International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 64...
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 64...
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 64...
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 64...
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 64...
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 64...
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 64...
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 64...
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 64...
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 64...
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Effect of non uniform heat source for the ucm fluid over a stretching sheet wit-2

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Effect of non uniform heat source for the ucm fluid over a stretching sheet wit-2

  1. 1. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 6, September – October (2013), © IAEME 40 EFFECT OF NON-UNIFORM HEAT SOURCE FOR THE UCM FLUID OVER A STRETCHING SHEET WITH MAGNETIC FIELD Anand H. Agadi1* , M. Subhas Abel2 , Jagadish V. Tawade3 and Ishwar Maharudrappa4 1 Department of Mathematics, Basaveshwar Engineering College, Bagalkot-587102, INDIA 2 Department of Mathematics, Gulbarga University, Gulbarga- 585 106, INDIA 3 Department of Mathematics, Bheemanna Khandre Institute of Technology, Bhalki-585328 4 Department of Mathematics, Basaveshwar Engineering College, Bagalkot-587102. ABSTRACT This article is concerned effect of Non-uniform heat source for the UCM fluid over a stretching sheet with the combined effect of external magnetic field and non-uniform heat source/sink. By means of similarity transformations, the non-linear equations governing the flow are reduced to an ordinary differential equation using similarity transformations. These equations are solved numerically by using standard fourth order Runge–Kutta method with the given set of parameters. The results are compared with the earlier published results and our results are better in agreements under some limiting cases. The effect of several parameters controlling the velocity and temperature profiles are shown graphically and discussed briefly. Key words: Boundary layer, Elastic Parameter, Eckert number, Maxwell fluid, Magnetic parameter, Non-uniform heat source. 1. INTRODUCTION It is generally recognized that rheological properties of material are specified by their constitutive equations. Recently, non-Newtonian fluids have been receiving a great deal of research focus and interest due to their engineering applications in a number of processes. The familiar examples are the extrusion of polymer fluids, solidification of liquid crystals, animal bloods, exotic lubricants and colloidal and suspension solutions. Because of the complexity of these fluids, there is not a single constitutive equation which exhibits all properties of non-Newtonian fluids. A steady INTERNATIONAL JOURNAL OF ADVANCED RESEARCH IN ENGINEERING AND TECHNOLOGY (IJARET) ISSN 0976 - 6480 (Print) ISSN 0976 - 6499 (Online) Volume 4, Issue 6, September – October 2013, pp. 40-49 © IAEME: www.iaeme.com/ijaret.asp Journal Impact Factor (2013): 5.8376 (Calculated by GISI) www.jifactor.com IJARET © I A E M E
  2. 2. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 6, September – October (2013), © IAEME 41 two-dimensional laminar flow of an incompressible, electrically conducting MHD Visco-elastic liquid (Walter’s liquid B model) due to a stretching sheet is considered. Fig 1. Schematic of the two-dimensional stretching sheet problem The sheet lies in the plane y = 0 with the flow being confined to y > 0. The coordinate x is being taken along the stretching sheet and y is normal to the surfaced, two equal and opposite forces are applied along the x-axis, so that the sheet is stretched, keeping the origin fixed. Under the boundary layer approximation and the assumption that the contribution due to the normal stress is of the same order of magnitude as the shear stress. The pioneering work due to stretching sheets is done by Sakiadis ([1,2]), Sarpakaya ([3]) was the first researcher to study the MHD flow a of non-Newtonian fluid. Prandtl’s boundary layer theory proved to be of great use in Newtonian fluids as Navier-Stokes equations can be converted into much simplified boundary layer equation which is easier to handle. Crane ([4]) was the first among others to consider the steady two-dimensional flow of a Newtonian fluid driven by a stretching elastic flat sheet which moves in its own plane with a velocity varying linearly with the distance from a fixed point. Subsequently, various aspects of the flow and/or heat transfer problems for stretching surfaces moving in the finite fluid medium have been explored in many investigations, (e.g. Refs. Dutta et al[5], Chakrabarti et al[6], M.S.Abel et al[7]). Extrusion of molten polymers through a slit die for the production of plastic sheets is an important process in polymer industry. In a typical