 World Journal of Mechanics, 2013, 3, 203-214 doi:10.4236/wjm.2013.34020 Published Online July 2013 (http://www.scirp.org/journal/wjm) Thermal Radiation, Heat Source/Sink and Work Done by Deformation Impacts on MHD Viscoelastic Fluid over a Nonlinear Stretching Sheet F. M. Hady1, R. A. Mohamed2, Hillal M. ElShehabey2,3* 1Mathematics Department, Faculty of Science, Assiut University, Assiut, Egypt 2Mathematics Department, Faculty of Science, South Valley University, Qena, Egypt 3Institute of Mathematics and Scientific Computing, University of Graz, Graz, Austria Email: happliedmath@yahoo.com Received December 16, 2012; revised February 16, 2013; accepted February 23, 2013 Copyright © 2013 F. M. Hady et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT This work is focused on the effects of heat source/sink, viscous dissipation, radiation and work done by deformation on flow and heat transfer of a viscoelastic fluid over a nonlinear stretching sheet. The similarity transformations have been used to convert the governing partial differential equations into a set of nonlinear ordinary differential equations. These equations are then solved numerically using a very efficient implicit finite difference method. Favorable comparison with previously published work is performed and it is found to be in excellent agreement. The results of this parametric study are shown in several plots and tables and the physical aspects of the problem are highlighted and discussed. Keywords: Flow and Heat Transfer; Second Grade Fluid; Nonlinear Stretching Sheet; Heat Source; Radiation 1. Introduction The study of fluids is different to that of solids as there are differences in physical structure of fluids and solids. The nature of fluids and the way of study are two aspects of fluid mechanics which make it different to solid me- chanics. Furthermore, the fluids are categorized to New- tonian and non-Newtonian fluids. The fluids of low mo- lecular weight fall into the Newtonian class and are com- pletely characterized by the Navier-Stokes theory. There is a large variety of materials such as geological materi- als, liquid foams, polymeric liquids and food products etc. which are capable of flowing but which exhibit flow characteristic that cannot be adequately described by the Navier-Stokes theory. This inadequacy of the Navier- Stokes theory has led to the development of several theo- ries of non-Newtonian fluids. Unlike Navier-Stokes fluids, there is not a single mo- del which can completely describe all the properties of the non-Newtonian fluids. They cannot be described in a single model as for Newtonian fluids and there has been much confusion over the classification of non-Newtonian fluids. There are many models describing the properties but not all of non-Newtonian fluids. These models or constitutive equations, however, cannot describe all the behaviors of non-Newtonian fluids, e.g., the normal stress relaxation, the elastic effects, and the memory effects. The constitutive equations describing the behaviors of non-Newtonian fluids are more complicated and non lin- ear than those of Newtonian fluids. As non-Newtonian fluid model Rivlin-Ericksen fluids gained much acceptance from both theorists and experi- menters. The special cases of the model, which is the fluid of second grade, are extensively used and a lot of works have been done on the subject. These investiga- tions have been for non-Newtonian fluids of the differen- tial type . In the case of fluids of differential type, the equations of motion are an order higher than the Navier- Stokes equations, and thus the adherence boundary con- dition is insufficient to determine the solution completely [2-4] for a detailed discussion of the relevant issues. The same is also true for the approximate boundary layer ap- proximations