OBLIQUE STAGNATION POINT FLOW OF A NON-NEWTONIAN NANOFLUID OVER STRETCHING SURFACE WITH RADIATION A Numerical Study

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1 THERMAL SCIENCE, Year 7, Vol., No. 5, pp OBLIQUE STAGNATION POINT FLOW OF A NON-NEWTONIAN NANOFLUID OVER STRETCHING SURFACE WITH RADIATION A Numerical Stud b Abuzar GHAFFARI a*,**, Tariq JAVED a, and Fotini LABROPULU b a Department of Mathematics and Statistic, International Islamic Universit, Islamabad, Pakistan b Department of Mathematics, Luther College, Universit of Regina, Regina, Canada Original scientific paper In this stud, we discussed the enhancement of thermal conductivit of elasticoviscous fluid filled with nanoparticles, due to the implementation of radiation and convective boundar condition. The flow is considered impinging obliquel in the region of oblique stagnation point on the stretching surface. The obtained governing partial differential equations are transformed into a sstem of ordinar differential equations b emploing a suitable transformation. The solution of the resulting equations is computed numericall using Chebshev spectral newton iterative scheme. An excellent agreement with the results available in literature is obtained and shown through tables. The effects of involving parameters on the fluid flow and heat transfer are observed and shown through graphs. It is importantl noted that the larger values of Biot number impl the enhancement in heat transfer, thermal boundar laer thickness, and concentration boundar laer thickness. Ke words: thermal conductivit, elastico-viscous fluid,oblique stagnation point, spectral method Introduction Oblique stagnation-point flow appears when fluid from an source impinges obliquel on a rigid wall at an arbitrar angle of incidence as shown in fig.. Man researchers have studied the stead -D oblique stagnation-point flow of a Newtonian fluid. Stuart [] did the pioneer work in this field, later studied b Tamada [] and Dorrepaal [3]. Recentl, Reza and Gupta [] generalized the problem of Chiam [5] b introducing a stretching surface. In their paper, the ignored the displacement thickness and pressure gradient. This was partiall rectified in a paper b Lok et al. [6]. Ver recentl, Reza and Gupta [7] gave a correct solution to the problem b fixing the errors in [] and [6]. Drazin and Rile [8], Tooke and Blth [9] reviewed the problem and included a free parameter associated with the shear flow component related to the pressure gradient. Weidman and Putkaradze [, ] studied the stead-oblique stagnation-point flow impinging on a circular clinder. The flow is described using a coupled set of ordinar differential equations. Recentl, Erfani et al. [], Husain et al. [3], Mahapatra et al. [], Lok et al. [5], Yajun and Liancun [6], and Javed et al. [7] have done notable work on orthogonal and oblique stagnation point flow. * Corresponding author, abuzar.iiui@gmail.com ** Now in Department of Mathematics, Universit of Education, Lahore, Pakistan