sheet production process the extrudate starts to solidify as soon as it exits from the die. The sheet is then brought into a required shape by a wind-up roll upon solidification (see Fig. 1). An important aspect of the flow is the extensibility of the sheet which can be employed effectively to improve its mechanical properties along the sheet. To further improve sheet mechanical properties, it is necessary to control its cooling rate. Physical properties of the cooling medium, e.g., its thermal conductivity, can play a decisive role in this regard. The success of the whole operation can be argued to depend also on the rheological properties of the fluid above the sheet as it is the fluid viscosity which determines the (drag) force required to pull the sheet. Boundary Layer 0 B 0B 0B 0B Stretching sheet y Slit x Force
  3. 3. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 6, September – October (2013), © IAEME 42 Problems involving fluid flow over a stretching sheet can be found in many manufacturing processes such as polymer extrusion, wire and fiber coating, foodstuff processing, etc. Essentially, the quality of the final product depends on the rate of cooling in the process which is significantly influenced by the fluid flow and heat transfer mechanism. Water is amongst the most-widely used fluids to be used as the cooling medium. However, the rate of cooling achievable with water is often realized to be too excessive for certain sheet materials. To have a better control on the rate of cooling, in recent years it has been proposed that it might be advantageous for water to be made more or less viscoelastic, say, through the use of polymeric additives ([9]). The idea is to alter flow kinematics in such a way that it leads to a slower rate of solidification with the price being paid that fluid’s viscosity is normally increased by such additives. The radiative heat transfer properties of the cooling medium may also be manipulated to judiciously influence the rate of cooling ([10,11]). In recent years, MHD flows of viscoelastic fluids above stretching sheets (with and without heat transfer involved) has also been addressed by various researchers (Pahlavan et al [12], Renardy [13], Rao and Rajgopal [14], Pahlavan and Sadeghy [15]). Although there is no doubt about the importance of the theoretical studies cited above, but they are not above reproach. For example, the viscoelastic fluid models used in these works are simple models such as second-order model and/or Watler’s B model which are known to be good only for weakly elastic fluids subject to slow and/or slowly-varying flows (Pahlavan and Sadeghy [15]). To this should be added the fact that these two fluid models are known to violate certain rules of thermodynamics (Aliakbar[16]). A non-Newtonian second grade fluid does not give meaning full results for highly elastic fluids (polymer melts) which occur at high Deborah numbers (Cibeci [17] and Rajgopal [18]). Therefore, the significance of the results reported in the above works are limited, at least as far as polymer industry is concerned. Obviously, for the theoretical results to become of any industrial significance, more realistic viscoelastic fluid models such as upper-convected Maxwell model or Oldroyd-B model should be invoked in the analysis. Indeed, these two fluid models have recently been used to study the flow of viscoelastic fluids above stretching and non-stretching sheets but with no heat transfer effects involved (Sadegy et. al [11], Pahlavan[12] and Renardy[13]). Motivated by all the above, in this study, the MHD flow of UCM fluid over a stretching sheet with the combined effects of Magnetic field and non-uniform heat source is numerically studied using Runge-Kutta fourth order method with efficient shooting technique. The effects various parameters of flow and heat transfer coefficients are shown through several plots. It is shown that the heat fluxes from the liquid to the elastic sheet decreases with S for Pr 0.1≤ and increases with S forPr 1≥ . The important observation in this study is that, the non-uniform heat sink is one better suited for effective cooling purpose as heat source enhance the temperature in the boundary layer. On the other hand it is disclosed that large values of elastic parameter β increase the magnitude of the skin friction coefficient. 2. MATHEMATICAL FORMULATION The equations governing the transfer of heat and momentum between a stretching sheet and the surrounding fluid (see fig.1) can be significantly simplified if it can be assumed that boundary layer approximations are applicable to both momentum and energy equations. Although this theory is incomplete for viscoelastic fluids, but has been recently discussed by Renardy [13], it is more plausible for Maxwell fluids as compared to other viscoelastic fluid models. For MHD flow of an incompressible Maxwell fluid resting above a stretching sheet, the equations governing transport of heat and momentum can be written as Pahlavan and Sadeghy [15].