of the equations of motion. In the absence of a clear means obtaining additional boundary condi- tions, Beard and Walters , in their study of an incom- pressible fluid of second grade, suggested a method for overcoming this difficulty. They suggested a perturbation approach in which the velocity and the pressure fields were expanded in a series in terms of a small parame- *Corresponding author. Copyright © 2013 SciRes. WJM F. M. HADY ET AL. 204 ter ε. Danberg and Fansler  studied the solution for the boundary layer flow past a wall that is stretched with a speed proportional to the distance along the wall. Ra- jagopal et al.  independently examined the same flow as in  and obtained similarity solutions of the bound- ary layer equations numerically for the case of small vis- coelastic parameter λ. It is shown that skin-friction de- creases with increase in λ. Dandapat and Gupta  ex- amined the same problem with heat transfer, where Hady and Gorla  studied the effect of uniform suction or injection on flow and heat transfer from a continuous surface in a parallel free stream of viscoelastic second- order fluid. The effect of radiation on viscoelastic boun- dary-layer flow and heat transfer problems can be quite significant at high operating temperature. Very recently, researches in these fields have been conducted by many investigators [10-14]. On the other hand, another physical phenomenon is the case in which the sheet stretched in a nonlinear fashion. On this domain, Mahdy and Elshehabey  studied the flow and heat transfer in a viscous fluid over a nonlinear stretching sheet utilizing nanofluid where, effects of vis- cous dissipation and radiation on the thermal boundary layer over a nonlinearly stretching sheet were studied by Cortell . Vajravelu  studied viscous flow over a nonlinearly stretching sheet, where viscous flow and heat transfer over a nonlinearly stretching sheet were obtained by Cortell  then, series solution of flow over non- linearly stretching sheet with chemical reaction and mag- netic field was investigated by employing the Adomian decomposition method by Kechil and Hashim  where, Ziabakhsh et al.  used homotopy analysis method to present flow and diffusion of chemically reactive species over a nonlinearly stretching sheet immersed in a porous medium. Muhaimin et al.  studied the effect of che- mical reaction, heat and mass transfer on nonlinear boun- dary layer past a porous shrinking sheet in the presence of suction and, Robert  discussed high-order nonlin- ear boundary value problems admitting multiple exact solutions with application to the fluid flow over a sheet. Cortell  studied heat and fluid flow due to non-line- arly stretching surfaces where, existence and uniqueness results for a nonlinear differential equation arising in vis- cous flow over a nonlinearly stretching sheet were ob- tained by Robert et al. . Finally, Vajravelu et al.  studied the diffusion of a chemically reactive species of a power-law fluid past a stretching surface. In this paper, as motivated by the previous studies and the study of Cortell  which investigated the effects of heat source/sink, radiation and work done by deforma- tion on flow and heat transfer of a viscoelastic fluid over a stretching sheet, we consider viscoelastic fluid with an- other physical phenomenon in which the sheet stretched in a nonlinear fashion. Also, the effects of work due to deformation on viscoelastic flows and heat transfer in the presence of radiation, viscous dissipation and heat source/ sink have been studied. 