2 THERMAL SCIENCE, Year 7, Vol., No. 5, pp U e =ax+b In last few decades, heat transfer in nanofluids has become a topic of major interest. Man researchers contributed in this area due its significance in pharmaceutical and food processes, hperthermia, fuel cells, micro-electronics, hbridpowered engines, coolants for advanced nuclear Stagnation point power plants [8] and man others. The basic x idea of using nanosized particle to enhance the thermal conductivit of the fluid was given b Figure. Phsical Model Maxwell [9]. Choi [] was the first who introduce the term nanofluid in 995. He studied the characteristics of nanofluids and deduced that the thermal conductivit of the base fluid (water, oil, biofluids, organic liquids, ethlene glcol, etc.) can be enhanced b introducing metallic particles (average size about nanometers). Nanoparticles are made of different metals (Al, Cu, Ag, Au, Fe), metal carbides (SiC) nonmetals (graphite carbon nanotubes), oxides (Al O 3, CuO, TiO ), nitrides (AlN, SiN), etc. In 6, Buongiorno [] has studied the convective transport in fluid and he considered seven slip mechanisms (thermophoresis, diffusiophoresis, Brownian diffusion, inertia, Magnus effect, gravit, and fluid drainage) to discuss the relative velocit of the fluid and nanoparticles and he concluded that among these seven slip mechanisms onl two are important. Recentl, Kuznetsov and Nield [, 3] studied the double-diffusive and natural convective boundar-laer flow of a nanofluid past a vertical plate, the found the analtical solution of the problems. Makinde and Aziz [] studied the convective heat transfer in nanofluid past a stretching sheet and the discussed Brownian motion and thermophoresis effects in detail. There is extensive literature available on the topic through different aspects. Few representative recent studies on the topic ma be seen in the refs. [5-38]. Literature surve witnessed that much attention in the past has been accorded to the flow of viscous nanofluids. However, in real situation the base fluids in the nanomaterials is not viscous. No doubt, the base fluid in realit is viscoelastic. Mention ma be made to some viscoelastic nanofluids like ethlene glcol-cuo, ethlene glcol-al O 3, ethlene glcol-zno. Keeping such preference in view the viscoelastic nanofluid is considered in this paper. Man viscoelastic fluids models have been proposed but here constitutive equations of Walter-B fluid [39-] are emploed in the mathematical formulation. Our intention here is to compute the oblique stagnation point flow of viscoelastic nanofluid. To the best of our knowledge, such problem has not been attempted before. An efficient approach namel the Chebshev Spectral Newton Iterative Scheme (CSNIS) is implemented for the numerical solution. The graphical results are interpreted with respect to various parameters of interest. A comparison with the previousl published results in limiting sense is given. Heat transfer rate and mass diffusion flux are also analzed. Problem formulation We consider the stead -D laminar flow of Walter-B nanofluid impinging obliquel on a stretching surface, which is placed at and the fluid occupies the upper half plane as shown in fig.. The surface is heated convectivel, b convective heating process, which is characterized b a temperature, T f, and a heat transfer coefficient, h f. We neglect the viscous dissipation to estimate accuratel the effect of convective boundar condition because viscous dissipation would disturb the thermal boundar conditions. The velocit of the outer flow far awa from the surface is Ue ( x, ) ax b. The flow, energ and concentration equations are, see Beard and Walters [].

3 THERMAL SCIENCE, Year 7, Vol., No. 5, pp u x v () k o u u p u u u u u v v u u v u v x x x x x x x xx v u u u v v u v x x x x v v p v v k o u u v v v u u v u v x x x x x x x u v u v v v u v x x T T k T T qr C T C T D T T T u v D B () x Cp x Cp x x T x C C C C D T T T u v DB (5) x x T x In previous equations, ux (, ) and vx (, ) are the velocit components in x - and - directions, Cx (, ) the concentration, T( x, ) the temperature, and px (, ) the pressure of the fluid flow. Also, the kinematic viscosit, ρ the densit, k o the elasticit of fluid, C p the specific heat, k the thermal conductivit of the fluid, and q r the radiative heat flux. The D T and D B are the Brownian motion coefficient and thermophoretic diffusion coefficient, respectivel, and τ = (ρc) p /(ρc) f is the ratio of effective heat capacit of nanoparticles materials to heat capacit of the fluid. The boundar conditions of the problem can be defined: :,, T u cx v k hf Tf T, C Cw (6) : u ax b, T T, C C in which a, b, and c are positive constants having the dimension of inverse time, the ambient temperature, and h f the heat transfer coefficient. The radiative heat flux can be modeled b using Rosseland s approximation: T qr (7) 3( ) r where σ is the Stefan-Boltzmann constant, α r the Rosseland mean absorption coefficient, and σ s the scattering coefficient. Assuming that the temperature difference within the flow is sufficient small so that T ma be expressed as linear function T such that: thus eq. (7) takes the following form: 3 3 s T T T T (8) () (3)

