Int. J. of Applied Mechanics and Engineering, 2018, vol.23, No.1, pp DOI: /ijame

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1 Int. J. of Applied Mechanics and Engineering, 018, vol.3, No.1, pp DOI: /ijae EFFECTS OF HEAT SOURCE/SINK AND CHEMICAL REACTION ON MHD MAXWELL NANOFLUID FLOW OVER A CONVECTIVELY HEATED EXPONENTIALLY STRETCHING SHEET USING HOMOTOPY ANALYSIS METHOD C.S. SRAVANTHI * Departent of Matheatics, The M.S. University of Baroda Vadodara, Gujarat-39000, INDIA E-ail: srinivasan_sravanthi@yahoo.co R.S.R. GORLA Departent of Mechanical Engineering Cleveland State University Cleveland-44114, OHIO, USA The ai of this paper is to study the effects of cheical reaction and heat source/sink on a steady MHD (agnetohydrodynaic) two-diensional ixed convective boundary layer flow of a Maxwell nanofluid over a porous exponentially stretching sheet in the presence of suction/blowing. Convective boundary conditions of teperature and nanoparticle concentration are eployed in the forulation. Siilarity transforations are used to convert the governing partial differential equations into non-linear ordinary differential equations. The resulting non-linear syste has been solved analytically using an efficient technique, naely: the hootopy analysis ethod (HAM). Expressions for velocity, teperature and nanoparticle concentration fields are developed in series for. Convergence of the constructed solution is verified. A coparison is ade with the available results in the literature and our results are in very good agreeent with the known results. The obtained results are presented through graphs for several sets of values of the paraeters and salient features of the solutions are analyzed. Nuerical values of the local skin-friction, Nusselt nuber and nanoparticle Sherwood nuber are coputed and analyzed. Key words: HAM, cheical reaction, heat source/sink, Maxwell nanofluid, porous exponentially stretching sheet, convective boundary conditions. 1. Introduction The boundary layer flows of non-newtonian fluids driven by stretching surfaces have attracted uch research interest [1 4] during the last few years due to several iportant engineering and industrial applications such as electronic chips, glass-fiber and paper production, food processing, etc. Single constitutive equation exhibiting all the properties of non-newtonian fluids is not available due to the diversity of these fluids in their constitutive behavior, siultaneous viscous and elastic properties. Thus, several odels of non-newtonian fluids have been proposed to fit well with the experiental observations [5]. The non-newtonian fluids can be classified into the following three types, i.e., (i) differential type (ii) rate type and (iii) integral type. The Maxwell fluid odel, a siplest subclass of rate type fluids, describes the characteristics of the relaxation tie. Theroplastic polyers in the vicinity of their elting teperature, fresh concrete (neglecting its aging), nuerous etals at a teperature close to their elting point, geoaterials, etc. behave typically as Maxwell fluids. The exact solutions of a fractional Maxwell * To who correspondence should be addressed Download Date 9/7/18 1:0 AM