2. Problem Formulation Consider a steady two-dimensional flow of an incom- pressible second grade fluid through a porous medium over a wall coinciding with the plane , the flow being confined to . Two equal and opposite forces are applied along the x-axis so that the wall is stretched keeping the origin fixed. Thus, the basic boundary layer equations, governing the flow and heat transfer in pres- ence of radiation, with a temperature-dependent heat source/sink in the flow region, viscous dissipation, and taking into account the work due to deformation, are given in usual notation by 0y0y0,uvxy (1) 22002223123,Buu uuv uxy yuuu uuvxyxyyy    (2) 2221(1.rpppTTuvxyqQTTTucy cycyuuuuvcyy xy)p     (3) where, the power-law heat flux on the wall is considered in the form 0,,at 0,nwswwux bxvvTqk Dxyy  (4) 0,0,as ,uuTTy  (5) where ,xy denotes the Cartesian coordinates along the sheet and normal to it, u and v are the velocity com- ponents of the fluid in the x and directions, re- spectively, b and n are parameters related to the surface stretching speed, v is the kinematic viscosity, y is the thermal diffusivity, pc is the specific heat at constant pressure, r is the radiative heat flux and Q the volu- metric rate of heat generation/absorption. The radiative heat flux term by using the Rosseland approximation is given by  q44,3rTqyk  (6) Copyright © 2013 SciRes. WJM F. M. HADY ET AL. 205sidwhere and are the Stefanand the mean abs rption coefficiation (3) reduces to k o-Boltzmann constant ent, respectively. We can coner that the temperature differences within the flux are sufficiently small such that the term 4T may be expressed as a linear function of temperature by expand- ing 4T in a Taylor series about T and neglecting higher-order terms we get  443 34.443 .TTTT TTT  (7) using Equations (6) and (7), EquT23221163,ppppTTuvxyTTucyck yQT Tuuuuvccyyxy     (8) where it can be seen that the effect of radiation is to en- hance the thermal diffusivity. Defining the following dimensionless function u, v and g, which related to the similarity variable  as 12,2nsTTgDvxx (9) 1sbn1211,21nbn nvxfn  f (10) where,  is the free stream function that Equation (1) with satisfies ,,uvyx (11) Then we have the transformed momequations together with the boundary cEquentum and energy onditions given by ations (5) and (11) in the form 22211nMffff fnn 231 12022ivnnfff ff   (12) 5122233(2 1)34 34132134334311 022RRRRRRnsRcRNnNggfsn gfNNnNNgnNNExNnnffff ff     (13) If 51,2ns we find from (13) that 23343434 13213 4331 134 220RRRRRRRcRNNnggf gfNNnNNgnNNnnEf fffffN    (14) e transformThed boundary conditions are ,1,(0)1at00,0,0as fRf gffg  (15) Here the prime denotes differentiation with respect to the independent similarity variable Moreover, .is the viscoelastic parameter, 0201nBMbx 11nbx is the magnetic parameter, 0121nRv nbx isuction parameter, s the  is the Prandtl number, 212cpDc vbnkbE is the Eckert number, 1npQNcbx is the heat source/sink parameter, 34RkkNT is the radiation parameter and 03RRN01k34kN (with thermal radiation);  (with- n). The shear stress at the stretched surface is defined as out thermal radiatiowwuy, (16) using Equations ( 9) and (10) we have,  31210nbnbxf 2wv (17) 3. Results as- fer- rous stretching sheet, in the pris examined in this paper. Strdary, viscous dissipation, temperature dependent heat source/sink and thermal radiation are taken into consid- nd Discussion A boundary layer problem for momentum and heat tran in a viscoelastic fluid flow over a non-isothermal poesence of thermal radiation, etching of the porous boun- Copyright © 2013 SciRes. WJM F. M. HADY ET AL. Copyright © 2013 SciRes. WJM 206 dary layer partial differential equations, which are highly non-linear, have been converted into a set of nonlinear orditial equations by applying suitable similarity transforma- ns are obtained with a ements the three-stage mation is taken in account and this is also true for the second case, where ,,Rn and  have an opposite behavior. The plots in Fures 1(a) and (b) show the temera- ture distribution eration in this study. The