  4. 4. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 6, September – October (2013), © IAEME 43 0, u v x y ∂ ∂ + = ∂ ∂ (1) 22 2 2 2 2 2 0 2 2 2 2 , Bu u u u u u u v u v uv u x y y x y x y σ υ λ ρ  ∂ ∂ ∂ ∂ ∂ ∂ + = − + + − ∂ ∂ ∂ ∂ ∂ ∂ ∂  (2) 2 2 . p p T T k T q u v x y C y Cρ ρ ′′′∂ ∂ ∂ + = + ∂ ∂ ∂ (3) where 0B , is the strength of the magnetic field, υ is the kinematic viscosity and λ is the relaxation time Parameter, k is the thermal conductivity, ρ is the density, T is the temperature, pC is the specific heat at constant pressure and q′′′ is the space and temperature dependent internal heat generation/absorption which is modeled as ( ) [ *( ) ( ) ( ) *],w w ku x q A T T f T T B x η υ ∞ ∞   ′′′ = − + −    (4) Where A* and B* are the coefficients of space and temperature dependent internal heat generation/absorption respectively. Here we make a note that the case 0*,0* >> BA corresponds to internal heat generation and that 0*,0* << BA corresponds to internal heat absorption. As to the boundary conditions, we are going to assume that the sheet is being stretched linearly. Therefore the appropriate boundary conditions on the flow are , 0 at 0, 0 as u Bx v y u y = = = → → ∞ (5) where B>0, is the stretching rate. Here x and y are, respectively, the directions along and perpendicular to the sheet, u and v are the velocity components along x and y directions. The flow is caused solely by the stretching of the sheet, the free stream velocity being zero. Equations (1) and (2) admit a selt-similar solution of the form 1 2 ( ), ( ), , B u Bxf v B f yη ν η η ν  ′= = =     (6) Where superscript ' denotes the differentiation with respect to η . Clearly u and v satisfy Equation (1) identically. Substituting these new variables in Eq. (2), we have ( ) ( ) 22 2 0,f M f f f ff f ffβ′′′ ′ ′ ′′ ′ ′′ ′′′− − + + − = (7) Here 2 2 0 and B M B B σ β λ ρ = = are magnetic and Elastic parameters.
  5. 5. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 6, September – October (2013), © IAEME 44 The boundary conditions (4) become (0) 1, (0) 0 at 0 ( ) 0, (0) 0 as f f f f η η ′ = = = ′ ′′∞ → → → ∞ (8) We define the dimensionless temperature as 2 ( ) , where = ( ) ( )w w T T x T T b PST Case T T l θ η θ η∞ ∞ ∞ −   = −   −   (9a) 2 2 - ( ) , where ( ) 1 w T T D x g T T PHF Case k l bx b l k b υ η ν ∞ ∞   = − =          (9b) The thermal boundary conditions depend upon the type of the heating process being considered. Here, we are considering two general cases of heating namely, (i) Prescribed surface temperature and (ii) prescribed wall heat flux, varying with the distance. (i) Governing equation for the prescribed surface temperature case For this heating process, the prescribed temperature is assumed to be a quadratic function of x is given by 2 0, 0, ( ) 0. 0, w s x u Bx v T T x T T at y l u T T as y∞   = = = = − =    = → → ∞ (10) where l is the characteristic length. Using (5), (6) and (10), the dimensionless temperature variable θ given by (9a), satisfies [ ] * * Pr 2 ( ) ,f f A f Bθ θ θ θ′ ′ ′ ′′− − + = (11) Where Pr pc k µ = is the Prandtl number and corresponding boundary conditions are (0) 1 at 0 ( ) 0 as θ η θ η = = ∞ = → ∞ (12) (ii) Governing equation for the prescribed heat flux case The power law heat flux on the wall surface is considered to be a quadratic power of x in the form 2 , 0 0, . w w T x u Bx k q D at y y l u T T as y∞  ∂   = − = = =    ∂    → → → ∞ (13)