2 138 C.S.Sravanthi and R.S.R.Gorla odel for a flow between two-sided wall perpendicular to a plate were presented by Vieru et al. [6], using Fourier and Laplace transfors. The unsteady flow in Maxwell fluids with the flow being induced by oscillating/accelerated nature of the rigid body was studied by Fetecau et al. [7-8]. Hayat et al. [9] studied the MHD unsteady flow of a Maxwell fluid in a rotating frae of reference and porous ediu. Suction or injection of a fluid through the bounding surface can significantly change the flow field. Injection of a fluid through a porous bounding wall is of general interest in practical probles involving boundary layer control applications such as fil cooling, polyer fiber coating, and coating of wires. The process of suction and injection has also its iportance in any engineering activities such as in the design of thrust bearings and radial diffusers, and theral oil recovery. In cheical processes, suction is applied to reove reactants whereas injection is used to add reactants and reduce the drag. The study of heat source/sink effects on heat transfer is very iportant because their effects are crucial in controlling the heat transfer. Bataller [10] studied the effects of heat source/sink, radiation and work done by deforation on flow and heat transfer of a viscoelastic fluid over a stretching sheet. Abel and Nandeppanavar [11] exained the effects of a non-unifor heat source in a viscoelastic boundary layer flow over a stretching sheet. Mukhopadhyay [1] ade heat transfer analysis for an unsteady flow of a Maxwell fluid over a stretching surface in the presence of heat source/sink by using the shooting ethod. The ass transfer phenoenon has applications in various scientific disciplines for different systes and echaniss that involve olecular and convective transport of atos and olecules. The transport of ass and oentu with cheical reactive species in the flow caused by a linear stretching sheet is discussed by Andersson et al. [13]. Cortell [14] investigated ass transfer with cheically reactive species for two classes of viscoelastic fluids over a porous stretching sheet. The effect of ass transfer on the stagnation point flow of an upper-convected Maxwell fluid is investigated by Hayat et al. [15]. Mukhopadhyay and Bhattacharyya [16] discussed the ass transfer effects on a Maxwell fluid flow past an unsteady stretching sheet. The theral conductivity rate of ordinary base fluids including water, ethylene glycol and oil is very low. But, nowadays the cooling of electronic devices is the ajor industrial requireent. To achieve this, the nanoscale solid particles are uniforly ixed with host fluids which change the therophysical characteristics of these fluids and enhance the heat transfer rate draatically. These fluids are said to be nanofluids. Choi [17]was the first who identified this colloidal suspension. The nanofluids have applications in cooling of electronics, heat exchangers, nuclear reactor safety, hypertheria, bioedicine, engine cooling, vehicle theral anageent and any others. Further, agneto nanofluids are useful in the anufacturing processes of gastric edications, bioaterials for wound treatent, sterilized devices, etc. Ethylene glycol- Alo3, ethylene glycol-cuo and ethylene glycol-zno are the exaples of visco-elastic nanofluids. A bulk of research articles on nanofluids is available in the literature in which a few can be seen in the Refs. [18-4]. For any of the above entioned probles, nuerical techniques have been developed for years to obtain the accurate solution. But, due to soe restrictions [5], scientists have considered analytical approaches as an alternative. The Hootopy Analysis Method (HAM) proposed by Liao [6], is a general analytical approach to obtain series solutions of strongly nonlinear equations which can provide us a siple way to ensure the convergence of solution series. Unlike nuerical ethods, it can be ipleented with the boundary conditions at infinity. In nuerical ethods, the far field boundary conditions denoted by ax ust be chosen approxiately for all coputations as the coputational doain has to be finite whereas the physical doain is unbounded. No attept has been ade so far to analyze the MHD flow of a Maxwell nanofluid past an exponentially stretching surface with heat source/sink, cheical reaction effects under suction/blowing and convective boundary conditions. The present work ais to fill the gap in the existing literature. Hence, otivated by the above entioned applications, an attept is ade to study the boundary layer MHD heat and ass transfer flow of a Maxwell nanofluid past an exponentially stretching sheet with heat source/sink, cheical reaction effects under suction/blowing and convective boundary conditions analytically via the Hootopy Analysis Method (HAM). The present proble (in the absence of the above entioned effects) has recently been investigated nuerically by Mustafa et. al. [7] only for a liited range of physical Download Date 9/7/18 1:0 AM

