Numerical Solution of Thermal Radiation and Joule-Heating Effects on an Unsteady MHD with Heat and Mass Transfer with Chemical Reaction

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1 Numerical Solution o Thermal Radiation and Joule-Heating Eects on an Unsteady MHD with Heat and Mass Transer with Chemical Reaction Adel A. Megahed, Ali A.Hallool, Hamed.A. El Mky Department o Mathematics and Engineering Physics, Faculty o Engineering, Cairo University, Egypt.. Department o Physics and Engineering Mathematics, Higher Institute or Engineering in 5 May, Helwan, Cairo, Egypt. Department o Mathematics, Faculty o Science, Aswan University, Egypt. ABSTRACT: The radiation and chemical reaction eects on an unsteady two-dimensional magnto-hydrodynamics ree convection low with heat and mass transer rom a vertical porous stretching surace with joule-heating. The luid is assumed to be viscous, electrically conducting and incompressible. The non-linear partial dierential equations, governing the low ield under consideration, have been transormed by a similarity transormation into a system o non-linear ordinary dierential equations and then solved numerically using the Runge-Kutta integration scheme with a modiied version o the Newton-Raphson shooting method. Resulting non-dimensional velocity, temperature and concentration proiles are then presented graphically or dierent values o the parameters o physical including the Prandtl numberpr, Eckert number Ec, Schmidt number Sc, Chemical Reaction G, unsteadiness parameter A, mixed convection parameter, suction parameter 0, thermal radiation parameter R and magnetic parameter M. KEYWORDS: heat and mass transer, thermal radiation, joule heating, chemical reaction. Nomenclature C C C w C 0 k Pr Re x Concentration the luid Concentration in luid ar away rom the stretching surace Concentration at the surace Reerence concentration Coeicient o thermal conductivity o the luid Prandtl number local Reynolds number q r Radiation heat lux q T local heat transer coeicient q C local mass transer coeicient t Time T Fluid temperature T w Surace temperature T 0 Reerence temperature T Temperature or away rom the stretching surace u,v Velocity components along x and y axes u w Velocity o the moving sheet x, y Cartesian coordinates along the sheet and normal to it respectively. V w Velocity o suction k Thermal diusivity o the luid Copyright to IJARSET 577

2 c, Positive constants Similarity variable g Gravity ield Kinematic viscosity Volumetric coeicient o thermal expansion Electrical conductivity c p Speciic heat at constant pressure D Diusivity coeicient k Chemical reaction B 0 Uniorm magnetic ield Density s Stean-Boltzman constant A Unsteadiness parameter Mixed convection parameter M Magnetic ield parameter Pr Prandtl number k* Absorption coeicient. R thermal radiation parameter Ec Eckert number Sc Schmidt number Stream unction G Chemical reaction parameter 0 Suction parameter F' Dimensionless velocity θ Dimensionless temperature φ Dimensionless concentration C Skin riction Nu x local Nusselt number Sh x local Sherwood number Superscripts Dierentiation with respect to I. INTRODUCTION Boundary layer low, heat and mass transer over a linearly stretched surace has received considerable attention in recent years. This is because o the various possible engineering and metallurgical applications such as hot rolling, wire drawing, metal and plastic extrusion, continuous casting, glass iber production, crystal growing and paper production. These are discussed in a book published by Gebhart et al. []. A.M. Rashad [], ocused on the study o unsteady magneto hydrodynamics boundary-layer low and heat transer or a viscous laminar incompressible electrically conducting and rotating luid due to a stretching surace embedded in a saturated porous medium with a temperaturedependent viscosity in the presence o a magnetic ield and thermal radiation eects. But EL-Kabeir et al. [3], studies the unsteady two-dimensional low o an electrically-conducting incompressible luid over a continuous moving vertical stretching sheet embedded in a luid-saturated porous medium and the low is permeated by a uniorm transverse magnetic ield. The unsteady heat transer problems over a stretching surace, which is started impulsively rom rest or is stretched with a velocity that depends on time, are considered. Abo-Eldahab et al. [4], investigated the eects o Heat Transer on MHD Flow Over an Unsteady Stretching Surace in a Micropolar Fluid and a Porous Medium with Prescribed Surace Heat Flux.Elbashbeshy and Bazid [5], presented an exact similarity solution or unsteady momentum and heat transer low whose motion is caused solely by the linear stretching o a horizontal Copyright to IJARSET 578