  6. 6. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 6, September – October (2013), © IAEME 45 Here D is constant, k is thermal conductivity. Using (5), (6) and (13), the dimensionless temperature variable g given by (9b), satisfies [ ] * * Pr 2 ( ) ,f g g f A f B g g′ ′ ′ ′′− − + = (14) The corresponding boundary conditions are ( ) 1 0 ( ) 0 . g at g as η η η η ′ = − → = → ∞ (15) The rate of heat transfer between the surface and the fluid conventionally expressed in dimensionless form as a local Nusselt number and is given by 0 Re '(0)x w y x T Nu x T T y θ ∞ =  ∂ ≡ − = −  − ∂  (16) Similarly, momentum equation is simplified and exact analytic solutions can be derived for the skin-friction coefficient or frictional drag coefficient as 0 2 1 (0) ( ) R e y f x u dy C f Bx µ ρ =  ∂     ′′≡ = − (17) Where 2 Rex Bxρ µ = is known as local Reynolds number. 3. NUMERICAL SOLUTION We adopt the most effective shooting method (see Refs. Cebeci [17]) with fourth order Runge-Kutta integration scheme to solve boundary value problems in PST and PHF cases mentioned in the previous section. The non-linear equations (6) and (11) in the PST case are transformed into a system of five first order differential equations as follows: ( ) [ ] 0 1 1 2 2 2 1 1 0 2 0 1 22 2 0 0 1 * *1 1 0 1 0 , , 2 , 1 , Pr 2 ( ). df f d df f d f M f f f f f fdf d f d d d f f A f B d η η β η β θ θ η θ θ θ θ η = = + − − = − = ′= − − + (18)
  7. 7. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 6, September – October (2013), © IAEME 46 Subsequently the boundary conditions in (7) and (12) take the form, 0 1 1 2 0 0 (0) 0, (0) 1, ( ) 0, (0) 0, (0) 0, ( ) 0. f f f f θ θ = = ∞ = = = ∞ = (19) Here 0 0( ) and ( ).f f η θ θ η= = Aforementioned boundary value problem is first converted into an initial value problem by appropriately guessing the missing slopes 2 1(0) and (0)f θ . The resulting IVP is solved by shooting method for a set of parameters appearing in the governing equations with a known value of 2 1(0) and (0)f θ . The convergence criterion largely depends on fairly good guesses of the initial conditions in the shooting technique. Once the convergence is achieved we integrate the resultant ordinary differential equations using standard fourth order Runge–Kutta method to obtain the required solution. 4. RESULTS AND DISCUSSION The exact solution do not seem feasible for a complete set of equations (6)-(11) because of the non linear form of the momentum and thermal boundary layer equations. This fact forces one to obtain the solution of the problem numerically. Present results are compared with Hayat ([10]) some limiting cases are shown in Table 1. The effect of several parameters controlling the velocity and temperature profiles are shown graphically and discussed briefly. 0 2 4 6 8 0.0 0.2 0.4 0.6 0.8 1.0 Fig.2(a).Theeffectof MagneticparameterMontemperaturedestributionθ(η) θ(η) η PST-Case β=1.0 Pr=1.0 A * =0.5 B * =0.5 M=0,1,2 0 2 4 6 8 0.0 0.2 0.4 0.6 0.8 1.0 Fig.2(b).TheeffectofMagneticparameterMontemperaturedestributiong(η) g(η) η PHF-Case β=1.0 Pr=1.0 A * =0.5 B * =0.5 M=0,1,2 Figs. 2(a) and 2(b) show the effect of magnetic parameter on the temperature profiles above the sheet for both PST and PHF cases. An increase in the magnetic parameter is seen to increase the fluid temperature above the sheet. That is, the thermal boundary layer becomes thicker for larger the magnetic parameter.