3 Effects of heat source/sink and cheical reaction paraeters by assuing ax. It is not surprising that their results are liited to only special paraeters that are consistent with ax. However, current results are independent to the value of and cover a wide range of physical paraeters. In addition, the effects of non-diensional paraeters such as the agnetic paraeter, heat source/sink paraeter, cheical reaction paraeter, skin friction, Nusselt nuber and nanoparticle Sherwood nubers have been discussed in detail.. Forulation of the proble A steady two-diensional incopressible boundary layer flow of a Maxwell nanofluid past a porous exponentially stretching sheet subject to a transverse agnetic field of strength B 0 in the presence of cheical reaction and heat source/sink effects is considered. The flow is assued to be generated by stretching of the elastic boundary sheet fro a slit with a large force such that the velocity of the boundary sheet is of an exponential order of the flow direction coordinate x. The x-axis is directed along the continuous stretching surface and points in the direction of the fluid otion while the y-axis is perpendicular to the surface. The flow is confined to y 0 (See Fig.1) [7, 31]. Here, the effect of the induced agnetic field is neglected. The Hall and electric field effects are neglected. Therophorosis and Brownian otion effects are taken into account. We assue that the surface of the sheet is heated by convection fro a hot fluid placed at the botto of the sheet at teperature T f and concentration C f with heat and ass transfer coefficients h 1 and h. The equations governing such type of flow are written as [16, 7, 8, 3] u v 0, (.1) x y u u u u u u B0 u u v 1 u v uv u 1v, x y x y x y y f y (.) T T k T C T DT T u v DB Q( T T ) x y, c y y T y f y (.3) C C C DT T u v DB kr( CC ) x y T. (.4) y y The boundary conditions for the present proble are [7, 33, 34] T C u uwx, vvwx, k h1tf T, DB hcf C at y0, y y Here, uw u0, T T, CC as y (.5) x L Ue is the stretching velocity, V x 0 w 0 x w is the velocity of suction and velocity of blowing, V ( x) V el [35] is a special type of velocity at the wall, h 1 f V x 0 is the w x L h e is the heat Download Date 9/7/18 1:0 AM

4 140 C.S.Sravanthi and R.S.R.Gorla x L he 1 transfer coefficient, h is the ass transfer coefficient, Q Q0 e is the variable voluetric rate of heat generation (i.e., heat source) or heat absorption (i.e., heat sink), where Q 0 is a constant having the x L 1 sae diension as Q, kr k0e [35] is the variable rate of cheical conversion of the first order irreversible reaction, where k 0 is a constant having the sae diension as k, the diffusivity of the species can either be destroyed or generated in the reaction. x L.1. Method of solution Fig.1. Physical odel and coordinate syste of the proble. Introducing the siilarity variables [7] as x x x y 0 L ' L L 0 U e, u Uf e, ve f f L L T T C C, T T C C f 0 f 0 U ', (.6) Using Eqs (.6), Eqs (.)-(.4) reduce to ' 1 ' 3 ' ' f ff f 3 ff f f f f f f M f f f f 0 (.7) 1 Nb Nt f QH 0, (.8) Pr Download Date 9/7/18 1:0 AM

5 Effects of heat source/sink and cheical reaction Nt Sc f Sc 0, (.9) Nb and the corresponding boundary conditions are f 1, f S, 1, ( 1) at 0, 1 f 0, 0, 0 as (.10) 0 x L LB Re where M is the local agnetic paraeter, x 1 DBC is the local Deborah nuber, Nb L Ue DT Tf T ( c) f is the Brownian otion paraeter, Nt is the therophoresis paraeter, Pr is the T k V0 KL 0 Prandtl nuber, S is the suction (S 0) /blowing (S 0) paraeter, is the cheical U U L reaction paraeter where 0 indicates a destructive cheical reaction, 0 denotes a generative QL 0 cheical reaction, and 0 represents the nonreactive species, QH is the heat source QH 0 /sink U0 h ( QH 0) paraeter, Sc 1 is the Schidt nuber, 1 is the heat transfer Biot nuber, D B U k L x h Ue L L ass transfer Biot nuber and Re is the local Reynolds nuber. U DB L The local skin-friction, Nusselt nuber and nanoparticle Sherwood nuber can be defined [7, 3] as C w xqw xq, Nu, Sh U k T T D C C f w f B f Here, w is the surface shear stress and q w, q are the heat flux and nanoparticle ass flux at the surface, respectively, and are defined as follows u T w, qw k y y, B y0 y0. C q D. y It is worth entioning that using diensionless variables, the skin friction and Nusselt nuber and nanoparticle ass transfer can be written as y 0 Download Date 9/7/18 1:0 AM