basic bounnary differen- ig pg different values of the heat source/sink parameith two cases of the radiation; in the case of existence of the thermal radiation and absence of the thermal radiatio and whout work done by deformatioforr wn withvioupostions and their numerical solutiofinite difference code which implteLobatto IIIa formula is used to solve that system [29-31]. In order to verify the accuracy of the present numerical method, the results are compared with those reported earlier by  for the case of linear stretching sheet. The results of these comparisons are shown in Table 1. It can be seen from this table that excellent agreement between itn. It is obs that, the effect of increasing thope strength of the heat sink is to increase the temperature profile, and the ite behavior is seen for a heat source. In contrast of thermal radiation, the existence the work done by deformation is to decrease the temperature profile. From Figures 2(a) and (b), we can see that the effect of increasing values of Prandtl number is to decrease temperature at a point in the flow field, as there would be a thinning of the thermal boundary layer as a result of reduced thermal conductivity. On the other hand Figures 3(a) andthe results exists. This lends confidence to the numerical results. The effects of viscous dissipation, work due to defor- mation, internal heat generation/absorption and thermal radiation are considered in the energy equation and the variations of dimensionless surface temperature, dimen- sionless velocity profiles as well as the heat transfer cha- racteristics with various values of non-dimensional vis- coelastic parameter , nonlinear stretching parameter n, heat source/sink parameter N, magnetic parameter M, suction parameter R, Prandtl number , Eckert number cE and radiation parameter RN shown in Figures 1-10. Also, the values of wall temperature g(0) for various va (b), demonse effect non linear pa- ratrate thof the meter n and it is evident that, the temperature profile decreases with increasing the values of the non linear stretching parameter. Moreover, we can see the effect of increasing the Eckert number cE from Figu res 4(a) and (b) for the same cases disused in Figures 1(a) and (b), which is to increase the temperature distribution. Figures 5(a) and (b) depict the effect of the magnetic field M, by analyzing these graphs, we see that the ef- fect of increasing values of M is to increase the tem- perature distribution in the bodary layer. This is be- cause of the fact that the introduction of transverse mag- netic field to an electrically conducting fluid givelues of physical parameters are shown in Tables 2 and 3 with and without taking the work done by deformation at the energy equation, respectively. There are many re- sults which can be obtained from those tables. For more details, Increasing the heat source parameter N or, the Eckert number cE, or the magnetic parameter M, or the radiation paramete uns rise to a enf σrRN leads to an increasing in the wall temperature 0g when twork done by defor- resistive force, known as Lorentz force. This force has a tendcy to slow down the motion of the fluid in the , NR and Ec wi th n = 0.0 (Non lin ear stretching sheet). g(0) he Table 1. Comparison results of g(0) for various values oλ Ec NR σ Nβ Cortell  Present Result 0.2 0.02 1.0 3.0 0.05 0.671732 0.6548280 0.5 0.651127 0.6421121 0.2 0.4 0.757167 0.7573403 1 1 7 3. −05 0.−05 0.5 0.25 0.716127 0.7079965 0.0 0.646621 0.6494326 0.02 5.0 0.462286 0.4647484 8.0 0.441677 0.4443301 .0.01.291859 1.2864327 .00.406628 0.4087103 0 .2597791 0.5987765 .10.614045 0.6153823 0.00.641036 0.6440087 10.693218 0.6790583 F. M. HADY ET AL. 207Table 2. Wall temperature (0) with work dormation. g(0) gne by defoNσ n Ec M R λ iation β With Radiation Without Rad−08.0 1/3 0.02 1.0 1 0.15 4259812 5 .2 0.0.0.2648700. 