3 stretching surace. EL-Hakiem et al.[6] presented Group method analysis or the eect o radiation on MHD coupled heat and mass transer natural convection low or water vapor over a vertical cone through porous medium. E. M.A.Elbashbeshy, D.A.Aldawody[7], examined the eects o Thermal Radiation and Magnetic Field on Unsteady Mixed Convection Flow and Heat Transer Over a Porous Stretching Surace. M.A. EL-Hakiem, A.M. Rashad [8], carried out an Eect o radiation on nondarcy ree convection rom a vertical cylinder embedded in a luid-saturated porous medium with a temperature-dependent viscosity. The present trend in the ield o chemical reaction analysis is to give a mathematical model or the system to predict the reactor perormance. A large amount o research work has been reported in this ield. In particular, the study o heat and mass transer with chemical reaction is o considerable importance in chemical and hydrometallurgical industries. In order to study the thermal stratiication eects over the above-mentioned problem, an attempt has been made have to analyze the nonlinear hydro magnetic low with heat and mass transer over a vertical stretching surace with chemical reaction and thermal stratiication eects. Among some o the interesting problems which were studied are the analyses o laminar orced convection mass transer with homogeneous chemical reaction by J.D. Goddard, A. Acrivos [9]. Lakshmisha et al. [0] presented numerical solutions or the unsteady three-dimensional low with heat and surace mass transer over a stretching surace. Gorla and Sidawi [] presented numerical solutions or the problem o steady three-dimensional ree convection low over a stretching surace with suction and blowing. Chamkha [] studied the problem o the hydro magnetic steady three-dimensional ree convective low on a vertical stretching surace with heat generation or absorption. Eldabe et al.[3] presented a Numerical study o viscous dissipation eect on ree convection heat and mass transer o MHD non-newtonian luid low through a porous medium. In recent years, numerous investigations have been conducted on the magneto-hydrodynamic lows and heat transer because o its important applications in metallurgical industry. In the presence o a transverse magnetic ield, the low and heat transer over a stretching surace have been investigated by [4-8]. All the above mention studies consider the steady state problem.abo- El_dahab et al. [9] studied the joule heating eect on hydrody magnetic three dimensional low over a stretching surace with heat and mass transer. Recently, Abo-Eldahab et al. [0] studied the Eects o Heat and Mass Transer on MHDFlow Over a Vertical Stretching Surace with Free Convection and Joule-Heating. The aim o the present work is to study the eects o thermal radiation,chemical Reaction and magnetic ield on unsteady mixed convection low, heat and mass transer over a vertical stretching surace with viscous dissipation in the presence o wall suction. II. MATHEMATICAL FORMULATION Consider an unsteady two-dimensional MHD ree convection laminar boundary layer low o a viscous incompressible and electrically conducting luid with heat and mass transer under the inluence o thermal radiation over a vertical porous stretching surace and moving with velocityu cx t distribution w where c and are constants and with temperature Tw T T0c x t, where T 0 is a reerence temperature such that 0T 0 T w and the concentration distribution C C C c x t w 0, C 0 is a reerence concentration such that 0C 0 C w.introducing the Cartesian coordinate system, the x-axis is taken along the stretching surace in the vertically upward direction and the y-axis is taken as normal to the surace. Two equal and opposite orces areintroduced along the x-axis, so that the surace is stretched keeping the origin ixed.a uniorm magnetic ield o strength B 0 is applied normal to the stretching surace which produces magnetic eect in the x-axis. The radiative heat lux in the x-direction is considered negligible in comparison to the y-direction. This work is an extension o[7].under the above assumptions, the governing boundary layer equations are: Copyright to IJARSET 579

4 u x v 0 y u u u u B u v g ( T T ) u t x y y 0 r p y p p T T T k T q J u v t x y c c y c C C C C u v D K C C t x y y With the boundary condition u Uw, v V w, T Tw, C Cw at y 0 u 0, T T, C C at y Where V w c t is the velocity o suction (V w 0), u and v are the velocity components along x and y axes,respectively, t is the time, is the kinematics viscosity,g is the gravity ield, is the volumetric coeicient o thermal expansion, is the electrical conductivity, B 0 is the uniorm magnetic ield, c p is the speciic heat at constant pressure, is the density, kis the coeicient o thermal conductivity o the luid, D is the diusivity coeicient andk is the chemical reaction (reaction rat).t,t and T w are the temperature, the temperature or away rom the stretching surace and surace temperature respectively.c, C and C w are the species concentration in luid, species concentration in luid ar away rom the stretching surace and species concentration at the surace.q r the radiation heat lux. By using Rossel and diusion approximation or radiation q r 4 s T 3 k * y 4 where s and k*are the Stean-Boltzman constant and absorption coeicient.let the temperature within the low is such that T 4 may be expanded in a Taylor, s series. Expanding T abutst and neglecting higher order we gett 4T T 3T. Now eq. (3) becomes: 3 T T T k 6 st T J u v * t x y c p 3c pk y c p The continuity equation is satisied i we choose a stream unction (x, y) such that u, v we used the y x similarity variable: 6 c c y, x, y x, J E v B t t c x c x T TT0, C C C 0 t 3 t From the eq.(7) in eqs. (,,4and 6) the governing equations inally reduce to: 7 Copyright to IJARSET 580