6 14 C.S.Sravanthi and R.S.R.Gorla C Re f ( 0), f Nu L () 0, Re x Sh L () 0. Re x Nu L In the present context, C f Re, and Sh L are referred to as the reduced skin friction Re x Re x coefficient, reduced Nusselt nuber and reduced nanoparticle Sherwood nuber which are represented by '' ' f 0, ( 0) and ' ( 0). Our task is to investigate how the values of f '' ( 0 ), θ'( ) and '( ) vary with the pertinent paraeters of the present proble. 3. Series solution by HAM The first step in the HAM is to find a set of base functions to express the solution of the proble under investigation. Here, due to boundary layer flows which are decaying exponentially at infinity, we assue that f, ( ) and ( ) can be expressed by a set of base functions in the for k { exp n k / k 0, n 0}, (3.1), k,, exp( ) f a a n 00 kn k0n1 k b, exp( n ), (3.) kn k0n0 k c, exp( n ) kn k0n0 where akn,, b kn, and c kn, are coefficients. Thus, all the approxiations of f, ( ) and ( ) ust obey the above expressions. This is called the rule of solution expression for f, ( ) and ( ). According to the boundary conditions (.10) and the rule of solution expression defined by Eq.(3.), we choose 0 exp( ) f S 1, 1 0 exp( ), (3.3) 1 0 exp( ). 1 1 as the initial approxiations of f, ( ) and ( ). Besides, we select the auxiliary linear operators L f, L and f L, which are the linear parts of the respective equations, as Download Date 9/7/18 1:0 AM

7 Effects of heat source/sink and cheical reaction L L L f 3 d f df d, d f 3 d, (3.4) d d, d satisfying the following properties exp L C C exp C 0, f 1 3 exp L C4exp C5 0, (3.5) exp L C6 exp C7 0 where C, i1 7, are constants. i Zeroth-order deforation proble 1 p Lf f ; p f0 p f Nf[ f,, ], 1 p L ; p 0 p N[ f,, ], (3.6) 1 p L ; p 0 p N[ f,, ] where p [ 01, ] denotes the ebedding paraeter and indicates the non-zero auxiliary paraeter and 3,, f(, p) f p f p N f f,, f, p 3 f, p f, p f, p f, p 3f, p f, p f, p f, p 3 f, p f, p f, p M f, p, Download Date 9/7/18 1:0 AM

8 144 C.S.Sravanthi and R.S.R.Gorla, p, p, p 1 N f,, Nb Pr, p, p Nt f, p QH, p, (3.7) Nt, p, p, p N f,, Sc f, p Sc, p Nb Subject to the following boundary conditions ' ',,,,, f 0pSf 0p1 f p 0, ' 0p, 1( 1 0, p ),, p 0, (3.8) ' 0p, ( 1 0p, ),, p 0, th-order deforation probles Differentiating -ties the zeroth-order deforation probles (3.6) with respect to p, then dividing by! and setting p 0, we get the following higher-order deforation probles and the boundary conditions are where f Lf f f1 f R, L L R, (3.9) 1 R, 1 ' ',, f 0 0 f 0 0 f 0, ' 0 0, 0, (3.10) 1 ' 0 0, 0 f 1 f (, p)!, 1 (, p)!, 1 (, p), (3.11)!. Download Date 9/7/18 1:0 AM

