are made for different cases on the tube. Aruna Kumari et al.,[] studied the effect of heat transfer on MHD oscillator flow of Jeffre fluid in a channel with slip effect at a lower wall where the epressions for the velocit and temperature are obtained analticall. Israel-Cooke et al.,[0] investigated the combined effects of radiative heat transfer and a transverse magnetic field on stead flow of an electricall conducting opticall thin fluid through a horizontal channel filled with porous medium and non-uniform temperatures at the walls. Closed form analtical solutions for the problem. The unstead MHD free convective flow through porous medium sandwiched between electricall conducting viscous incompressible fluids in a horizontal channel with isothermal walls temperature using Brinkman model has been investigated b Kumar et al.,[]. The forced convection in a horizontal double passage with uniform wall heat flu has been studied b Joseph et al.,[] b taking into account the effect of magnetic parameter where the flow of the fluid is assumed to be laminar, two dimensional, stead and full developed. The fluid is incompressible and the phsical properties are constants and the walls are kept at uniform heat flu. Bodosa and Borkakati [7] analsed the problem of an unstead two dimensional flow of a viscous incompressible and electricall conducting fluid between two parallel plates in the presence of uniform transverse magnetic field. The lower plate is a stretched sheet while the upper one is an oscillating porous plate, which is oscillating in its own plane. The effect of radiation on unstead free convection flow bounded b an oscillating plate with variable wall temperature was studied b Pathak and Maheshwari [8]. The flow and heat transfer between two horizontal parallel plates, where the lower plate is a stretching sheet and the upper one is a porous solid plate in the presence of transverse magnetic field was discussed b Bharali and Borkakati []. Also Bharali and Borkakati [] studied the heat transfer in an aismmetric flow between two parallel porous disk under the effect of a transverse magnetic field. Sharma and Deka [] studied the effects of plate temperature oscillation on the unstead conducting Paper ID: 7090

2 fluid along a semi-infinite vertical porous plate subjected to a transverse magnetic field in the presence of a first order chemical reaction and thermal radiation. Israel-Cooke and Nwaigwe [9] studied the unstead MHD flow of a radiating fluid over a moving heated plate with time dependent suction. Kim [] discussed the unstead MHD convective heat transfer past a semi-infinite vertical porous moving plate with variable suction. The problem of unstead MHD periodic flow of viscous fluid through a planar channel in porous medium using perturbation techniques was considered b Kumar et al.,[]. Makinde and Mhone [7] studied the heat transfer to MHD oscillator flow in a channel filled with porous medium. Rita et al.,[0] investigated the visco-elastic unstead MHD flow between two horizontal parallel plates taking into hall current account. Maen Al-Rashdan [] presented an analtical investigation to the problem of full developed natural convective heat mass transfer through porous medium in a channel in the presence of a first order chemical reaction.. Mathematical formulation and solution of the Problem As in man other similar theoretical studies, the formulation analsis that follows, we use Cartesian coordinates. The flow is considered smmetric about the ais of the channel and driven b the stretching of the channel wall, such that u p u u B0 t z z t k the velocit of each wall is proportional to the aial coordinate. In order to stud the second-order effects of unstead MHD flow of non-newtonian, let us first consider the flow of a second-order fluid between two parallel plates at z=0 and z=h, where the -ais is taken parallel length of plates and z-ais along a direction perpendicular to the plates. A magnetic field of constant intensit B 0 is considered to be applied in the -direction. The phsical configuration of the problem is as shown in Fig.. Figure : Phsical configuration of the Problem The unstead hdromagnetic equations of the momentum, heat transfer and mass transfer for the MHD oscillator flow of second grade fluid through a porous medium in the parallel plate sstem are considered in the form, u ug( T T ) g ( CC ) v p v v B0 v v t z z t k T K T qr t Cp z Cp z C C D K ( c CC0) t z The boundar conditions for the problem under consideration are given b the corresponding boundar conditions are u v it it u, v, T T0 ( Tw T0) e, C C0 ( Cw C0) e at z h (.) z z u u u, v, T T0, C C at z 0 (.) 0 z z Using Rosseland approimation [9], the radiative transfer term q r in Equation (.) ma be epressed as * T qr (.7) r z We assume that the temperature differences within the flow are such that T can be epressed as a linear function of the temperature T. This is accomplished b epanding T in a Talor series about T 0 (which is assumed to be independent of z) and neglecting powers of T higher than the first. Thus we have qt TT 0 T (.8) 0 Then the heat transfer equation becomes q T K T *T0 T (.9) t Cp z Cpr z Combining the equations (.) and (.), q u iv and we obtain t q p q q B0 q qg( T T ) g ( C C ) (.0) t z z t k We now introduce the following non-dimensional variables: z q tu0 T T0 *, *, z*, q*, t*,, h h h U h T T 0 w 0 Paper ID: 7090 (.) (.) (.) (.)