4 THERMAL SCIENCE, Year 7, Vol., No. 5, pp q r 3 6 T T 3( ) r Upon using non-dimensional variables and stream function, ψ, this satisfies the continuit equation such: c c T T C C x x,, u u, v v, p p, T, C, c c c Tf T Cw C u, v x and then eliminating pressure from eqs. () and (3), eqs. ()-(6) take the following new form in term of ψ:,, We x, x, s (9) () () 3 T T T T 6 T T C T C T N b x x Pr x 3 k( r s) x x x T T N t C C C C N t T T Sc x x x N b x () (3) T : x,, Bi T, C a : x, T, C c where We = k o c / ρν is the Weissenberg number, Pr = C p / k the Prandtl number, Sc = ν / D B the Schmidt number, N t = D T τ(t f T )/ T ν the thermophoresis parameter, N b = D B τ(c w C ) / ν the Brownian motion parameter, Bi = (h f / k)( f / c) / the Biot number, and bcrepresents shear in the free stream. Suppose the solution of eqs. (-) is of the form: () xf () g(), T (), C () (5) where the functions f() and g() are normal and oblique component of the flows. Using the eq. (5) in eqs. ()-(), and after comparing the coefficient of x and x, we get: iv ''' ' '' v ' iv f ff f f We ff f f (6) iv ''' ' '' v ' iv g fg g f We fg g f (7) '' ' ' ' ' Rd Pr f Nb Nt 3 (8)

5 THERMAL SCIENCE, Year 7, Vol., No. 5, pp '' ' Nt '' Sc f (9) N ' ' ' : f( ), f ( ), g( ) g ( ), ( ) Bi ( ), ( ) () ' ' : f ( ) a c, g ( ), ( ), ( ) 3 where Rd T k( r s) is the radiation parameter and prime denotes the differentiation with respect to. After integrating eqs. (6) and (7) the resulting constants of integration can be evaluated b emploing the boundar conditions at infinit and we get: iv iv b ''' '' ' ' ''' '' a f ff f We f f f f () c ''' '' ' ' ' ''' '' '' ''' ' g fg g f We fg f g g f f g A () where A = A(a / c, We) is a constant which measures the boundar laer displacement. Constant A at free stream behave as (a /c) which also corresponds to the behavior of f() at the free stream. For simplicit, introducing a new variable, g () = γh(), then eq. () with boundar conditions is written: '' ' ' ''' ' '' ' '' ''' h f h f hwe fh f h h f f h A (3) h() h'( ) () Thus the sstem of non-linear ordinar equations becomes: ''' '' ' iv ' ''' '' a f ff ( f ) We ff f f ( f ) c '' ' ' ''' ' '' ' '' ''' (5) h fh f hwe fh f h h f f h A (6) '' ' ' ' ' Rd Pr f Nb Nt 3 (7) with boundar conditions '' ' Nt '' Sc f (8) N : f( ), f '( ), h( ), '( ) Bi ( ), ( ) : f '( ) a c, h'( ), ( ), ( ) To solve the fourth order ordinar differential eqs. (5) and (6), we used two extra boundar conditions f () = and h () = as. These conditions are called augmented boundar conditions [, 3]. The dimensionless components of velocit are: u xf '( ) g'( ), b v f ( ) x (9) (3)