5 F ''' AF '' F '' F F ' A M F ' 0 8 Pr A '' F ' F ' ' 4 M Ec F ' 0 9 R '' F A ' A G F ' 0 0 Sc And the boundary conditions (5) are transormed to: t F (0) V, F '(0), 0, 0 at 0 c F ' 0, 0, 0 at w 0 where the parameter that measures the unsteadiness is A=c, the mixed convection parameter is =gt 0 c, Ec=cxT 0 c p is the Eckert number, Sc=D is the Schmidt number, M= B 0 (-t)cis the magnetic ield parameter,pr=c p kis the Prandtl number,g= (-t)c is the Chemical Reaction, 0 t c Vw is the suction 3 xuw parameter,r= 6 s T 3kk* is the thermal radiation parameter and Rex is the local Reynolds number. v III. NUMERICAL METHOD FOR SOLUTION The transormed equations (8) - (0) subject to the boundary conditions () orm a nonlinear two-point boundary value problem, which has been solved numerically using the Runge-Kutta integration scheme with a modiied version o the Newton-Raphson shooting method. First o all, the higher order non-linear dierential equations (8) - (0) are converted into simultaneous linear dierential equation o irst order and they are urther transormed into initial value problem by applying the shooting technique. The resultant initial value problem is solved by employing Runge-Kutta ourth order method. The step size =0.00 is used to obtain the numerical solution with six decimal accuracy as criterion o convergence. The above mentioned third order and second order equations are written in terms o irst order equations asollows: F' z, z' p ' q p' ( A F ) p z A M z Pr A q' q 4 z Fq M Ec z R 3 ' r r' Sc ( A G ) z (F A) r 4 With boundary conditions F(0) 0, F(0), (0), (0) (5) In order to integrate equations ()-(4) as initial value problem we require a value or p (0) i.e. F (0) and q (0) i.e. (0) but no such values are given in the boundary. The suitable guess values or F (0) and (0) are chosen and Copyright to IJARSET 58