9 Effects of heat source/sink and cheical reaction and k f '' ' R f 1 f1nfn f 1nfn 3f1kfklfl f ' 1kf ' klfl n0 n0 k0l0 1 1 f 1 k f kl f l f ' 1 k f k l f l M f ' 1 f 1 n f n f 1 n f n 0, n '' ' R 1 Nb ' 1nn Nt ' 1n' n f1n' nqh10 Pr, n0 n0 n0 1 ' Nt 1 Sc 1n n 1 Sc 1 Nb n0 R f 0, (3.1) (.13) For p 0 and p 1, we can write f (, 0) = f, 0 (, 0) =, 1 0 (, 0) =, 1 0 f, 1 f( ),,, ( ) (3.14),, ( ) and with variation of p fro 0 to 1, f, p,, p and, p vary fro initial solutions f0, 0 and 0 to final solutions f, ( ) and ( ) respectively. By Taylor s series we have = f0 + f p, f f (, p) 1 1 f(, p)! p0, (3.15) (, p) = (, p) = 0 + p, p, 1 1 (, p)! 1 (, p)! p0 p0, (3.16). (3.17) The value of the auxiliary paraeter is chosen in such a way that these three series (3.15)-(3.17) are convergent at p 1. Eqs (3.9) represents the syste of non-hoogeneous linear differential equations whose general solutions are the su of copleentary and particular solutions which can be expressed as Download Date 9/7/18 1:0 AM

10 146 C.S.Sravanthi and R.S.R.Gorla * 1 3 f f C C e C e, * C4 e 5 C e, (3.18) * C6 e 7 C e i where the constants C ( i 1,,.. 7), considering the boundary conditions Eqs (3.10) are 4 6 C C C 0, 1 3 * C C f 0, 3 Convergence analysis * ' C f 0, (3.19) 1 C 0 0, 5 * ' 1 * C * ' * 1 In the HAM ethod, it is essential to ensure the convergence of our series solution. As pointed by Liao [30], the convergence rate of approxiation for the HAM solution strongly depends on the value of the auxiliary paraeter. This auxiliary paraeter provides us great freedo to adjust and control the convergence region of the series solution. Hence, in order to seek the perissible values of f, and '' ' the functions of f 0, ( 0) and ' ( 0) are plotted at 0 th -order of approxiations. The values of f, and are selected in such a way that curves are parallel to the horizontal axis i.e., axis [30]. Figures -4, clearly depict the acceptable range for values of f ( 06. f 01. ), ( ) and ( ). The current calculations are based on the value of f 04.. In order to ensure the convergence of solutions, Tab.1 is given. This table clearly shows that the convergence is obtained at 35 th order of approxiations. Download Date 9/7/18 1:0 AM

11 Effects of heat source/sink and cheical reaction Fig.. curve for f at 0 th order of approxiation. Fig.3. curve for at 0 th order of approxiation. Fig.4. curve for at 0 th order of approxiation. Download Date 9/7/18 1:0 AM

12 148 C.S.Sravanthi and R.S.R.Gorla Table 1. Convergence of HAM solution for '' ' f 0, ( 0) and ' ( 0) when 0.,M 0.,Pr 10, Nb Nt 05., QH 0., Sc 0, 05., S0, 0., Results and discussion 1 f Order of f 0 ( 0) ( 0) approxiation To clearly provide a physical insight to the heat and ass transfer of a Maxwell nanofluid in the presence of heat source/sink and cheical reaction effects, coputations have been carried out using the ethod described in the previous section for variations in the governing paraeters and the results are illustrated graphically in Figs 5-0. In the present study, the following default paraeter values are adopted for coputations: 1, M 0.,Pr 1 10, Nb 0., 5 Nt 0., 5 QH 0., 5 Sc 0, 05., S 0, 1 05., 05.. All graphs and tables as therefore correspond to these values unless otherwise stated. For the verification of accuracy of the applied HAM, a coparison of the present results corresponding to the values of 0 with the available published results of Magyari and Keller [36], Mustafa et al. [7] in the case of a regular fluid is ade nuerically and presented in Tab.. The results are in excellent agreeent. Therefore, we are confident that the present results are accurate. Table. Coparison of nuerical values of 0. 0 with previous studies in the case of regular fluid when Pr Magyari and Keller [36] Mustafa et al.[7] Present Figures 5-7 shows the profiles of velocity, teperature and nanoparticle volue fraction for different values of the local agnetic paraeter M. It is clear fro Fig.5 that an increase in M decreases velocity, throughout the boundary layer flow field. It is because the application of a transverse agnetic field will result in a resistive type force (Lorentz force which opposes the flow) and thus reduces the velocity. It is evident fro Fig.6 and Fig.7 that an increase in the local agnetic paraeter M increases the teperature as well as nanoparticle volue fraction. This is due to the fact that Lorentz force tends to resist the flow and this resistance offered to the flow is responsible for broadening the theral and concentration boundary layer thicknesses. Download Date 9/7/18 1:0 AM