0.4802939 0. 5710018 1 −04.0 6397695 0 7.0 0.2850470 10.0 0.3750452 0.2345641 1/3 0.0. 0.0. 0.0.1.1.0. 0.1.1.0.0. 0.0.0.0 2 0.2925684 0.3321800..2 0.0.390038 0.4600843 8.0 1.0 0.3496030 0.2987571 0.2190434 0.1891062 3.0 5.0 0.2848280 0.1811730 0 0.4116075 0.2478815 05 0.4475416 0.2903539 1 0.4834757 0.3328264 025 0.4139565 0.2572655 8 0.4214058 0.2619547 5 0.4364144 0.2716140 0 05 0.4438452 0.2808115 5 0.3132555 0.1747897 0 0.2270719 0.1181463 5 0.1753241 0.0886570 1 050.4271187 0.2653735 1 0.4265433 0.2651201 3 0.4243760 0.2641472 7 0.4206671 0.2624275 Table 3. Wall tempre gith wor by de g(0) eratu (0) wk doneformation.Nβ W Without Radiation σ n Ec M R λ ith Radiation−2 81/3 0. 1.0. 0..0 020 10.15 0.5083353 0.2642551 00−0.2 4.0 0.3894706 7.0 0.2844400 1 0.4613835 0.2339356 8.0 1.0 5.1/3 0.0. 0.0. 0.0.1.1.0. 0.1.1.0.0. 0.0.0..0 0.5798098 0.2918004 .2 0.7017604 0.3311588 0.7102198 0.5400828 0.0 0.4018430 0.2172851 3.0 0.3274262 0.1856562 0 0.3050635 0.1767095 0 0.4116075 0.2478815 05 0.6534269 0.2888155 1 0.8952464 0.3297495 025 0.4755726 0.2568651 8 0.4957535 0.2614253 5 0.5374969 0.2707835 0 05 0.5272657 0.2803169 5 0.3862716 0.1732892 0 0.2889979 0.1157326 5 0.2281186 0.0854866 1 050.5126295 0.2651608 1 0.5023476 0.2629805 3 0.4888003 0.2600536 7 0.5083353 0.2642551 Copyright © 2013 SciRes. WJM F. M. HADY ET AL. Copyright © 2013 SciRes. WJM 208 00.5 11.5 22.5000.8.20.40.6 ( a )g (  )00.5 11.5 22.500.20.40.60.8 ( b )g (  )- - - - - Withermal radi0,- - - - - With thermal radiation NR = 1.0,.... Without thermal ........... Without thermal radiation. = 0.M = 1. = 8. = 0.15, R = 0.1,M = 1.0, n = 1/3, = 8.0, Ec = 0.02. thation NR = 1........ radiation.15, R = 0.1,0, n = 1/3,0, Ec = 0.02.N = 0.2, 0. -0.2.0,N = 0.2, 0.0, -0.2. Figure 1. Temperature profiles for several values of Nβ (a) with and (b) without work done by deformation. 00.5 11.5 22.500.10.20.30.40.5 ( a )g (  )00.5 11.5 22.500.10.20.30.40.50.6 ( b )g (  )- - - - - With thermal radiation NR = 1.0,- - - - - With thermal radiation NR = 1.0,........... Without thermal radiation. = 0.15, R = 0.1, = 0.15, R = 0.1,M = 1.0, n = 1/3,N = -0.2, Ec = 0.02............ Without thermal radiation. = 7.0, 8.0,10.0.M = 1.0, n = 1/3,N = -0.2, Ec = 0.02. = 7.0, 8.0,10.0. Figure 2. Temperature profiles for several values of σ (a) with and (b) without work done by deformation. boundary layer and to increase the temperature distribu- tion, where the effect of the suction parameter is plotted in Figures 6(a) and (b) and the effect of the viscoelastic parameter  is clearly shown in Figures 7(a) and (b) with all cases. Velocity profiles with for various values of 0.5,3.0nMis shown in Figure 8(a) where the same values is pln Figure 8(b) but fo Also, the same val of Figures 8(a) and (b) otted for different vaes of the viscoelastic paraFigures 9(a) and (b) We can see from those figufaster motion is red when the viscincreases otted iueslu. consider1.0.R are Plmeter in res that, a oelastic parameter F. M. HADY ET AL. 209 00.5 11.5 22.500.10.20.30.40.5 ( a )g (  )00.5 11.5 22.500.10.20.30.40.50.6 g (  )( b )- - -.... - -t- - - - - With thermal radiation NR = 1.0,............... Without thermal radiation. With thermal radia ion NR = 1.0,... Without thermal radiation. = 0.15, R = 0.1,M = 1.0,  = 8.0,N = -0.2, Ec = 0.02.n = 1/3, 1.0, 3.0.n = 1/3, 1.0, 3.0. = 0.15, R = 0.1,M = 1.0,  = 8.0,N = -0.2, Ec = 0.02. Figure 3. Temperature profiles for several values of n (a) with and (b) without work done by deformation. 00.5 11.5 22.500.10.20.30.40.5 ( a )g (  )00.5 11.5 22.500.20.40.60.81 ( b )g (  )- - - - - With thermal radiation NR = 1.0,- - - - - With thermal radiation NR = 1.0,........... Without thermal radiation............ Without thermal radiation. = 0.15, R = 0.1,M = 1.0,  = 8.0,N = -0.2, n = 1/3. = 0.15, R = 0.1,M = 1.0,  = 8.0,N = -0.2, n = 1/3.Ec = 0.1, 0.05, 0.0.Ec = 0.1, 0.05, 0.0. Figure 4. Temperature profiles for several values of Ec (a) with and (b) without work done by deformation. whereas it is slower when the suction parameter and magnetic parameter increase. Finally, Figure 10 shows variation of skin fraction coefficient against R for differ- ent values of ,,,Mn from which we can say that, with an increasing in the nonlinear stretching parameter or the viscoelastic parameter n t reprends to a decreasing the wall shear stress whichesented in terms of in0f profileas defined by Equatio7) i.e. the velocity increases, but the opposite effect is seen for the magnetic parameter M. n (1Copyright © 2013 SciRes. WJM F. M. HADY ET AL. 210 00.5 11.5 200.10.20.30.40.5 ( a )g (  )00.5 11.5 200.10.20.30.40.50.6 ( b )g (  )- - - - - With thermal radiation NR = 1.0,- - - - - With thermal radiation NR = 1.0,on............ Without thermal radiation. = 0.15, R = 0.1,n = 1/3,  = 8.0,N = -0.2, Ec = 0.02. = 0.15, R = 0.1,n = 1/3,  = 8.0,N = -0.2, Ec = 0.02............ Without thermal radiatiM = 1.5, 0.5.M = 1.5, 0.5.M ( Figure 5. Temperature profiles for several values of a) with and (b) without work done by deformation. 00.5 11.500.10.20.30.4 ( a )g (  )00.5 11.500.10.20.30.40.50.6 ( b )g (  )- - - - - With thermal radiation NR = 1.0,........... Without thermal radiation.- - - - - With thermal radiation NR = 1.0,........... Without thermal radiation.R = 0.05, 0.5, 1.5.R = 0.05, 0.5, 1.5. = 0.15, N = -0.2,M = 1.0, n = 1/3, = 8.0, Ec = 0.02. = 0.15, N = -0.2,M = 1.0, n = 1/3, = 8.0, Ec = 0.02. Figure 6. Temperature profiles for several values of R (a) with and (b) without work done by deformation. Copyright © 2013 SciRes. WJM F. M. HADY ET AL. 21100.5 11.500.10.20.30.40.5 ( a )g (  )00.5 11.500.10.20.30.40.50.6 ( b )g (  )- - - - - With thermal radiation NR = 1.0,- - - - - With thermal radiation NR = 1.0,........... Without thermal radiation............ Without thermal radiation. = 0.05, 0.7.R = 0.1, N = -0.2,M = 1.0, n = 1/3, = 8.0, Ec = 0.02.R = 0.1, N = -0.2,M = 1.0, n = 1/3, = 8.0, Ec = 0.02. = 0.05, 0.7. Figure 7. Temperature profiles for several values of λ (a) with and (b) without work done by deformation. 01 2 3 4 500.20.40.60.81 ( a )f ' (  )01 2 3 4 500.20.40.60.81 ( b )f ' (  )........... n = 3.0............ n = 3.0.R = 0.05,  = 0.15. M = 0.8,R = 1.0,  = 0.15. 1.5.M = 0.8, 1.5.- - - - - n = 0.5,- - - - - n = 0.5, Figure 8. Velocity profiles with n = 0.5, 3.0 for various values of M and R. Copyright © 2013 SciRes. WJM F. M. HADY ET AL. Copyright © 2013 SciRes. WJM 212 01 2 3 4 500.20.40.60.81 ( a )f ' (  )01 2 3 4 500.20.40.60.81 ( b )f ' (  )- - - - - n = 0.5,- - - - - n = 0.5,........... n = 3.0.R = 0.05, M = 1.0. R = 1.0, M = 1.0. ........... n = 3.0. = 0.3, 0.1. = 0.3, 0.1. Figure 9. Velocity profiles with n = 0.5, 3.0 for various values of λ and R. 00.5 11.511.522.5R- f '' ( 0 ) = 0.01 = 0.3M = 0.5.............. M = 1.0, n = 1/3,  = 0.2, M = 1.0 .- - - - - - -  = 0.2, n = 1/3,M = 1.0n = 3.0n = 0.2 Figure 10. Variation of skin fraction coefficient against R for different values of M, n, and λ. 4. Conclusions The effects of heat source/sink, radiation and work done by deformation on flow and heat transfer of a viscoelastic fluid over a nonlinear stretching sheet have been investi-gated. Numerical solutions for momentum and heat transfer are obtained. In the light of the numerical results the following conclusions may be drawn:  A faster motion is considered when the viscoelastic parameter or the nonlinear stretching parameter in- creases whereas it is slower when the suction pa- e magnetic force increase.  The effect of increasing values of magnetic parameter is to increase the temperature distribution in the boun- dary layer and so does the Eckert number.  