3 C C0 h tw0 p, *, t*, *, p* = Cw C0 U0 h U0 Making use of non-dimensional quantities (dropping asterisks), the governing equation (.0), (.) and (.) can be written as q p q q Re M qgr + Gm (.) t z z t K Pr ( R) (.) t z Sc Kc (.) t z The corresponding non-dimensional boundar conditions assume the form q ( q it it q, e, e at z (.) i ) Rei M q P Gr Gm z K z (.0) q q, 0, 0 at z 0 (.) ( R) Pri 0 (.) z z From Eq. (.), it follows that, p is a function of t (.) ( SciKc) 0 z onl. We consider it to be of the form, With corresponding boundar conditions p Pe (.) q q,, at z (.) z To solve Eqs. (.), (.) and (.) subject to the q boundar conditions (.) and (.), we further write the q, 0, 0 at z 0 (.) z velocit, temperature and concentration as Solving (.0) (.) subject to the conditions (.) and q( z,t) qe (.7) (.), we have velocit field, temperature, concentration ( z,t) e (.8) respectivel, where the epressions for the constants m ( z,t) e i ( i,,..., ) and ai ( i,,..., ) (.9) Substituting these epressions (.7), (.8) and (.9) in (.), (.) and (.) respectivel and comparing the coefficients of like terms we have the equations. P Gr e e qzt (, )= ae ae m m Re i M ( K) e e a a Gm e e it e (.) m m e e a a it ( zt, )= m e e e (.) m e e it ( zt, )= m e e e (.7) m e e Skin friction: The wall shear stress at the wall of the upper plate is found as m m q q m m Gr me me w = = ame ame m m z zt e e a z a m m Gm me me it (.8) ( i ) e m m e e a a Nusselt number: The rates of heat transfer across the upper plate (upper wall) are calculated as m m it Nu = = m me me e m z z e e (.9) Sherwood number: The rates of mass transfer across the upper plate (upper wall) are calculated as = z e e Sh = m e m e e z. Results and Discussion m m it m m (.0) We have considered the unstead two dimensional MHD oscillator flow of non-newtonian fluid through a porous medium in a parallel plate channel under the influence of uniform transverse magnetic field. The governing equations for the velocit profile, wall shear stress and rates of heat Paper ID: 7090

4 and mass transfer have been obtained using perturbation technique and computationall discussed with respect to the different parameters namel viz. M is the Hartmann number (Magnetic field parameter), K is the Permeabilit parameter, is the second grade fluid parameter, Gr is the thermal Grashof number, Gm is the mass Grashof number, Pr is Prandtl parameter, R is the Radiation parameter, Kc chemical reaction parameter and Sc is the Schmidt number. The Figures (-) represent the velocit profiles for u and v; the Figure (-) represent the temperature profiles for ; the Figures (-7) represent the concentration profiles for. From the Figures (-) we noticed that, both the velocit components u and v reduces with increasing the intensit of the magnetic field or Hartmann number M. Also we have been seen that the resultant velocit q is eperiences retardation throughout the fluid region. The velocit component u increases and v reduces with increasing permeabilit parameter K. The resultant velocit enhances with increasing K in the flow field. We also noticed that lower the permeabilit lesser the fluid speed is observed the entire fluid region (Figures -). The similar behaviour is observed for the velocit components with radiation parameter R (Figures -). The magnitude of the velocit components u and v as well as resultant velocit reduces in the entire fluid region with increasing second grade fluid parameter, Pr, Sc and Kc (Figures -9, -7 & - ). From the