13 Effects of heat source/sink and cheical reaction Fig.5. Effect of M on velocity. Fig.6. Effect of M on teperature. Fig.7. Effect of M on nanoparticle volue fraction. Download Date 9/7/18 1:0 AM

14 150 C.S.Sravanthi and R.S.R.Gorla Figures 8-10 display the effect of the suction/blowing paraeter S on velocity, teperature and volue fraction of the nanofluid, respectively, in the presence of convection at the boundary for an exponentially stretching sheet. It is observed that velocity decreases significantly with increasing suction ( S 0) but increases with increasing blowing ( S 0) (Fig.8). When the wall suction ( S 0) is considered, this causes a decrease in the boundary layer thickness and the velocity field is reduced. S 0 represents the case of a non-porous stretching sheet. It is known that iposition of wall suction ( S 0) has the tendency to reduce the oentu boundary layer thickness, which causes reduction in the velocity. The opposite is noted for blowing ( S 0). This is due to the fact that when stronger blowing is provided, the fluid is pushed farther fro the wall where due to a saller influence of viscosity, the flow is accelerated. This effect increases the axiu velocity within the boundary layer. The sae principle operates but in the reverse direction in the case of suction. Figures 9 and 10 represent the profiles of teperature and nanoparticle volue fraction, respectively, for the variable suction/blowing paraeter S. It is evident that the fluid velocity, teperature and volue fractions are larger in the case of blowing copared to suction. It is also noted that the velocity, teperature and the volue fractions of the nanofluid in the absence of suction/blowing are greater copared to the case of a nanofluid with suction but saller copared to the case of a nanofluid with injection. Fig.8. Effect of S on velocity. Fig.9. Effect of S on teperature. Download Date 9/7/18 1:0 AM

15 Effects of heat source/sink and cheical reaction Figure 11 illustrates the influence of the heat generation (source) ( Q 0) /absorption (sink) QH 0 paraeter Q H on the teperature. It is seen fro the figure that the teperature decreases with increasing heat absorption ( QH 0) but increases with increasing heat generation ( QH 0). This is due to the fact that the theral boundary layer generates energy, which causes the teperature profiles to increase with increasing values of QH 0. However, in the case of QH 0, the boundary layer energy is absorbed, resulting in a considerable teperature fall with the decreasing values of Q H. H Fig.10. Effect of S on nanoparticle volue fraction. Fig.11. Effect of Q H on teperature. The effect of Brownian otion paraeter Nb on teperature is plotted in Fig.1. As the Brownian otion paraeter Nb increases, the rando otion of the fluid particles increases which results in ore heat. Hence the teperature increases.the chracteristics of the therophoresis paraeter Nt on the teperature is presented in Fig.13.Therophoresis is a echanis in which sall particles are pulled away fro a hot surface to a cold one. As a result it raises the teperature.the behavior of the Brownian otion paraeter Nb on the nanoparticle volue fraction is displayed in Fig.14. The increasse in Nb the rando otion and also collision of the acroscopic particles of the fluid increase which reduces the concentration of the fluid. Figure 15 describes the variation of the therophoresis paraeter Nt on the nanoparticle volue fraction. The bigger value of Nt coincides with the stronger therophoretic diffusion which blows the Download Date 9/7/18 1:0 AM

16 15 C.S.Sravanthi and R.S.R.Gorla nanoparticles away fro the hot surface towards the cold abient fluid. As a result, the nanoparticle volue fraction is thicker for larger values of Nt. Fig.1. Effect of Nb on teperature. Fig.13. Effect of Nt on teperature. Fig.14. Effect of Nb on nanoparticle volue fraction. Download Date 9/7/18 1:0 AM