The influence of the work due to deformation has sig- nificance effect of the temperature distribution.  The presence of the thermal radiation term in the en- ergy equation yields an augment in the fluid’s tem- perature.  The internal heat generation/absorption enhances or damps the heat transport. Finally, increasing suction rameter and th F. M. HADY ET AL. 213parameter or viscoelaber is to decrease the temperature distribution in the flow region. REFERENCES  A. C. Truesdell and W. Noll, “The Non-Linear Field Theories of Mechanics,” In: S. Flugge, Ed., Encyclopedia of Physics, Springer, Berlin, 1965, pp. 1-591.  K. R. Rajagopal, “On Boundary Conditions for Fluids of the Differential Type,” In: A. Sequeira, Ed., Navier— Stokes Equations and Related Non-linear Problems, Ple- num Press, New York, 1995, pp. 273-278.  K. R. Rajagopal and P. N. Kaloni, “Some Remarks on Boundary Conditions for Fluids of the Differential Type,” In: G. A. C. Graham and S. K. Malik , Eds., Continuum Mechanics and Its Applications, Hemisphere, New York, 1989, pp. 935-942.  K. R. Rajagopal and A. S. Gupta, “An Exact Solution for the Flow of a Non-Newtonian Fluid past an Infinite Plate,” Meccanica, Vol. 19, No. 2, 1984, pp. 158-160. doi:10.1007/BF01560464stic parameter or Prandtl num-  D. W. Beard and K. Walters, “Elastico-Viscous Boundary Layer Flows,” Proceedings of the Cambridge Philosophi- cal Society, Vol. 60, 1964, pp. 667-674. doi:10.1017/S0305004100038147  J. E. Danberg and K. SWall Boundary-LayerMathematics, Vol. 34, 1976, pp. 305-309.  K. R. Rajagopal, T. Y. Na and A. S. Gupta, “Flow of a Viscoelastic Fluid over a Stregica Acta, Vol. 23, No. 2, 1984, pp. 213-215. doi:10.1007/BF01332078. Fansler, “A Non Similar Moving Problem,” Quarterly of Applied tching Sheet,” Rheolo  B. S. Dandapat and A. S. Gupta, “Flow and Heat Transfer in a Viscoelastic Fluid over a Stretching Sheet,” Interna- tional Journal of Non-Linear Mechanics, Vol. 24, No. 3, 1989, pp. 215-219. doi:10.1016/0020-7462(89)90040-1  F. M. Hady and R. S. R. Gorla, “Heat Transfer from a Continuous Surface in a Parallel Free Stream of Viscoe- lastic Fluid,” Acta Mechanica, Vol. 128, No. 3, 1998, pp. 201-208. doi:10.1007/BF01251890  C. I. Cookey, A. Ogulu and V. B. Omubo-Pepple, “In- fluence of Viscous Dissipation and Radiation on Un- steady MHD Free-Convection Flow past an Infinite Heated Vertical Plate in a Porous Medium with Time-Dependent Suction,” Infer, Vol. 46, No. 13, 2003, pp. 2305-2311. 17-9310(02)00544-6ternational Journal of Heat and Mass Trans- doi:10.1016/S00  M. Kumari and G. Nath, “Radiation Effect on Mixed Convection from a Horizontal Surface in a Porous Me- dium,” Mechanics Research Communications, Vol. 31, No. 4, 2004, pp. 483-491. doi:10.1016/j.mechrescom.2003.11.006  M. A. Abd El-Naby, E. M. E. Elbarbary and N. Y. Abde- lazem, “Finite Difference Solution of Radiation Effects on MHD Unsteady Free-Convection Flow over Vertical Porous Plate,” Applied Mathematics and Computation, Vol. 151, No. 2, 2004, pp. 327-346. doi:10.1016/S0096-3003(03)00344-8  S. Abel, K. V. Prasad and A. Mahaboob, “Bouyancy Force and Thermal Radiation Effects in MHD Boundary- scoelastic Fluid Flow over Continuously Moving Stretching Surface,” International Journal of Thermal Sciences, Vol. 44, No. 5, 2005, pp. 465-476. doi:10.1016/j.ijthermalsci.2004.08.005Layer Vi  S. K. Khan, “Heat Transfer in a Viscoelastic Fluid Flow over a Stretching Surface with Heat Source/Sink, Suction/ Blowing and Radiation,” International Journal of Heat and Mass Transfer, Vol. 49, No. 3-4, 2006, pp. 628-639. doi:10.1016/j.ijheatmasstransfer.2005.07.049  Mahdy and H. M. ElShehabey, “Uncertainties in Physical Property Effects on Viscous Flow and Heat Transfer over a Nonlinearly Stretching Sheet with Nanofluids,” Inter- national Journal of Heat and Mass