  8. 8. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 6, September – October (2013), © IAEME 47 0 1 2 3 4 0.00 0.25 0.50 0.75 1.00 1.25 Fig. 3(a). Effect of Prandtl number on temperature destributionθ(η) PST-Case M = 1.0 A* = 0.5 B * = 0.5 Pr = 0.01, 0.1, 1, 5 θ(η) η 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 PHF-Case β = 1.0 M=1.0 A * = 0.5 B* = 0.5 Fig. 3(b). Effect of Prandtl number on temperature destribution g(η) Pr = 0.01, 0.1, 1.0, 5 g(η) η Figs.3(a) and 3(b) show the temperature profile ( )θ η and ( )g η versus η from the sheet, for different values of Pr. We infer from these figures that temperature decreases with increase in Pr which implies viscous boundary layer is thicker than the thermal boundary layer. Temperature in both PST and PHF cases asymptotically approaches to zero in free stream region. 0.0 0.4 0.8 1.2 1.6 2.0 0.0 0.4 0.8 1.2 Fig.4(a). The effect of Space dependent A * on temperature destribution θ(η) PST-Case β 1.0 M = 1.0 B * = 0.5 Pr = 1.0 A * = -0.5, 0, 0.5 θ(η) η 0.0 0.4 0.8 1.2 1.6 2.0 0.0 0.4 0.8 1.2 1.6 2.0 Fig.4(b). The effect of space dependent A * on temperature destribution g(η) PHF-Case β = 1.0 M = 1.0 B * = 0.5 Pr = 1.0A * = -0.5, 0, 0.5 g(η) η Figs. 4(a) and 4(b), are graphs of temperature profiles ( )θ η and ( )g η versus distance η for different values of A*. For A* > 0, it can be seen that the thermal boundary layer generates the energy, and this causes the temperature ( )θ η and ( )g η of the fluid to increase with increase in the value of A* > 0 (heat source), where as for A*<0 (absorption) the temperature ( )θ η decreases with increase in the value of A*.
  9. 9. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 6, September – October (2013), © IAEME 48 0 1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 Fig.5(a). The effect of temperature dependent B * on temperature dstribution θ(η) PST-Case M = 1.0 A * = 0.5 Pr = 1.0 η θ(η) B* = -0.5, 0.0, 0.5 0 2 4 6 0.0 0.4 0.8 1.2 1.6 Fig.5(b). The effect of Temperature dependent B * on temperature destribution g(η) PHF-Case β =1.0 M= 1.0 A* = 0.5 Pr = 1.0 B* =-0.05, -0.09, 0 g(η) η Figs. 5(a) and 5(b), depicts the temperature profiles ( )θ η and ( )g η versus distance η , for different values of B*. The explanation is similar to that given for A*. The present work analyses, the MHD flow and heat transfer within a boundary layer of UCM fluid above a stretching sheet in presence of non-uniform heat source. Numerical results are presented to illustrate the details of the flow and heat transfer characteristics and their dependence on the various parameters. The results of PST and PHF cases infer that the boundary layer temperature is quantitatively higher in PST case as compared to PHF case and the results are in tune with what happens in regions away from the sheet. Table 1: Comparison of values of skin friction coefficient ( )0f ′′ with M= 0.0 and M= 0.2 S Hayat et. al [10] Present Results M=0.0 M=0.2 M=0.0 M=0.2 0.0 -1.90250 -1.94211 -0.999962 -1.095445 0.4 -2.19206 -2.23023 -1.101850 -1.188270 0.8 -2.50598 -2.55180 -1.196692 -1.275878 1.2 -2.89841 -2.96086 -1.285257 -1.358733 1.6 -3.42262 -3.51050 -1.368641 -1.437369 2.0 -4.13099 -4.25324 -1.447617 -1.512280