17 Effects of heat source/sink and cheical reaction Fig.15. Effect of Nt on nanoparticle volue fraction. Figure 16 depicts the nanoparticle volue fraction profiles for the variable cheical reaction paraeter.it can be seen that the nanoparticle volue fraction decreases when the cheical reaction paraeter increases. The destructive cheical reaction ( 0) causes a decrease of the nanoparticle volue fraction and the constructive cheical reaction ( 0) results in an increase of the nanoparticle volue fraction. This is because the conversion of the species takes place as a result of a cheical reaction and thereby reduces the concentration in the boundary layer. Fig.16. Effect of on nanoparticle volue fraction. Figure 17 shows the influence of the heat transfer Biot nuber 1 on the teperature distribution. The heat transfer Biot nuber is defined as a ratio of convection heat transfer to the conduction heat transfer at the surface. It generally depends on the characteristic length of the surface, theral conductivity of the surface and convective heat transfer coefficient of the hot fluid below the surface. A higher heat transfer Biot nuber indicates a less conductive substance such as plastic, paper, polyer, etc. on the other hand, the Biot nuber is sall for a higher conductive aterials which include aluinu, ion and steel, etc. For increasing values of the heat transfer Biot nuber 1, the heat transfer coefficient h 1 increases which yields heat which leads to an incease of teperature. Figure 18 depects the effect of the heat transfer Biot nuber 1 on the nanoparticle volue fraction. The higher surface teperature corresponding to a larger heat transfer Biot nuber energizes nanoparticles in the vicinity of the sheet. In order to release that additional energy, the Download Date 9/7/18 1:0 AM

18 154 C.S.Sravanthi and R.S.R.Gorla nanoparticles travel away fro the stretching wall. This results in a larger penetration depth of nanoparticle concentration. Fig.17. Effect of 1 on teperature. Fig.18. Effect of 1 on nanoparticle volue fraction. The influence of the ass transfer Biot nuber on the teperature is exained in Fig.19. Here the teperature is inceased by increasing the ass transfer the Biot nuber. Biot nuber depends on the ass transfer coefficient h. The ass transfer coefficient h is increased when we increase the values of the Biot nuber due to which the teperature and theral boundary layer thickness are enhanced. Figure 0 shows elucidate that the nanoparticle volue fraction is increasing when the values of the ass transfer Biot nuber are increased. Download Date 9/7/18 1:0 AM

19 Effects of heat source/sink and cheical reaction Fig.19. Effect of on teperature. Fig.0. Effect of on nanoparticle volue fraction. Table 3 exhibits the nature of the reduced skin friction, Nusselt nuber and nanoparticle Sherwood nuber with the local agnetic paraeter M. It is found that the reduced skin-friction increases whereas the reduced Nusselt nuber and nanoparticle Sherwood nuber decrease with increasing the local agnetic paraeter M. Table 4 presents the effect of heat source/sink Q H on the reduced Nusselt nuber. It is clear that the Nusselt nuber increases with increasing the heat absorption (sink) ( QH 0) but decreases with increasing the heat generation (source) ( QH 0). The effect of the cheical reaction paraeter on the reduced nanoparticle Sherwood nuber is presented in Tab.5. It can be observed that the nanoparticle Sherwood nuber decreases with increasing the constructive cheical reaction ( 0) and increases with increasing the destructive cheical reaction ( 0). Table 3. Effect of the local agnetic paraeter M on the reduced skin-friction, Nusselt nuber and nanoparticle Sherwoodnuber. M '' ' f 0 0 ' ( 0) Download Date 9/7/18 1:0 AM