6 then integration is carried out. We take the series o values or F (0), (0) and apply the ourth order Runge-Kutta methodwith dierent step-sizes ( 0.0,0.00, etc. ) so that the numerical results obtained are independent o. The above procedure is repeated until we get the results up to the desired degree o accuracy0. IV. RESULTS AND DISCUSSION The computations have been carried out or various governing low parameters where some observations on this system can clearly be noticed. Eq. (8) is coupled to Eqs. (9) and (0) through the parameter A and so the velocity F' is not aected by the Eckert number Ec, the Schmidt number Sc and the chemical reaction parameter G. Also Eqs. (9) and (0) are coupled to each other only through the unsteadiness parameter A. Thus the temperature is not aected by Schmidt number Sc and the chemical reaction parameter G and the concentration is not aected by the Eckert number.the results are given or various values o the Prandtl numberpr, Schmidt number Sc, suction parameter 0, chemical reaction parameterg, Eckert number Ec, thermal radiation parameter R, mixed convection parameter, magnetic ield parametermand the unsteadiness A. The parameters eects on the velocity, temperature and concentration proiles are displayed graphically in Figures. The dimensionless velocity, temperature and concentration proiles or dierent values o unsteadiness parameter A with constant 0, Sc,Ec, Pr, R,,M and G are shown in Figs. to3 respectively. It is clear that the velocity, temperature andconcentration o the luid decreasing with the increase unsteadiness parameter. Figure 4 present the behavior o the temperature proiles various values o the Eckert number EcwithPr=0.7, G= -0., =0., M=, R=, 0 = 0, A=0. and Sc=.From this igure it is shown that the temperature o the luid decreases greatly with the increase o the Eckert number Ec. The eect o the chemical reaction parameter G with constant 0, sc, Ec, Pr, R,, M and A, and Schmidt number Sc with constant 0, G, Ec, Pr, R,, M and Aon the concentration o the luid is illustrated in Figures 5 and 6, it is obvious that the concentration o the luid decreases accordingly the chemical reaction parameter G and Schmidt number Sc increases. The velocity proiles, temperature proiles and concentration proiles or dierent values o the Prandtl number Prwith A=0., G= -0., =0., M=, R=, Ec=0.7, 0 = 0, and Sc=are depicted in Figs. 7 to 9. Fig 7 and 9 demonstrate that the velocity and concentration o the luid decreases with the increase o the Prandtl number Pr. Fig. 8Shows that the temperature o the luid increases with the increase o the Prandtl number Pr. Figs. 0 to presents velocity proiles, temperature proiles and concentration proiles or various values o magnetic ield parameter M when Pr=0.7, G= -0., =0., A=0., R=, Ec=0.7, 0 = 0,and Sc=. The velocityo the luiddecreases with increasing values o magnetic ield parameterm, while the temperature and concentration o the luid decreases withincreasing values o magnetic ield parameter M.The eect o the thermal radiation parameterrwith Pr=0.7, G= -0., =0., M=, A=0., Ec=0.7, 0 = 0,and Sc= on the velocity, temperature and concentration o the luid are illustrated in Figs. 3to5. It is ound that the velocity and temperature increases as the thermal radiation parameter Rincreases, this is agreement with the physical act that the thermal boundary layer thickness increases with increasing R.But the concentrationdecreases as thermal radiation parameter R increases. Figs.6-8 presents the behavior o the velocity, temperature and concentration o the luid proiles various values o the suction parameter 0 with Pr=0.7, G= -0., =0., M=, R=, Ec=0.7, A=0.,and Sc=. From thefig. 6it is shown that with increasing suction parameter, velocity o the luid is ound to decrease, i.e. suction causes to decrease the velocity o the luid in the boundary layer region. This eect acts to decrease the wall shear stress. Increase in suction causes progressive thinning o the boundary layer. The physical explanation or such a behavior is as ollows; in case o suction the heated luid is pushed towards the wall where the buoyancy orces can act to retard the luid due to high inluence o the viscosity. This eect acts to decrease the wall shear stress. From the Figs.7 and 8 thetemperature and concentration in the boundary layer are also decreases with increasing suction parameter ( 0 > 0).The thermal boundary thickness decreases with suction parameter ( 0 > 0), which causes an 6 Copyright to IJARSET 58

7 increase in the rate o heat transer. The physical explanation or such a behavior is that the luid is brought closer to the surace and reduces the thermal boundary layer thickness in case o suction. Figs. 9 to presentsthe velocity, temperature and concentration proiles or various valueso mixed convection parameter λwhenpr=0.7, G= -0., A=0., M=, R=, Ec=0.7, 0 = 0and Sc=.Fig.9 The velocity increases with increasing values o mixed convection parameter λ,while rom Figs.0 and the temperature and concentration decreases with increasing values o mixed convection Parameter λ, at λ = 0 gives the result o orced convection case. F' A=0, 0., 0.,0.3, 0.5, Fig: Velocity proiles or various values o A with Pr=0.7, G= -0., =0., M=, R=, Ec=0.7, 0 = 0 and Sc=. Copyright to IJARSET 583

8 θ A=0, 0., 0.,0. 3, 0.5, Fig.: Temperature proiles or various values o A with Pr=0.7, G= -0., =0., M=, R=, Ec=0.7, 0 = 0 and Sc=. φ A=0, 0., 0.,0. 3, 0.5, Fig 3: Concentration proiles or various values o A with Pr =0.7, G= -0., =0., M=, R=, 0 = 0, Ec=0.7 and Sc=. Copyright to IJARSET 584

9 θ Ec =0.,,, 0.7, Fig.4: Temperature proiles or various values o Ec with Pr=0.7, G= -0., =0., M=, R=, 0 = 0, A=0. and Sc=. φ G= -0., 0, 0.,,.4, Fig.5: Concentration proiles or various values o G with Pr=0.7, A=0., =0., M=, R=, Ec=0.7, 0 = 0 and Sc=. Sc= 0., 0.3,,,.6,. Copyright to IJARSET 585