  10. 10. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 6, September – October (2013), © IAEME 49 REFERENCES [1]. Sakiadis BC., Boundary layer behavior on continuous solid surfaces: I Boundary layer equations for two dimensional and axisymmetric flow, AICHE. J.; 1961:7: 26-8. [2]. Sakiadis BC, Boundary layer behavior on continuous solid surfaces: II Boundary layer on a c flat surface, AICHE. J; 1961; 7: 221-25. [3]. T. Sarpakaya, Flow of non-Newtonian fluids in a magnetic field, AICHE. J. 7 (1961) 324-328. [4]. L.J. Crane, flow past a stretching plate, Z. Angrew. Math. Phys. 21 (1970) 645-647. [5]. B.K. Dutta, A.S. Gupta, cooling of a stretching sheet in a various flow, Ind. Eng. Chem. Res. 26 (1987) 333-336. [6]. A. Chakrabarti, A.S. Gupta, Hydromagnetic flow and heat transfer over a stretching sheet, Q. Appl. Math. 37 (1979) 73-78. [7]. M.S. Abel, P.G. Siddheshwar, Mahantesh M. Nandeppanavar, Heat transfer in a viscoelastic boundary layer flow over a stretching sheet with viscous dissipation and non-uniform heat source. Int. J. Heat Mass Transfer, 50(2007), 960-966. [8]. M.S. Abel and N.Mahesha, Heat transfer in MHD viscoelastic fluid over a stretching sheet with variable thermal conductivity, non-uniform heat source and radiation, Applied Mathematical Modeling. 32 (10) (2007) 1965-1983. [9]. D. N. Schulz, J. E. Glass, editors. Polymers as rheology modifiers. ACS symposium series, 462. Washington (DC): American Chemical Society; 1991 [10]. T. Hayat, Z. Abbas, M. Sajid. Series solution for the upper-convected Maxwell fluid over a porous stretching plate. Phys Lett A 358 (2006) 396-403. [11]. K. Sadeghy, A.H.Najafi, M.Saffaripour. Sakiadis flow of an upper convected Maxwell fluid Int J Non-Linear Mech 40 (2005) 1220. [12]. A. Alizadeh-Pahlavan, V. Aliakbar, F. Vakili-Farahani, K. Sadeghy. MHD flows of UCM fluids above porous stretching sheets using two-axillary-parameter homotopy analysis method. Commun. Ninlinear Sci Numer Simulat, doi:10. 1016/ j.cnsns. 2007. 09.011. [13]. M. Renardy. High Weissenberg number boundary layers for the Upper Convected Maxwell fluid. J. Non-Newtonian Fluid Mech. 68 (1997) 125. [14]. I.J. Rao and K. R. Rajgopal. On a new interpretation of the classical Maxwell model. Mechanics Research Communications. 34 (2007) 509-514. [15]. A. Alizadeh-Pahlavan, K. Sadeghy. On the use of homotopy analysis method for solving unsteady MHD flow of Maxwellian fluids above impulsively stretching sheets. Commun Ninlinear Sci Numer Simulat. In press(2008). [16]. V. Aliakbar, A. Alizadeh-Pahlavan, K. Sadeghy. The influence of thermal radiation on MHD flow of Maxwellian fluids above stretching sheets. Commun Ninlinear Sci Numer Simulat. In press(2008). [17]. T. Cebeci, P. Bradshaw, Physical and computational aspects of convective heat transfer, Springer- Verlag, New York, 1984. [18]. K.R.Rajagopal, Boundary Layers in non-Newtonian fluids in M.D.P.Montieivo Marques, J.F.Rodrigues(Eds). [19]. Rajneesh Kakar, Kanwaljeet Kaur and K. C. Gupta, “Viscoelastic Modeling of Aortic Excessive Enlargement of an Artery”, International Journal of Mechanical Engineering & Technology (IJMET), Volume 4, Issue 2, 2013, pp. 479 - 493, ISSN Print: 0976 – 6340, ISSN Online: 0976 – 6359. [21]. Dr P.Ravinder Reddy, Dr K.Srihari and Dr S. Raji Reddy, “Combined Heat and Mass Transfer in MHD Three-Dimensional Porous Flow with Periodic Permeability & Heat Absorption”, International Journal of Mechanical Engineering & Technology (IJMET), Volume 3, Issue 2, 2012, pp. 573 - 593, ISSN Print: 0976 – 6340, ISSN Online: 0976 – 6359. [20]. M N Raja Shekar and Shaik Magbul Hussain, “Effect of Viscous Dissipation on MHD Flow and Heat Transfer of a Non-Newtonian Power-Law Fluid Past a Stretching Sheet with Suction/Injection”, International Journal of Advanced Research in Engineering & Technology (IJARET), Volume 4, Issue 3, 2013, pp. 296 - 301, ISSN Print: 0976-6480, ISSN Online: 0976-6499.

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