20 156 C.S.Sravanthi and R.S.R.Gorla Table 4. Effect of the heat source/sink paraeter Q H on the reduced Nusselt nuber. Q H ' Table 5. Effect of the cheical reaction paraeter on the reduced nanoparticle Sherwood nuber. 5. Conclusions ' ( 0) The governing equations for the steady MHD ixed convective heat and ass transfer flow over a porous exponentially stretching sheet with heat source/sink and cheical reaction effect under suction/blowing and convective boundary conditions were forulated. The resulting partial differential equations were transfored into a set of ordinary differential equations using the siilarity transforations. These non-linear coupled ordinary differential equations were solved by eploying the hootopy analysis ethod. The conclusions of the study are as follows: 1. The velocity, skin-friction, Nusselt nuber and nanoparticle Sherwood nuber decrease, whereas the teperature and nanoparticle volue fraction increase with an increase in the agnetic paraeter M.. An increase in suction S 0 reduces the velocity, teperature and nanoparticle volue fraction, whereas an increase in blowing S 0 increases the velocity, teperature and nanoparticle volue fraction. 3. Both the teperature and nanoparticle volue fraction increase with an increase in either the heat transfer Biot nuber 1 or ass transfer Biot nuber. 4. An increase in heat source ( QH 0) increases the teperature and decreases the Nusselt nuber, whereas an increase in heat sink ( QH 0) increases the Nusselt nuber and decreases the teperature. 5. Both the nanoparticle volue fraction and nanoparticle Sherwood nuber decrease with an increase in the constructive cheical rection ( 0) and increase with an increase of the destructive cheical reaction ( 0). Noenclature C nanoparticle concentration [Kg -3 ] C nanoparticles concentration far away fro the surface [Kg -3 ] D B Brownian diffusion coefficient [ s -1 ] D T therophoretic diffusion coefficient [ s -1 ] Download Date 9/7/18 1:0 AM

21 Effects of heat source/sink and cheical reaction k theral conductivity [W -1 K -1 ] k r cheical reaction paraeter Q diensional heat generation/absorption coefficient q r radiative heat flux [W - ] T fluid teperature [K] T teperature far away fro the surface [K] U reference velocity [s -1 ] u, v coponents of velocity in the x and y directions [s -1 ] V initial strength of suction 0 1 relaxation tie of the Maxwell fluid [s -1 ] dynaic viscosity f density of the base fluid [Kg -3 ] electrical conductivity of the fluid [s -1 ] cp c ratio of the effective heat capacity of the nanoparticle aterial to the effective heat capacity of the f fluid / f kineatic viscosity superscript subscripts References ' pries denote differentiation with respect to f base fluid p nano particle [1] Anderson H.I., Hansen O.R. and Holedal B. (1994): Diffusion of a cheically reactive species fro a stretching sheet. Int. J. of Heat Mass Transfer, vol.37, No.4, pp [] Xu H. and Liao S.J. (009): Lainar flow and heat transfer in the boundary-layer of non-newtonian fluids over a stretching flat sheet. Coput. Math. Appl., vol.57, pp [3] Mukhopadhyay S. (013): Casson fluid flow and heat transfer over a nonlinearly stretching surface. Chin. Phys. B, No.7, [4] Prasad K.V., Santhi., S.R. and Datti P.S. (01): Non-Newtonian power-law fluid flow and heat transfer over a non-linearly stretching surface. Appl. Math., vol.3, No.5, pp [5] Wu J. and Thopson M.C. (1996): Non-Newtonian shear-thinning flows past a flat plate. J. Non-Newton. Fluid Mech., vol.66, pp [6] Vieru D., Fetecau C. and Fetecau C. (008): Flow of a viscoelastic fluid with fractional Maxwell odel between two side walls perpendicular to a plate. Appl. Math. Coput., vol.00, pp [7] Fetecau C., Jail M., Fetecau C. and Siddique I. (009): A note on the second proble of Stokes for Maxwell fluids. Int. J. of Non-Linear Mech., vol.44, pp [8] Fetecau C., Athar M. and Fetecau C. (009): Unsteady flow of generalized Maxwell fluid with fractional derivative due to a constantly accelerating plate. Coput. Math. Appl., vol.57, pp [9] Hayat T., Fetecau C. and Sajid M. (008): On MHD transient flow of a Maxwell fluid in a porous ediu and rotating frae. Phys. Lett. A 37, pp Download Date 9/7/18 1:0 AM

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