Figures (0- & 0-) the velocit components u and v as well as resultant velocit increase with increasing thermal Grashof number Gr, mass Grashof number Gm or. We also find that the magnitude of the velocit component u reduces and v enhances with increasing the frequenc of oscillation. The resultant velocit reduces throughout the fluid region with increasing the frequenc of oscillation (Figures 8-9). We noticed that from the Figures (-) the magnitude of the temperature reduces with increasing radiation parameter, where as the reversal behaviour is observed throughout the fluid region with increasing Prandtl number. Also we found that from the Figures (-7) the magnitude of the concentration increases with increasing Schmidt number Sc, where as the reversal behaviour is observed throughout the fluid region with increasing chemical reaction parameter Kc. The frictional force is determined from the velocit at the upper plate onl. The magnitude of the stress components and enhances with increasing K, Gr, Gm and. The opposite nature is observed for the same components with increasing M and Kc. The magnitude of the stress component reduces and increases with increasing, and R.The reversal behaviour is for the components and with increasing Pr and Sc (Table ).We also noticed that from the Table () the Nusselt number Nu enhances with increasing Radiation parameter R and Prandtl number Pr. Likewise the rate of mass transfer is reduces with increasing Schmidt number Sc and increases with increasing chemical reaction parameter Kc (Table ). Paper ID: 7090

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6 Table : Skin Friction M K R Pr Gr Gm Sc Kc Table : Nusselt number R Pr Nu Table : Sherwood number Sc Kc Sh Conclusions We have discussed the unstead two dimensional MHD oscillator flow of non-newtonian fluid through a porous medium under the influence of uniform transverse magnetic field. The conclusions are made as the following.. The resultant velocit q is eperiences retardation throughout the fluid region with increase in M.. The velocit component u increases and v reduces with increasing permeabilit parameter K or R. The resultant velocit enhances with increasing K or R in the flow field.. Lower the permeabilit lesser the fluid speed is observed the entire fluid region.. The resultant velocit reduces in the entire fluid region with increasing second grade fluid parameter, Pr, Sc and Kc Paper ID:

7 . The resultant velocit increase with increasing thermal Grashof number Gr, mass Grashof number Gm or.. The resultant velocit reduces throughout the fluid region with increasing the frequenc of oscillation. 7. The temperature reduces with increasing radiation parameter R, where as the reversal behaviour is observed throughout the fluid region with increasing Prandtl number Pr. 8. The concentration increases with increasing Schmidt number Sc, where as the reversal behaviour is observed throughout the fluid region with increasing chemical reaction parameter Kc. 9. The stress components and enhances with increasing K, Gr, Gm and. 0. and reduces with increasing M and Kc.. The stress component reduces and increases with increasing, and R. The reversal behaviour is for the components and with increasing Pr and Sc.. The Nusselt number Nu enhances with increasing Radiation parameter R and Prandtl number Pr.. The Sherwood number is reduces with increasing Schmidt number Sc and increases with increasing chemical reaction parameter Kc. References [] Aruna Kumari