6 THERMAL SCIENCE, Year 7, Vol., No. 5, pp The quantities of phsical interest are the skin friction coefficients, C f, the local Nusselt number, Nu x, and the local Sherwood number, Sh x, can be expressed: C x q q xq, Nu, Sh ( ) ( ) w w r m f x x uw ktf T DB Cw C where τ W is shear stress at the wall, q r the radiative heat flux, q w and q m represents local heat flux, and local mass diffusion flux, respectivel, at the wall are: (3) u v k uv uv x x v u vx ux u vx vu vx uux vxx w x T C T q k, q D, q w m B r 3( ) r s (3) After using eqs. () and (5), the skin friction coefficients, C f, the local Nusselt number, Nu x, and the local Sherwood number, Sh x, takes the following form: where Re x u w x. Re xcf x 3We f ''() ( We) h'() 3 / / Rex Nu x Rd '(), Rex Sh x '() Numerical method Exact solutions of the non-linear differential eqs. (5)-(8) subject to the boundar conditions (9) are ver rare due to the non-linearit. Some authors have used analtical semianaltical techniques to solve these eqs. (33) and (39). In the present stud, we used a numerical technique named as CSNIS. In this scheme, we first convert the sstem of non-linear differential equation into a linear form b using Newton iterative scheme. For (i+) th iterates, we write: i i i i i i i i i (33) f f f,,, (3) for all dependent variables, where δf i, δθ i and δ i, represents a ver small change in f i, θ i, and i respectivel. The equations (5)-(8) in linearized form are: iv ''' '' ', i i, i i, i i 3, i i, i i, i a ff a f a f a f a f R ''' '' ' ''' '' ', i i, i i, i i 3, i, i i 5, i i 6, i i 7, i i, i b f b f b f b f b h b h b h b h R '' ' ', i, ii, ii 3, ii 3, i c f c c c R '' '' '', i, ii, ii 3, ii R, i d f d d d (35) subject to boundar conditions ' ' ' ' '' '' i i i i i i i i f () f (), f () a c f (), f ( ) f ( ), f ( ) f ( ) ' ' '' '' i i i i i i h () h (), h ( ) h ( ), h ( ) h ( ) ' ' i i i i i i () Bi () () Bi (), ( ) ( ) (36)

7 THERMAL SCIENCE, Year 7, Vol., No. 5, pp The sstem of linear eq. (35) subject to boundar conditions (36) is solved using the Chebshev spectral collocation method [, 5]. For this purpose, the phsical domain [, ] is truncated to finite domain [, L], where L is chosen sufficientl large. The reduced domain is transformed to [, ] b using transformation ξ = η L. Nodes from to are defined as ξ j = cos(πj / N), j =,,, N, which are known as Gauss-Lobatto collocation points. The Chebshev spectral collocation method is based on differentiation matrix [D], which can be computed in different was. Here we used [D] as suggested b Trefethen [6]. The coefficients a j,i, b j,i, c j,i, d j,i, and R j,i ( j =,,, 3...) are ' '' ' ''' '' iv, i We i,, i We i,, i i We i, 3, i i We i,, i i We i ' '' ' ''' ', i We i,, i We i,, i i We i, 3, i i We i,, i We i, 5, i We i a f a f a f f a f f a f f b h b h b h h b h h b f b f '' ' ''' ' ' b6, i fi We fi, b7, i fi We fi, c, i Pr i, c, i Rd, 3, Pr 3 c i Nbi ' ' c Pr f N ' N, d Sc, d N N, d, d Sc f where iv, i We i i i i i i i i i ''' ' '' '' ' ''' '' ' ' R, i We fihi fihi fi hi fi hihi fihi fihi '' ' ' ' ' R 3, i Rd i Pr fii Nb i i Nt i, i i b i t i, i i, i t b, i 3, i i ' ''' ' ''' '' ' R f f f f f f f f f a c 3 N R ' t '', i i '' Sc fii i Nb A Appling collocation method to eqs. (35) and (36), the following matrix is obtained: A A A3 A fi R, i A A A3 A hi R, i A3 A3 A33 A3 i R 3, i A A A3 A i R, i 3 A a, id a, id a, id a3, ida, ii, A, A3, A 3 3, i, i, i 3, i, i 5, i 6, i 7, i 3 A3 c, ii, A3, A33 c, id c, id, A3 c3, id A d, ii, A, A3 d, id, A d, id d3, id A b D b D b Db I, A b D b D b Db I, A, A (39) where I is the identit matrix, a j,i, b j,i, c j,i, d i,j, and R j,i (j =,,,3...) are given in eq. (37). Results and discussion The non-linear differential eqs. (5)-(8) subject to the boundar conditions (9) are solved numericall for the different values of dimensionless parameters namel Weissenberg number, velocities ratio parameter, a/c, radiation, Rd, therm ophoresis, N t, Brownian motion, N b, Prandtl number, Schmidt number, and Biot number (tabs. -). The values of f ''(), '() and '() shown in limiting case through tabs. and and numerical values of A, eqs. (38) and (39), in tab. 3. It is found that the results are in excellent agreement with previous investigations published in the literature. The results in term of velocit profile, temperature profile, (37) (38)