10 φ Fig.6: Concentration proiles or various values o Sc with Pr=0.7, G= -0., =0., M=, R=, Ec=0.7, 0 = 0 and A=0.. F' Pr=0, 0.7,,.4, Fig.7: Velocity proiles or various values o Pr with A=0., G= -0., =0., M=, R=, Ec=0.7, 0 = 0 and Sc=. θ Pr=0, 0.7,,.4, Fig.8: Temperature proiles or various values o Pr with A=0., G= -0., =0., M=, R=, Ec=0.7, 0 = 0 and Sc=. Copyright to IJARSET 586

11 φ Pr=0, 0.7,,.4, Fig.9: Concentration proiles or various values o Pr with A=0., G= -0., =0., M=, R=, Ec=0.7, 0 = 0 and Sc=. F' M=0,0.,, Fig.0: Velocity proiles or various values o M with Pr=0.7, G= -0., =0.,A=0., R=, Ec=0.7, 0 = 0 and Sc=. Copyright to IJARSET 587

12 θ M=0,0.,, Fig.: Temperature proiles or various values o M with Pr=0.7, G= -0., =0.,A=0., R=, Ec=0.7, 0 = 0 and Sc=. φ M=0,0.,, Fig.: Concentration proiles or various values o M with Pr=0.7, G= -0., =0.,A=0., R=, Ec=0.7, 0 = 0 and Sc=. Copyright to IJARSET 588

13 F' R= 0,,,3,5, Fig.3: Velocity proiles or various values o R with Pr=0.7, G=0.= -0., =0., M=, A=0., Ec=0.7, 0 = 0, and Sc=. θ R= 0,,,3,5, Fig.4: Temperature proiles or various values o R with Pr=0.7, G= -0., =0., M=, A=0., Ec=0.7, 0 = 0 and Sc=. Copyright to IJARSET 589

14 φ R= 0,,,3,5, Fig.5: Concentration proiles or various values o R with Pr=0.7, G= -0., =0., M=,A=0., Ec=0.7, 0 = 0 and Sc=. F' 0 = 0,,.5,, 5, Fig.6: Velocity proiles or various values o 0 with Pr=0.7, G= -0., =0., M=, R=, Ec=0.7, A=0. and Sc=. Copyright to IJARSET 590

15 θ 0 = 0,,.5,, 5, 7 0. Fig.7: Temperature proiles or various values o 0 with Pr=0.7, G= -0., =0., M=, R=, Ec=0.7, A=0. and Sc= φ 0 = 0,,.5,, 5, Fig.8: Concentration proiles or various values o 0 with Pr=0.7, G= -0., =0., M=, R=, Ec=0.7,A=0. and Sc=. Copyright to IJARSET 59

16 F' =-0.,0,0.,0.3, Fig.9: Velocity proiles or various values o with Pr=0.7, G= -0., A=0., M=, R=, Ec=0.7, 0 = 0 and Sc=. θ =-0.,0,0.,0.3, Fig.0: temperature proiles or various values o with Pr=0.7, G= -0., A=0., M=, R=, Ec=0.7, 0 = 0 and Sc=. Copyright to IJARSET 59

17 φ =-0.,0,0.,0.3, Fig: Concentration proiles or various values o with Pr=0.7, G= -0., A=0., M=, R=, Ec=0.7, 0 = 0 and Sc=. V.CONCLUSION The purpose o the study in this paper is to present numerical solutions o unsteady mixed convection low and heat transer over a porous stretching surace taking into consideration, the eects o unsteadiness parameter A, mixed convection parameter λ, mixed concentration parameter λ, chemical reaction G, Eckert number Ec, Schmidt number Sc, magnetic parameter M, radiation parameter R, suction parameter 0 and Prandtl number Pr. With the help o similarity transormations, the governing time dependent boundary layer equations or momentum and thermal are reduced tocouple ordinary dierential equations which are then solved numerically using shooting method. The numerical solution indicated that: - The velocity increases with an increase in the value o mixed convection parameter λ and radiation parameter R while decreases with increase o magnetic parameter M, Prandtl number Pr, suction parameter 0 and unsteadiness parameter A. - The temperature decreases with an increase in the value o mixed convection parameter λ, Prandtl number Pr, suction parameter 0 and unsteadiness parameter Awhile increases with increase o Eckert number Ec,magnetic parameter M and radiation parameter R. 3- The concentration increases with an increase in the value o magnetic parameter M and Prandtl number Pr while decreases with increase ounsteadiness parameter A, chemical reaction G, Schmidt number Sc, suction parameter 0,convection parameter λ and radiation parameter R. REFERENCES []. B. Gebhart, Y. Jaluria, R.L. Mahajan, B. Sammakia,Buoyancy,Induced Flows and Transport, Springer-Verlag,Berlin, 988. []. A.M. Rashad, Eects o radiation and variable viscosity on unsteady MHD low o a rotating luid rom stretching suracein porous medium, Journal o the Egyptian Mathematical Society (04), [3]. S.M.M. EL-Kabeir, A.M. Rashad,Rama Subba Reddy Gorla,Unsteady MHD combined convection over a moving vertical sheet in a luid saturated porous medium with uniorm surace heat lux, Mathematical and Computer Modelling 46 (007) [4]. E. M. Abo-Eldahab, Ali A. Hallool and M.M. El-Kady, Eects oheat Transer on MHD Flow Over anunsteady Stretching Surace inamicropolarfluid and a Porous Medium with PrescribedSurace HeatFlux,Int.J.o Appl. Mathematics and Physics,4(), (0): [5]. E.M.A. Elbashbeshy and M.A.A. Bazid. Heat transer over anunsteady stretching surace. Heat Mass Transer,4(004):-4. [6]. M.A. EL-Hakiem, S.M.M. EL-Kabeir, A.M. Rashad, Group method analysis or the eect o radiation on MHD coupled heat and mass transer natural convection low or water vapor over a vertical cone through porous medium, Int. J. Appl. Math. Mech. 3 () (007) Copyright to IJARSET 593

18 [7]. E. M.A.Elbashbeshy, D.A.Aldawody,Eects o Thermal Radiationand Magnetic Field on Unsteady MixedConvection Flow and HeatTranser over a Porous stretching Surace,Int. J. o NonlinearScience 9(00)4, [8]. M.A. EL-Hakiem, A.M. Rashad, Eect o radiation on nondarcy ree convectionrom a vertical cylinder embedded in a luid-saturated porous medium with a temperature-dependent viscosity, J. Porous Media 0 () (007) [9].J.D. Goddard, A. Acrivos, Quart.,Analyses o laminar orcedconvection mass transer with homogeneous chemical reaction.mech. Appl. Math. 0(967) [0]. Lakshmisha KN, Venkateswaran S, Nath G., three dimensional unsteady lows with heat and mass transer cover a continuous stretching surace, J. HeatTranser Vol.0, pp , 988. []. Gorla RSR. Sidawi I., Free convection on a vertical stretching surace with suction and blowing, Appl. Sci. Res.Vol.5, pp.47 57, 994. []. Chamkha AJ (999) Hydromagnetic three-dimensional ree convection on avertical stretching surace with heat generation or absorption, Int. J. Heat Fluid Flow Vol.0, pp.84 9, 999. [3]. N.T.M. Eldabe, Sallam N. Sallam, Mohamed Y. Abou-zeid,Numerical study o viscous dissipation eect on ree convection heat and mass transer o MHD non-newtonian luid low through a porous medium, J.Egyptian Math. Soc. 0 (0)39-5. [4]. P. Chandran, N. C. Sacheti and A. K. Singh. Hydro-magnetic lowand heat transer past a continuously moving porousboundary.int.commun. Heat Mass Transer., 3(996): [5]. I. Pop and T. Y. Na. A note on MHD over a stretching permeablesurace. Mech. Res.Commun., 5(998): [6]. S. Mukhopadhyay, G. C. Layek and Sk. A. Samad. Study o MHDlow over a heated stretching sheet with variableviscosity. Int.J.Heat Mass Transer., 48(005): [7]. H. I. Andersson, K. H. Bech and B. S. Dandapat. MHD low o a power law luid over a stretching sheet. Int. J.Non-Linear Mech.,7(99): [8]. C. H. Chen. Eects o magnetic ield and suction/injection onconvection heat transer o non-newtonian power lawluid past astretched sheet with surace heat lux. Int. J. Therm. Sci., 47(008): [9]. EmadM.Abo El-dahab,M.Abd El-aziz,Ali A.Gaber, Joule heating eect on hydrodymagnetic three dimensional low over a stretching surace with heat and mass transer,int. J.Appl.Math.andPhys.(008),7-38. [0]. E. M. Abo El-dahab, M. M. El-Kady and Ali A. Hallool, Eects oheat and Mass Transer on MHDFlow Over a Vertical StretchingSurace with Free Convection and Joule-Heating, Int. J. o Appl. Math. and Physics, 3() (0): Copyright to IJARSET 594