Transfer, Vol. 39, No. 5, 2012, pp. 713-719. sstransfer.2012.03.019doi:10.1016/j.icheatma  R. Cortell, “Effects of Viscous Dissipation and Radiation on the Thermal Boundary Layer over a Nonlinearly Stretching Sheet,” Physics Letters A, Vol. 372, No. 13, 2008, pp. 631-636. doi:10.1016/j.physleta.2007.08.005  K. Vajravelu, “Viscous Flow over a Nonlinearly Stretch- ing Sheet,” Applied Mathematics and Computation, Vol. 124, No. 3, 2001, pp. 281-288. doi:10.1016/S0096-3003(00)00062-X eat Transfer over a Non- lied Mathematics and Com- putation, Vol. 184, No. 2, 2007, pp. 864-873. doi:10.1016/j.amc.2006.06.077 R. Cortell, “Viscous Flow and Hlinearly Stretching Sheet,” App  S. Awang Kechil and I Hashim, “Series Solution of Flow over Nonlinearly Stretching Sheet with Chemical Reac- tion and Magnetic Field,” Physics Letters A, Vol. 372, No. 13, 2008, pp. 2258-2263. doi:10.1016/j.physleta.2007.11.027  Z. Ziabakhsh, G. Domairry, H. Bararnia and H. Baba- zadehtical Solution of Flow and Diffusion of Cally Reactive Species over a Nonlinearly Stretch- ing Sheet Immersed in a Porous Medium,” Journal of the Taiwan Institute of Chemical Engineers, Vol. 41, No. 1, 2010, pp. 22-28. doi:10.1016/j.jtice.2009.04.011, “Analyhemic  Muhaimina, R. Kandasamya and I. Hashimb, “Effect of Chemical Reaction, Heat and Mass Transfer on Nonlinear Boundary Layer past a Porous Shrinking Sheet in the Presence of Suction,” Nuclear Engineering and Design, Vol. 240, No. 5, 2010, pp. 933-939. doi:10.1016/j.nucengdes.2009.12.024  R. Gorder, “High-Order Nonlinear Boundary Value Prob- lems Admitting Multiple Exact Solutions with Applica- tion to the Fluid Flow over a Sheet,” Applied Mathemat- ics and Computation, Vol. 216, No. 7, 2010, pp. 2177- 2182. doi:10.1016/j.amc.2010.03.053 3] R. Cortell, “Heat and Fluid Flow Due to N[2 on-Linearly Stretching Surfaces,” Applied Mathematics and Compu- tation, Vol. 271, 2011, pp. 7564-7564. doi:10.1016/j.amc.2011.02.029  R. Gorder, K. Vajravelu and F. T. Akyildiz, “Existence and Uniqueness Results for a Nonlinear Differential Copyright © 2013 SciRes. WJM F. M. HADY ET AL. Copyright © 2013 SciRes. WJM 214 Equation Arising in Viscous Flow over a Nonlinearly Stretching Sheet,” Applied Mathematics Letters, Vol. 24, No. 2, 2011, pp. 238-242. doi:10.1016/j.aml.2010.09.011  K. Vajravelu, K. V. Prasad and N. S. Prasanna, “Diffu- sion of a Chemically Reactive Species of a Power-Law Fluid past a Stretching Surface,” Computers & Mathe-matics with Applications, Vol. 62, No. 1, 2011, pp. 93-108. doi:10.10 16/j.camwa.2011.04.055  R. Cortell, “Effects of Heat Source/Sink, Radiation and Work Done by Deformation on Flow and Heat Transfer of a Viscoelastic Fluid over a Stretching Sheet,” Com- puters & Mathematics with Applications, Vol. 53, No. 2, 2007 pp. 305-316. doi:10.1016/j.camwa.2006.02.041  P. S. Datti, K. V. Prasad, M. Subhas Abel and A. Joshi, “MHD Viscoelastic Fluid Flow over a Non-Isothermal Stretching Sheet,” International Journal of Engineering Science, Vol. 42, No. 8-9, 2004, pp. 935-946. doi:10.1016/j.ijengsci.2003.09.008  M. M. Rahman and T. Sultana, “Radiative Heat Transfer Flow of Micropolar Fluid with VariaPorous Medium,” Nonlinear Analysble Heat Flux in a is: Modelling and Control, Vol. 13, No. 1, 2008, pp. 71-87  F. M. Hady, F. S. Ibrahim, H. M. El-Hawary and A. M. AbdElhady, “Forced Convection Flow of Nanofluids past Power Law Stretching Horizontal Plates,” Applied Mathe- matics, Vol. 3, No. 2, 2012, pp. 121-126. doi:10.4236/am.2012.32019  L. F. Shampine, M. W. Reichelt and J. Kierzenka, “Solv-ing Boundary Value Problems for Ordinary Differential Equations in MATLAB with bvp4c.” http://www.mathworks.com/bvp_tutorial L. F. Shampine, I. Gladwell and S. Thompso n, “Solving ODEs with MATLAB,” Cambridge University Press, Cambridge, 2003. doi:10.1017/CBO9780511615542