8 6 THERMAL SCIENCE, Year 7, Vol., No. 5, pp Table. Comparison of -θ'() for the various values of a/c and Prandtl number in the absence thermophoresis effects and Brownian motion of nanoparticles when We =, Rd =, and Bi a/c Present Present Present [7] [7] [7] work work work Pr = Pr = Pr = Table 3. Numerical values of A for various values of We and a/c a/c We Table. Comparison of () and () for the various values of Nt and Nb when We =, a/c =, Rd =, Pr =, Sc =, and Bi =. N t..3.5 N t..3.5 N t..3.5 N b =. () (.99).99 (.95).955 (.9).9 N b =.3 () (.769).7688 (.79).79 (.7).6697 N b =.5 () (.383).3833 (.69).69 (.8).8 () (.77).77 (.8).8 (.783).783 () (.399).399 (.39) (.79).793 () (.356).3563 (.576).576 (.535).535 The results in small brackets are reported b Makinde and Aziz [] Table. Numerical values of Rex / Nux and for wider range of Pr We a/c Rd Nt Nb Bi Sc Pr Rex / Nux Rex / Shx

9 THERMAL SCIENCE, Year 7, Vol., No. 5, pp concentration profile, f (), θ () and '() for sundr parameters are shown through graphs. In most cases, the values of the parameters are taken as Pr = 6.8, Sc =.5, Bi =.5, We =., N t = N b =.3, a/c =.3,., and Rd = or otherwise mentioned. The variation of f ''(), '() and '() against Weissenberg number for a/c =.8,.,., and. are shown in figs. -, respectivel. From these figures, it is observed that the similarit eqs. (5)-(8) subject to the boundar conditions (9) have dual solutions in some range of the parameter Weissenberg number. There exist unique solution in particular range of the parameter Weissenberg number and there exist a region where the solution of the equation does not exist. The solid lines show the stable solution and dashed lines show the unstable solution. For a/c >, the range of solution enhances due to increase in a/c and for a/c < the range of unstable solution become larger than the stable solution. There exist a unique solution at critical value We = We c, dual solution exist between the range We < We c and no solution exists for We < and We > We c. The critical values are We c =.39, We c =.36, We c3 =.58, and We c =.33 for different values of a/c as shown in figures. It is observed that unstable solution has higher values of f ''(), '(), and '() than that of the stable solution for given values of Weissenberg number. It is further noted that in stable solution (first solution) heat and mass transfer rate increase with increase in the values of a/c, where a reverse behavior has been observed for unstable solution (second solution). The stabilit analsis of multiple solutions has been discussed b man researcher [8-5]. The found that first solution is applicable phsicall while the second solution is not. In fig. 5 the velocit profile is plotted against for the different values of Weissenberg number, a/c, and γ. Here γ = and γ =. correspond to the case for orthogonal stagnation point flow and non-orthogonal stagnation point flow, respectivel. It is noted that the velocit of the fluid is increasing with increase in the values of We when a/c >. An opposite behavior is observed for the case when a/c <. It is also seen that with increase in the values of the velocit of the fluid increases. In figs. 6 and 7 the variation of local Nusselt, Re -/ x Nu x, and local f () 8 6 We c a/c =.8,.,.,. - Stable solution Unstable solution a/c =.8,.,. We c We c We c We Figure. Variation of f () against We for the different values of a/c () θ Stable solution Unstable solution a/c =.8,., We c We c We c a/c =.8,.,.,. We c We Figure 3. Variation of θ () against We for the different values of a/c θ ().6.. We c.8 a/c =.8,.,.,. Stable solution Unstable solution a/c =.8,.,. We c We c We c We Figure. Variation of () against We for the different values of a/c f () a/c =.3 a/c =. We =,.,.,.3 =. = 3 5 Figure 5. Velocit profile against for the different values of We, a/c a, and γ γ γ

10 8 THERMAL SCIENCE, Year 7, Vol., No. 5, pp Re x -/ Nux 8 6 a/c =.3 a/c =. Rd =, 5, Figure 6. Variation of Nusselt number against Nt for the different values of Rd and a/c Re x -/ Shx a/c =.3 a/c =. Rd =, 5, Rd =, 5, N t Figure 7. Variation of Sherwood number against Nt for the different values of Rd and a/c Re x -/ Nux 7 a/c =. a/c =.3 Rd =, 5, Figure 8. Variation of Nusselt number against Nb for the different values of Rd and a/c Re x -/ Shx.5.5 Rd =, 5, a/c =. a/c = Figure 9. Variation of Sherwood number against Nb for the different values of Rd and a/c N t N b N b Sherwood, Re x -/ Sh x, numbers are plotted against thermophoresis parameter, N t, for the different values of Rd and a/c. It is clear from fig. 6 that with increase in the values of N t, a ver slight decrease in local Nusselt number is observed for both cases of a/c (a/c >, a/c < ). Consequentl, temperature profile and thermal boundar laer thickness increase with increase of thermophoresis parameter, N t, near the wall. Figure 7 elucidates that the local Sherwood number decreases with increase of N t, as a consequence the concentration profile and concentration boundar laer increase with increase of N t. From figs. 6 and 7, an increase in local Nusselt and local Sherwood numbers is observed due to enhancement of radiation. In figs. 8 and 9, the values of local Nusselt, Re x -/ Nu x, and local Sherwood, Re x -/ Sh x, numbers are plotted against Brownian motion parameter, N b, for the different values of R d and a/c. It is seen that with increase in Brownian motion the local Nusselt number decreases but the local Sherwood number increases. This increase in local Sherwood number is ver rapid in the range < N b <.. This phenomenon leads to increase the temperature and thermal boundar laer thickness but decrease in concentration profile. In figs. and, the variation of local Nusselt (Re x -/ Nu x and local Sherwood Re x -/ Sh x ) numbers are plotted against Biot number (depending on the heat transfer coefficient) for the different values of Rd and a/c. It is seen that the local Nusselt number increases and local Sherwood number decreases for initial values of Biot number and for the larger values of Biot number both quantities become constant. Due to the larger values of Bi the surface become heated and heat transfer rate increases. In fact, the larger values of Biot number impl the strong surface convection result in high surface temperature. Therefore, increase in Biot number enhanced the temperature and thermal boundar-laer thickness. This behavior can be predicted from fig.. In fig. 3, concentration profile is plotted against for different values of Biot number when Rd = and a/c =.3. Concentration profile increases with increase in the values of Biot number because concentration distribution depends upon the temperature field hence the larger Biot number helps to increase the concentration of nanoparticles in fluid.

11 THERMAL SCIENCE, Year 7, Vol., No. 5, pp Re x -/ Nux 8 6 a/c =. a/c =.3 Rd =, 5,.5 Rd =, 5, Bi Bi Figure. Variation of Nusselt number against Bi for the different values of Rd and a/c Re x -/ Shx a/c =. a/c =.3 Figure. Variation of Sherwood number against Bi for the different values of Rd and a/c θ() Bi =,.,.5, Figure. Temperature profile against for the different values of Bi φ() Bi =,.,.5, Figure 3. Concentration profile against for the different values of Bi θ()...8 a/c =. a/c =.3.6 We =,.,.,.3 We =,.,.,.3. We. We Figure. Temperature profile against for the different values of We and a/c θ() φ( ) φ() Rd 5 a/c =. a/c =.3 Figure 5. Concentration profile against for the different values of We and a/c..5 Rd =,, 5, Figure 6. Temperature profile against for the different values of Rd.. Rd Figure 7. Concentration profile against for the different values of Rd

12 5 THERMAL SCIENCE, Year 7, Vol., No. 5, pp In figs. and 5 temperature and concentration profiles are plotted against for the different values of Weissenberg number and a/c when Rd =, Bi =.. For a/c <, it is observed that the temperature and concentration profiles are increasing functions of Weissenberg number but for a/c > an opposite behavior is noted. In figs. 6 and 7, temperature and concentration profiles are plotted against for the different values of Rd when Bi =. and a/c =.3. With increase in the values of radiation parameter, temperature of the fluid increases where the concentration profile decreases near the wall but for the larger value of, it increases with increase in the values of radiation parameter. Conclusions The combined effect of radiation and convective boundar condition in the region of oblique stagnation point flow of elastico-viscous fluid saturated with nanoparticles is considered. The governing partial differential equations are transformed into sstem of ordinar differential equations b using the similarit transformation. The obtained sstem of equations is solved numericall b using CSNIS. The present numerical results are in excellent agreement with the previousl obtained results. It is observed that the similarit eqs. (5)-(8) subject to the boundar conditions (9) have unique solution, dual solution and no solution in different region of the parameter Weissenberg number. For a/c >, the range of existence of solution increases due to increase in a/c and for a/c <, the range of unstable solution become larger than that of the stable solution. It is also concluded that. The velocit of the fluid intensifies due to increase in Weissenberg number when a/c > but an opposite behavior is observed for a/c <. The velocit of the fluid is found an increasing function of. Temperature profile and thermal boundar laer thickness enhance due to increase in the values thermophoresis parameter. Concentration profile and concentration boundar laer thickness increase with increase of thermophoresis parameter. Brownian motion enhanced the thermal boundar laer thickness. Brownian motion decreases the concentration boundar laer thickness. The larger values of Biot number impl the enhancement in heat transfer and thermal boundar laer thickness. Concentration profile increases with increase in the values of Biot number. Temperature is noted an increasing function of radiation. Acknowledgment The authors are grateful to the editor and anonmous reviewers for their valuable suggestions that helped to improve the manuscript. The first author is thankful to HEC for financial assistance. Nomenclature a, b, c positive constants Bi Biot number C solutal concentration Cf skin friction coefficient Cp specific heat constant Cw solutal concentration at the wall C ambient solutal concentration DB Brownian diffusion coefficient DT f hf k ko Nb Nt Nu thermophoretic diffusion coefficient dimensionless stream function convective heat transfer coefficient thermal conductivit of the nanofluid elasticit of fluid Brownian motion parameter thermophoresis parameter Nusselt number

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15 THERMAL SCIENCE, Year 7, Vol., No. 5, pp [7] Labropulu, F., et al., Non-Orthogonal Stagnation-Point Flow towards a Stretching Surface in a Non- Newtonian Fluid with Heat Transfer, Int. J. Therm. Sci., 9 (), 6, pp. -5 [8] Weidman, P. D., et al., The Effect of Transpiration on Self-Similar Boundar Laer Flow over Moving Surfaces, Int. J. Eng. Sci., (6), -, pp [9] Paullet, J., Weidman, P., Analsis of Stagnation Point Flow toward a Stretching Sheet, Int. J. Nonlinear Mech.,, (7), 9, pp. 8-9 [5] Rosca, A. V., Pop, I., Flow and Heat Transfer over a Vertical Permeable Stretching/ Shrinking Sheet with a Second Order Slip, Int. J. Heat Mass Transfer, 6 (3), Ma, pp Paper submitted: April, 5 Paper revised: October 6, 5 Paper accepted: October 8, 5 7 Societ of Thermal Engineers of Serbia. Published b the Vinča Institute of Nuclear Sciences, Belgrade, Serbia. This is an open access article distributed under the CC BY-NC-ND. terms and conditions.