B., Ramakrishna Prasad K., Kavita K. (0): Slip Effects on MHD Oscillator Flow of Jeffre Fluid in a Channel with Heat Transfer. Int. J. Math. Arch. (8):90 9. [] Asadullah M., Umar K., Nareed A., Raheela M. and Mohud Din S.T. (0): MHD Flow of a Jeffre Fluid in Converging and Diverging Channels. Int. J. Mod. Math. Sci. (): 9 0. [] Aziz A. and Na T.Y. (98): Perturbation Methods in Heat Transfer. Hemisphere, New York. [] Badari Naraana C.H, Sreenadh S. and Devaki P. (0): Oscillator Flow of Jeffre Fluid in Elastic tube of Variable Cross Section. Adv. Appl. Sci. Res. (): [] Bharali A. and Borkakati A.K. (980): The Flow and Heat Transfer between Two Horizontal Parallel Plates. Journal of the Phsical Societ of Japan. 9 (): 09. [] Bharali A. and Borkakati A.K. (98): The Heat Transfer in an Aismmetric Flow between Two Parallel Porous Disk under the Effect of a Transverse Magnetic Field. Journal of the Phsical Societ of Japan. ():. [7] Bodosa G. and Borkakati A.K. (00): MHD Couette Flow with Heat Transfer between Two Horizontal Plates in the Presence of Uniform Transverse Magnetic Field. Theor. Appl. Mech. 0 (): 9. [8] Devika B., Satga Naraana P.V. and Venkatarmana S. (0): MHD Oscillator Flow of a Viscous Elastic Fluid in a Porous Channel with Chemical Reaction. Int. J. Eng. Sci. Inv. ():. [9] Israel Cooke C., Nwaigwe C. (00): Unstead MHD Flow of a Radiating Fluid over a Moving Heated Porous Plate with Time Dependent Suction. Am. J. Sci. Ind. Res. (): [0]Israel Cooke C., Omubo Pepple V.B. and Tamunoberetonari I. (00): On Stead Hdromagnetic Flow of a Radiating Viscous Fluid through a Horizontal Channel in a Porous medium. Am. J. Sci. Ind. Res. (): []Joseph K.M., Onwubuoa C., Daniel K.S., and Nonum E.O. (0): MHD Forced Convection in a Horizontal Double Passage with Uniform Wall Heat Flu. Int. J. Math. Stat. Stud. (): 0 0. []Kavita K., Pasad R.K. and Kumari B. A. (0): Influence of Heat Transfer on MHD Oscillator Flow of Jeffre Fluid in a Channel. Adv. Appl. Sci. Res. ():. []Kim U.J. (000): Unstead MHD Convective Heat Transfer past a Semi Infinite Vertical Porous Moving Plate with Variable Suction. Int. J. Eng. Sci. 8: 8 8. []Kumar N., Jain T. and Gupta S. (0): Unstead MHD Free Convection Flow through Porous Medium Sandwiched between Viscous Fluids. Int. J. Ener. Tech. (7): -. []Kumar A., Varshne C. L., Lal S. (00):Perturbation Technique to Unstead Periodic Flow of Viscous Fluid through a Planar Channel. J. Eng. Tech. Res. (): 7 8. Effect Of Heat And Mass Transfer On Unstead Mhd Oscillator Flow Of Jeffre Fluid In A 87 Page []Maen Al Rashdan (0): Heat and Mass Full Developed Natural Viscous Flow with Chemical Reaction in Porous Medium. Adv. Theor. Mech. (): 9. [7]Makinde O.D., Mhone P.Y. (00): Heat Transfer to MHD Oscillator Flow in a Channel Filled with Porous Medium. Romanian J. Phsics. 0 (9 0): [8]Pathak G. and Maheshwari C.H. (00): Effects of Radiation on Unstead Free Convection Flow Bounded b an Oscillating Plate with Variable Wall Temperature. Int. J. Appl. Mech. Eng. (): 7 8. [9]Perdikis C, Raptis A:Heat transfer of a micropolar fluid b the presence of radiation, Heat Mass Transfer (99) 8 8. [0]Rita C. and Das K.S. (0): Visco-Elastic Unstead MHD Flow between Two Horizontal Parallel Plates with Hall Current. IOSR J. Math. (): 0 8. []Sharma S. and Deka R.K. (0): Thermal Radiation and Oscillating Plate Temperature Effects on MHD Unstead Flow past a Semi Infinite Vertical Plate under Suction and Chemical Reaction. Int. J. Phs. Math. Sci. ():. Paper ID: