Analysis of MHD Williamson Nano Fluid Flow over a Heated Surface

Transcription

1 Journal of Applied Fluid Mechanics, Vol. 9, No., pp , 016. Available online at ISSN , EISSN Analysis of MHD Williason Nano Fluid Flow over a Heated Surface S. Nadee 1 and S. T. Hussain 1 Departent of Matheatics, Quaid-I-AzaUniversity 4530, Islaabad 44000, Pakistan. DBS&H, CEME, National University of Sciences and Technology, Islaabad, Pakistan Corresponding Author Eail: snqau@hotail.co (Received Septeber 1, 013; accepted April 0, 015) ABSTRACT In the present article Williason nano fluid flow over a continuously oving surface is discussed when the surface is heated due to the presence of hot fluid under it. Governing equations have been developed and siplified using the suitable transforations. Matheatical analysis of various physical paraeters is presented and the percentage heat transfer enhanceent is discussed due to variation of these paraeters. We eployed Optial hootopy analysis ethod to obtain the solution. It is presented that initial guess optiization will provide us one ore degree of freedo to obtain the convergent and better solutions. Keywords: Nano fluid; Convective heat transfer; Optial hootopy analysis ethod (OHAM); Non-linearly oving surface. NOMENCLATURE a stretching paraeter B agnetic field ( N/(A)) b body force (N) Bi Biot nuber C nanoparticles voluetric fraction c specific heat (J/kg K) D B brownian diffusion coefficient (²/s) D T therophoretic diffusion coefficient (²/s) E electric field (N/C) h convective heat transfer coefficient (W/² K) I identity tensor J current density (A/²) i,j indexing variable k nano fluid theral conductivity (W/ K) M agnetic paraeter n stretching index Nu local Nusselt nuber p pressure (N/²) Pr Prandtl nuber Re local Reynolds nuber S cauchy stress tensor (N/²) Sc Schidt nuber Sh local Sherwood nuber T teperature of fluid (K) t tie (s) u horizontal coponent of velocity (/s) V velocity vector (/s) v vertical coponent of velocity (/s) x distance along the plate σ electrical conductivity (S/) ν kineatic viscosity (²/s) α theral diffusivity (²/s) ρ density (kg/³) η siilarity variable λ non Newtonian Williason paraeter subscripts B brownian otion T therophoresis infinity p nano particles w wall iteration nuber 1. INTRODUCTION Nano fluid is the cobination of siple fluid and nano sized particles uniforly suspended in the fluid. These nano sized particles can be etallic (Cu, Al, Hg, Ti, etc.) or nonetallic (ZnO, Al 0 3, TiO and several other etallic oxides). Nano particles have advantage over icro size particles due to negligible effects of gravitational settling and cluster foration during flow. Fluids are widely used in heat transfer phenoenon due to their strong convection properties. Nano fluids got the attention of researchers and industrialists due to their better perforance in heat transfer phenoenon. Several researchers

2 S. Nadee and S. T. Hussain /JAFM, Vol. 9, No., pp , 016. (Bang and Chang 005; Suganthi and Rajan 01; Kuznetsov 011) have discussed the Nano fluid flow for various physical phenoenon by considering water as base fluid. But suspension of solid particles in water up to large percentage of its volue will change it fro the Newtonian to non-newtonian fluid. Also we know that several non-newtonian fluids have better heat transfer properties as copared with water for e.g. liquid etals. So it s iportant to discuss the flow of non- Newtonian nano fluids due to their wide range use in industrial and cheical processes. Choi (1995) was the first one to use the ter nano fluid due to nano sized particle suspension in fluid. He discussed that nano particles (due to their sall size) are uch better as copared to icro sized particles and their incorporation can reduce the cooling cost several ties. Masuda et al. (1993) discussed the theral conductivity enhanceent due to addition of ultrafine particles in fluid. Buongiorno (006) discussed the already presented atheatical odels for nano fluid flow. He analyzed the possible effect of seven slip echaniss on heat transfer enhanceent and concluded that only Brownian otion and theroophoresis are the iportant slip echaniss. He odeled the governing equations for the nano fluid flow. Nield and Kuzentsov (009) discussed the theral instability in a porous ediu layer saturated by a nano fluid using the Buongiorno odel. In another paper (011), they discussed the double diffusive convection in a nano fluid flow. Nadee and Lee (01) obtained the analytical solutions for the boundary layer flow of a nano fluid over an exponentially oving surface. Khan and Pop (010), nuerically investigated the boundary layer flow of nano fluid past a stretching sheet. Recently Malvandi and Ganji (014 a,b,c) discussed in detail the effect of Brownian otion and therophoresis on the nano fluid flow in a icrochannel. The study of flow over a continuous oving surface is a popular area of research due to its enorous application in industrial anufacturing processes. Flows over continuously oving surface are widely discussed by researchers after the pioneering work of Sakiadis (1961). He first tie odelled the boundary layer equations for the flow over a continuous oving surface which has been widely discussed afterwards (Ziabakhsh et al. 010; Nadee et al. 01, 013; Saleh et al. 010). But any physical phenoenons involve the non-linear stretching for e.g. extrusion of plastic sheet, glass anufacturing etc. Cortell (007) initiated the study of flows over non-linearly stretching surface. He nuerically investigated the flow of viscous fluid over a non-linearly stretching surface. Raptis and Perdikis (006) discussed the flow of viscous fluid over a non-linearly stretching surface in the presence of agnetic field. Abbasbandy and Ghehsareh (01) presented the solutions for agnetohydrodynaic fluid over a nonlinear stretching sheet with the help of Hankel-Pade ethod. Soe studies of viscous nano fluid over non-linear stretching surface have been reported in literature (Rahan and Eltayeb 013; Rana and Bhargava 01; Hady et al. 01), but very little attention has been given to the flow of non-newtonian nano fluid. Therefore the present article discusses the flow of non-newtonian nano fluid with convective boundary conditions. Convective boundary conditions are generalized as copared to constant surface teperature condition and have application in cooling systes and heat exchangers. Recently, Mansur and Ishak (013) discussed the blasius flow for a copper water nano fluid with convective boundary conditions. They extended the work of Aziz (009), who discussed the lainar flow of viscous fluid with convective boundary conditions. Makinde and Aziz (011) investigated the boundary layer flow of nano fluid with convective boundary conditions. All of the aforeentioned studies consider the viscous fluid with convective boundary condition. Williason fluid odel describes the flow of shear thinning non-newtonian fluids. This odel was proposed by Williason (199) and later on used by several authors (Dapra and Scarpi 007; Vasudev et al. 01; Nadee and Akbar 010) to investigate fluid flow. We used Optial Hootopy Analysis Method (OHAM) to solve the governing syste of equation for Williason nano fluid flow. OHAM is the refined version of Hootopy Analysis Method (HAM) Liao (01). HAM has been widely used to solve the nonlinear differential equations (Malvandi et al. 014 a,b; Abbasbandy 007; Hayat and Qasi 010; Shehzad et al. 01). Recently Liao (010) has suggested that it is better to use OHAM instead of HAM for better rate of convergence. He discussed different types of OHAM and suggested that it s better to use basic OHAM due to its coputational efficiency. In OHAM discrete squared residual errors are optiized against the convergence control paraeters. Fan and You (013) discussed the global and step by step approaches to optiize the convergence control paraeters. Nadjafi and Jafari (011) copared Liao's optial hootopy analysis ethod with the Niu's one-step optial hootopy analysis ethod. They eployed both techniques to obtain solution of linear Volterra integro-differential equations and integro-differential equation and concluded that Liao's optial HAM has ore accuracy to deterine the convergence-control paraeter than the one-step optial HAM suggested by Niu and Wang (010). In the present article boundary layer equations for the 730

3 S. Nadee and S. T. Hussain /JAFM, Vol. 9, No., pp , 016. flow of MHD non-newtonian Williason nano fluid with convective boundary condition have been developed. Suitable transforations have been introduced to convert syste of partial differential equations to the syste of ordinary differential equations. The governing nonlinear coupled syste of ordinary differential equations has been solved with the help of OHAM. In order to exaine the validity of results they are copared with the nuerical results. Effects of iportant physical paraeters have been discussed through graphs and tables. To the best of author s knowledge no study has been reported for Williason nano fluid over a non-linearly stretching surface with convective boundary condition.. MATHEMATICAL FORMULATION We consider a steady two diensional flow of an incopressible MHD Williason nano fluid over a horizontal heated surface. Where x and y-axis are taken along horizontal and vertical direction respectively. So that nano fluid is confined to y > 0. The plate is stretched along x-axis with the velocity ax n, where a > 0 is stretching paraeter and n is the stretching index. It is assued that the surface is heated due to convection of hot fluid below the surface. A space varying agnetic field B(0, B(x), 0) is applied along the transverse direction of flow see Fig.l. The fluid is assued to be slightly conducting, so that the agnetic Reynolds nuber is uch less than unity and hence the induced agnetic field is negligible in coparison to the applied agnetic field. The general transport equations for nano fluid are (Boungiorono 006) div V 0, (1) dv div S b, dt () T c( V T) (3) t T T kt pcp( DBT DT ), T C T VC ( DB C DT ), t (4) Fig.1 1. Geoetry of of the proble. where in steady state the velocity vector is given by V(u(x, y), v(x, y), 0), ρ is nano fluid density and p p is nano particles density,s is Cauchy T stress tensor, b is body force vector, c and c p are heat capacities of nano fluid and nano particles respectively, T is teperature, k is nano fluid theral conductivity, D B is Brownian diffusion coefficient, C is nano particles voluetric fraction, D T is therophoretic diffusion coefficient and T is the abient fluid teperature. For Williason fluid odel Cauchy stress tensor S is defined as S pi, (5) ( 0 ) [ ] A1, (6) 1 in above equations is extra stress tensor, is 0 liiting viscosity at zero shear rate and is liiting viscosity at infinite shear rate, 0 is a tie constant, A is the first Rivlin-Erickson tensor 1 and is defined as follows [8-30] 1, trace( A ). 1 (7) Here, we considered the case for which 0 and 1. Thus Eq.(6) can be written as 0 [ ] A1, (8) 1 or by using binoial expansion we get [1 ] A. (9) 0 1 The interaction of agnetic field and velocity will give rise to the Lorentz force J B defined in Eq.(10), in which J is the current density and is the electrical conductivity. In the absence of electric field E 0. J=σ(E + V B) (10) Since we have considered the steady state velocity so all the derivatives w.r.t are zero. Making use of Eqs.5,9 and (10) in Eqs.1 4 the two diensional boundary layer equations governing the flow are given by u v 0, x y u x y y y y (1) (11) u u u. u u v B x u c u v ( D ), T T T p p C T DT T B x y y c y y T y C C C D T u v DB x y y T y T, (13) (14) where uxy (, ) and vxy (, ) are horizontal and 731

4 S. Nadee and S. T. Hussain /JAFM, Vol. 9, No., pp , 016. vertical coponents of velocity, is kineatic viscosity and is nano fluid theral diffusivity. n 1 The agnetic field is chosen as B( x) B x. The corresponding boundary conditions to the flow proble are n T uuwax ; v0 ; k ht ( f T), CCwat y0, y u0 ; TT, CC as y. (15) In boundary conditions k is theral conductivity, T is the teperature of the hot fluid and f h is the convective heat transfer coefficient. Introducing the following transforations in above equations n1 n an ( 1) x n1 uax f, v, f f n1 n1 an ( 1) x TT CC y,,. TwT CwC (16) With the help of above transforations, Eq.(11) is identically satisfied and Eqs.(1) to (14) along with boundary conditions (15) take the following for n 0, n 1 f f ff f f Mf Nc Nc Le LeNbt (17) Pr f 0, (18) 1 Scf 0, (19) Nbt The corresponding boundary conditions and the non-diensional paraeter are f 0, f1, Bi(1 ), 1 at 0, f 0, 0, 0 as. (0) Matheatically, Sc can be written as Sc Le Pr. (1) D D B B With the help of Eq.(1), Eq.(19) can be written as 1 Le Pr f 0. () Nbt paraeter), 3 3n1 a ( n1) x (non Newtonian Williason Pr (oentu diffusivity/nano fluid theral diffusivity), Le (nano fluid theral diffusivity/brownian D B diffusivity), Sc (oentu diffusivity/brownian D B diffusivity). M B an ( 1) (Magnetic paraeter) Bi hx (Surface convection paraeter or Re x k n1 reduced Biot nuber) Nc pcp ( w ) c C C (nano particles heat capacity/nano fluid heat capacity), DBT ( CwC ) Nbt (Brownian diffusivity/ DT ( TwT ) therophoretic diffusivity). Since we are interested in the study of heat transfer enhanceent, so it is better to introduce the effects of theral diffusivity in nano particles equation with the help of Le and Pr. Also it will enhance the coupling effects of heat and nano particles equation. Finally, Eqs. (17), (18) and () for the non-diensional syste of equation with corresponding boundary conditions given in (0). For 0, proble reduces to the one for Newtonian nano fluid and for DB DT 0 in Eq.(1), heat equation reduces to the classical boundary layer heat equation in the absence of viscous dissipation. Physical quantities of interest for present study are local Nusselt nuber Nu and local Sherwood nuber Sh. x T x C Nu, Sh, Tw T y C y 0 w C y y0 (3) or by introducing the transforations 16, we get Nu n 1 Sh n 1 (0), (0), Re Re x x (4) U where Re x = w x is local Reynolds nuber. Physical paraeters will be discussed later in the results section. 3. HOMOTOPY BASED SOLUTION TECHNIQUE HAM is a strong analytic technique to solve linear and non-linear, ordinary and partial differential equations. HAM was developed by Liao (003), This technique is better than the perturbation techniques as it can be equally applied to weak and strong nonlinear probles. It is also independent of sall and large physical paraeter restriction. The advantage of HAM lies in the choice of convergence control paraeter according to the given set values of input paraeters. It provides a 73

5 S. Nadee and S. T. Hussain /JAFM, Vol. 9, No., pp , 016. way to check and adjust the convergence of obtained solution with the help of auxiliary paraeters and base functions. To obtain the hootopy based solutions, we choose two sets of initial guess defined by (3), is a siple constant its value will be deterined in the optiization section. This initial guess reduces to the ost coonly used initial guess f 1exp( ), when 1. In the result 0 section we will show that 1is not the best value, but it is better to optiize for the best possible value. f0 (1 )exp( ) (1 )exp( ), Bi 0 exp, Bi 1 0 exp, (5) L L L 3 df df f (f) = -, 3 dη dη d θ dθ θ(θ) = +, dη dη d φ dφ φ(φ) = +, dη dη (6) Due to space constraint the detailed HAM procedure is not discussed here. The detailed HAM procedure can be followed fro [3]. We can calculate any order approxiation for 1,,3... with the help of Matheatica. 3.1 Optial Convergence Control Paraeters It is noteworthy that our solutions f, and ( ) will contain unknown convergence control (auxiliary) paraeters ( hf, h, h ). To find out the optial values of convergence control paraeters we define the exact squared residuals at the th order approxiation as follows f N f[ fi] d, 0 i0 i i i 0 i0 i0 i0 N [ f,, ] d, i i i 0 i0 i0 i0 (7) (8) N[ f,, ] d, (9) and t f, (30) where t is the total squared residual error and f, are the corresponding individual errors at th iteration. In above equations N, N f, N denotes the left hand side of the equations (17), (18) and (19) at the th order deforation respectively. It is obvious that quickly t approaches to zero, faster the corresponding solution converges. Thus at th iteration the optial values of convergence control paraeters are given by the iniu of t, corresponding to the set of these equations t t t t 0. (31) h h h f The total squared residual error defined by Eq.(30) takes too uch CPU tie to calculate the error even if the order of approxiation is not very high. Thus to increase the coputational efficiency we define the discrete squared residual error (as defined by Liao [38] ) at the th iteration by N f 1 E N f[ fi( j)], N 1 j0 i0 (3) N 1 E N [ fi( j), i( j), i( j)], N 1 j0 i0 i0 i0 (33) N 1 E N [ fi( j), i( j), i( j)], N 1 j0 i0 i0 i0 (34) where j j, 0.5 and N 0 in above equations. The total discrete squared residual error is defined as t f E E E E. t t t t E E E E (35) 0. h h h f In present paper total discrete squared residual error approach is used to obtain optial convergence control paraeters. In order to obtain the local optial convergence control paraeters, we directly eploy the iniize coand in coputational software Matheatica. 4. RESULTS AND DISCUSSION In hootopy analysis ethod, the convergence control paraeters are chosen fro the range of values, obtained through h-curves. After plotting the h-curves we randoly choose the values of convergence control paraeter fro the convergence region. But with the help of Optial Hootopy, we precisely choose the best possible values of the convergence control paraeters. 4.1 Optial Values of Convergence Paraeters In this section the Optial values of convergence control paraeters are shown when Bi 5,Pr, Nc 0.5, Nbt, Le, 0., n 0.5, M Table.1 shows the total discrete squared residual 733

6 S. Nadee and S. T. Hussain /JAFM, Vol. 9, No., pp , 016. Table 1 Average squared residual errors at different order of approxiation h f h θ h φ ε ⁴ ⁴ ³ ³ ⁶ ⁶ ⁵ ⁵ ⁷ ⁷ ⁶ ⁶ ⁸ ⁹ ⁷ ⁷ error at different order of approxiation. We split table.1 into two sections one for the case 1 (ost coonly used case) and other when is optiized instead of setting its particular value. Table Local Optial convergence control paraeters at different order of approxiation h f h θ h φ (ε=1) ² ⁴ ⁴ ⁵ h h h E f For the present study we found that total squared residual error can be reduced significantly if we optiize initial guess by introducing arbitrary paraeter. So we will use the optiized value of instead of choosing 1. Table.1 also gives the optial values of convergence paraeters at different order of approxiations. These optial values are used to obtain the discrete squared residual errors at different order of approxiation see Table.. It is observed that the average squared residual error and total squared residual error decrease as the nuber of iterations increased. This assures that better solutions are obtained at higher order approxiations. We choose, 9 th iteration set of optial values to plot figures and draw tables in the coing sections. Results will be siilar if we choose optial convergence paraeters value fro 9 th or any higher order approxiation. 4. Plots and Tables Fig. is drawn to exaine the effect of agnetic field on the nano fluid velocity. We observe that velocity profile decrease with increase in M. Eq. (10) involves the cross product of velocity vector and agnetic field vector, it will give rise to Lorentz force which is perpendicular to the velocity vector and agnetic field vector. This force will act as a resistive force to the fluid flow which will ultiately slow down the fluid otion. It is also clear that the oentu boundary layer thickness decreases with the increase in agnetic paraeter M. Fig.3 shows that teperature increases due to the increase in M. Nano fluid contains the nano size particles whose otion is affected by the applied agnetic field. These particles act as the energy carrier to the fluid causing the increase in nano fluid teperature. M is zero, when no external agnetic t field is applied. We can also see that theral boundary layer is increasing with the increase in M. 0. Le, Bi 5, Nc 0.5, Nbt, Pr, 0., n 0.5 M, 0.5,,.0 f ' Fig.. Velocity profile against M. Fig. Velocity profile against M Bi 5,Pr,Nc 0.5,Nbt,Le, 0.,n 0.5 M=, 0.5,, Fig. 3. Teperature profile against M. Fig.3 Teperature profile against M Fig.4 shows that the teperature profile against for different values of Biot nuber Bi. We can see that the teperature at wall is varying with Bi because of convective boundary conditions. When Bi approaches to infinity the teperature boundary condition at wall reduces to the case of constant wall teperature. Thus convective boundary conditions are ore generalized as copare to the constant wall teperature condition. The graph also depicts that as the Bi approaches to 100, attain the constant wall teperature condition (0) 1. Biot nuber is the ratio of the convection at the surface to the conduction within the surface. Thus large Biot nuber iplies stronger convection at the surface and sall Biot nuber iplies stronger conduction within the surface. Teperature graph against Bi also predicts this behavior, the fluid teperature increases with the increase in Bi. Also theral boundary layer increases with the increase in Bi. 734

7 S. Nadee and S. T. Hussain /JAFM, Vol. 9, No., pp , Le,M,Nc 0.5,Nbt,Pr, 0.,n 0.5 Bi= 0.5,,.0, Fig. Fig.4 4. Teperature profile against Bi. Figs.5 and 6 deonstrate the effects of Lewis nuber Le and diffusivity ratio Nbt, on the teperature profile when other paraeters are kept constant. Teperature profile as well as theral boundary layer thickness decrease with the increase in Le. Le and Nbt cannot be chosen equal to zero because Le appears in the denoinator of the Eq. (18) and Nbt appears in the denoinator of both Eq. (18) and (). Physically Le cannot equal to zero since it is ratio of theral diffusivity to Brownian diffusion. Buongiorno (006) defined the paraeter Nbt to discuss the relative effect of Brownian diffusion to therophoretic diffusion, we followed the sae paraeter. It is observed that teperature profile and theral boundary layer decrease with increase in Nbt. when Brownian diffusivity is very large as copared to therophoretic diffusivity, teperature profile only shows very sall variation. Fig.7 describes the effect of heat capacities ratio Nc on the teperature profile. It is observed that teperature and theral boundary layer thickness increases with the increase in Nc. If we look at the definition of Nc, it is ratio of heat capacity of nano particles and nano fluid. Usually the specific heat c p of nano particles is less than that of base fluid because typically specific heat of solid is less than that of liquids. So addition of solid particles will decrease the specific heat of base fluid, hence teperature profile decrease. Fig.8 shows that the teperature and theral boundary layer thickness decreases with the increase in Prandtl nuber Pr. As we know Pr controls the relative thickness of oentu and theral boundary layer. Large Pr eans saller theral boundary layer and larger oentu boundary layer. 0. Bi 5,M,Pr,Nbt,Le, 0.,n 0.5 Nc= 0.5,,.0, Fig. Fig.7 7. Teperature profile profile against against Nc Nc. Bi 5,M,Nc 0.5,Nbt,Le, 0.,n 0.5 Bi 5,M,Nc 0.5,Nbt,Pr, 0.,n 0.5 Le= 0.5,,.0, Pr=,,.0, Fig. Fig.5 5. Teperature profile profile against against Le Le. Bi 5,M,Nc 0.5,Le,Pr, 0.,n 0.5 Nbt= 0.5,,.0, Fig. Fig.6 6. Teperature profile against against Nbt Nbt Fig. Fig.8 8. Teperature profile profile against against Pr Pr. Figs are drawn to exaine the effects of iportant paraeters on the nano particles voluetric fraction. Although soe paraeters are not directly involved in the nano particles voluetric fraction equation, but there effects are appearing due to the coupling of Eqs.(18) and (). Eq. () is second order equation in both θ and φ. So value of Nbt will play the doinant rule when effects of paraeters involved in θ are discussed for φ. Fig.9 shows the variation of nano particles voluetric fraction against Bi. The nano particles voluetric fraction increases with the increase in Bi. Since φ is equal to 1 at wall so its value reains fixed for different values of Bi while θ (0) was varying with Bi values (see Fig.4). 735

8 S. Nadee and S. T. Hussain /JAFM, Vol. 9, No., pp , 016. Le,M,Nc 0.5,Nbt,Pr, 0.,n 0.5 Bi 5,M,Nc 0.5,Le,Pr, 0.,n 0.5 Bi= 0.5,,.0, 100 Nbt= 0.5,,.0, Fig.9 Fig. Nano 9. Nano particles particles voluetric voluetric fraction against fraction Bi against Bi Bi 5,M,Nc 0.5,Nbt,Pr, 0.,n 0.5 Fig.1 1. Nano Nano particles particles voluetric voluetric fraction fraction against against Nbt. Bi 5,M,Nc 0.5,Nbt,Le, 0.,n Le= 0.5,, 1.5, Fig.10 Nano 10. Teperature particles voluetric profile fraction against against Le. Le 0. Bi 5,Pr,Nc 0.5,Nbt,Le, 0.,n 0.5 M =,,, Fig.11 Fig. Nano 11. Teperature particles voluetric profile fraction against against M. M It is noticed that φ decreases with the increase in Le and decreases with the increase in M (see Figs.10 and 11). The concentration boundary layer thickness decreases quickly as Le increase fro 0.5 to 1. When the value of Le is less than 1, it iplies Brownian diffusions have larger value than the theral diffusion. Large Brownian diffusion will create larger penetration depth for concentration boundary layer which can be seen fro Fig.10 for Le=0.5. Concentration boundary layer increases with the increase in M as seen in Fig. 11. When the value of M increases it excites fluid particles otion which will diffuses quickly into the neighboring fluid layers due to the enhanced Brownianotion. 0. Pr=, 1., 1.6, Fig.13 Nano 13. Nano particles particles voluetric voluetric fraction fraction against Pr againt Pr. Figs.1 and 13 shows that φ decrease with the increase in Nbt and Pr. Since the Brownian and therophoretic diffusion both cause the dispersion of particles across the boundary layer, thus the concentration profile decrease with the increase in Nbt across the flow field. Its effects on the concentration profile are ore proinent as copared to the teperature profile due to the doinance of ass diffusion of nanoparticles. Nadee and Haq (014) have shown that concentration profile decreases with the increase in Prandtl nuber up to certain value of η and decrease afterwards. This effect is reoved if product of Le and Pr is introduced instead of Sc in the nano particles equation and graph shows uniforly decreasing behavior against Pr throughout the boundary layer. Tables.3 and 4 are drawn to exaine the effects of iportant paraeters on the local Nusselt and Sherwood nubers respectively. In order to copare the analytical results with nuerical results, we used built in Maple 16 algorith to obtain nuerical solution for the boundary value proble. It is found that nuerical results are in good coparison with the analytic results up to three decial places. It is observed that the reduced Nusselt nuber (Nu/ Re) decreases with the increase in M, when other paraeters are kept fixed while it increases with the increase in Pr. As the value of Pr increases fro to.0,reduced Nusselt nuber increases by 83%. 736

9 S. Nadee and S. T. Hussain /JAFM, Vol. 9, No., pp , 016. Table 3 Values of - θ (0). n+1 M Pr Nc Le Nbt Bi OHAM Nuerical It is noticed that increasing the value of Nc decreases the wall teperature gradient. When Nc value is increased fro 0.5 to, value of reduced Nusselt nuber is decreased by 17.06% and for the next 0.5 increase in Nc, it is decreased by 17.9%. So reduced Nusselt nuber decreases ore quickly for greater than 1 value of Nc in coparison to less than 1 values. Le is iportant paraeter due to involveent of theral diffusivity and Brownian diffusion coefficient. It is observed that reduced Nusselt nuber shows increasing behavior with the increase in Le. When Le=0.5, the value of reduced Nusselt nuber is 004. It can be calculated that when the theral diffusivity is increased fro half the value of Brownian diffusion coefficient to twice the value of Brownian diffusion coefficient, reduced Nusselt nuber shows an enorous increase of 9%. When the value of Nbt is increased beyond 1, Brownian diffusion coefficient becoes doinant over the therophoretic diffusion coefficient. This effects is visible in Table.3, reduced Nusselt nuber shows increasing behavior with the increasing value of Nbt. Due to convective boundary condition Bi is the iportant paraeter involved in heat transfer. We coputed reduced Nusselt nuber for several values of Bi to exaine the effect of Bi on heat transfer when values are less than and greater than 1. When Bi is increased fro 0.1 to 0., reduced Nusselt nuber is increased by 77.7%. For the next increase in Bi value, reduced Nusselt nuber is increased by 16.%. As Bi is increased fro to 10, reduced Nusselt nuber is increased by 55.7%. For the large increase in Bi fro 10-50, reduced Nusselt nuber just shows an increase of 5.1%. After exaining the effects of Bi on heat transfer, we can conclude that over all heat transfer rate increases with the increase in Bi. Increase in reduced Nusselt nuber is extreely significant for sall values of Bi less than 1 (specifically about 0.1) and this increase starts reducing as Bi is increased beyond 1 and specifically 10. Table.4 shows the variation in reduced Sherwood nuber (Sh/ Re) against M, Pr, Nc, Le and Nbt. Reduced Sherwood nuber decrease with the increase in M. When value of M is increased fro 0 to 1, values of reduced Nusselt nuber decrease by 10.1% while value of reduced Sherwood nuber decrease by 9.%. So M has ore effect on wall teperature gradient as copared to wall concentration gradient. Reduced Sherwood nuber shows increasing trend with the increase in Pr and Nc. Fro Table.3 and 4, it is noticed that Nc has opposite effects on the reduced Nusselt nuber and Sherwood nuber. Le shows the increasing effect on reduced sherwood nuber. Reduced Sherwood nuber is increased by 108% as Le value is increased fro to.0. It can be calculated that when Le is equal to 1.3 the value of reduced Sherwood nuber is For the increase of 0.5 in Le (-1.3), reduced Sherwood nuber is increased by 5% and for the sae increase of 0.5 (1.5-.0), it is increased by 3.6%. This shows that reduced Sherwood nuber is greatly affected by the value of Le. It is also observed that reduced Sherwood nuber increases with the increase in Nbt.Table.5 is drawn to copare the values of wall teperature gradient for different types of fluid when n=1/. It is noticed fro the table that wall teperature gradient decrease as fluid is changed fro viscous to Williason MHD nano fluid when soe of the paraeters have fixed values as entioned in front of the. By Fourier law of heat conduction =, k has inverse relation with the teperature gradient for fixed heat flux. Therefore, Williason MHD nano fluid has higher theral conductivity than all other fluids entioned in Table.5. Table 4 Values of - φ (0) n+1 M Pr Nc Le Nbt HAM Nuerical CONCLUDING REMARKS In present article flow of MHD Williason nano fluid over a non-linearly stretching surface with convective boundary conditions has been discussed in detail. The governing PDE's are transfored to coupled syste of nonlinear ODE's and OHAM is eployed to solve the resulting syste of ODE's. OHAM results have been copared with nuerical 737

10 S. Nadee and S. T. Hussain /JAFM, Vol. 9, No., pp , 016. results. The ain findings of the paper are as follows Better values of total squared residual errors are obtained by optiizing the initial guess. OHAM results are in good coparison with the nuerical results. Matheatically it is appropriate to introduce product of Lewis and Prandtl instead of Schidt nuber in the nano particles voluetric fraction equation. In the neighborhood of 0.1 a sall increental increase in reduced Biot nuber shows enorous increase in the reduced Nusselt nuber. Nc has opposite effect on the odified Nusselt and odified Sherwood nuber. Theral conductivity of MHD Williason nano fluid is higher than that of siple Williason fluid. Table 5Values of wall teperature gradient for different fluids Fluid type -θ'(0) Viscous fluid (λ=0, M=0, Pr=) 0.78 Williason fluid (λ=0.3, M=0, Pr=) 0.77 MHD viscous fluid (λ=0, M=, Pr=) MHD Williason fluid (λ=0.3, M=, Pr=) Williason nano fluid (λ=0.3, M=0, Pr=, Nc=0.5, Le=, Nbt=) 501 ACKNOWLEDGEMENT This research was supported by the Higher Education Coission of Pakistan (PhD indigenous schee PIN no PS1-576). REFERENCES Abbasbandy, S. (007). The application of hootopy analysis ethod to solve a generalized HirotaSatsua coupled KdV equation.physics Letters A 361(6), Abbasbandy, S. and H. R. Ghehsareh (01). Solutions of the agnetohydrodynaic flow over a nonlinear stretching sheet and nano boundary layers over stretching surfaces.international Journal for Nuerical Methods in Fluids 70, Aziz, A. (009). A siilarity solution for lainar theral boundary layer over a flat plate with a convective surface boundary condition. Counications in Nonlinear Science and Nuerical Siulation 14, Bang, I. C. and S. H. Chang (005). Boiling heat transfer perforance and phenoena of Al 0 3 water nano-fluids fro a plain surface in a pool, International Journal of Heat and Mass Transfer 48(1), Buongiorno, J. (006). Convective transport in nanofluids.journal of Heat Transfer18, Choi, S. (1995). Enhancing Theral Conductivity of Fluids With nanoparticles in Developents and Applications of non-newtonian Flows.D. A.Siginer and H. P. Wang eds., ASME. 66, Cortell, R. (007). Viscous flow and heat transfer over a nonlinearly stretching sheet, Applied Matheatics and Coputation 184, Dapra, I. and G. Scarpi (007). Perturbation solution for pulsatile flow of a non-newtonian Williason fluid in a rock fracture, International Journal of Rock Mechanics and Mining Sciences 44, Fan, T. and X. You(013). Optial hootopy analysis ethod for nonlinear differential equations in the boundary layer, Nuerical Algoriths 6, Hady, F. M., F. S. Ibrahi, S. M. Abdel-Gaied and M. R. Eid(01). Radiation effect on viscous flow of a nanofluid and heat transfer over a nonlinearly stretching sheet.nanoscale Research Letters 7, 9. Hayat, T. and M. Qasi(010). Influence of theral radiation and Joule heating on MHD flow of a Maxwell fluid in the presence of therophoresis.international Journal of Heat and Mass Transfer 53, Khan, W. A. and I. Pop(010). Boundary-layer flow of a nanofluid past a stretching sheet.international Journal of Heat and Mass Transfer 53, Kuznetsov, A.V. (011). Nanofluid bioconvection in water-based suspensions containing nanoparticles and oxytactic icroorganiss: oscillatory instability. Nanoscale Research Letters 6, 100. Liao, S. (003).Beyond Perturbation: Introduction to the Hootopy Analysis Method, Chapan and Hall/CRC Liao, S. (010). An optial hootopy-analysis approach for strongly nonlinear differential equations.counications in Nonlinear Science and Nuerical Siulation 15, Liao, S. (01).Hootopy Analysis Method in Nonlinear Differential Equations. Springer &Higher Education Press, Heidelber. Makinde,O.D and A. Aziz(011). Boundary layer flow of a nanofluid past a stretching sheet with a convective boundary condition.international Journal of Theral Sciences50(7), Malvandi, A.and D.D. Ganji(014). Brownian otion and therophoresis effects on slip flow of aluina/water nanofluid inside a circular icrochannel in the presence of a agnetic field.international Journal of Theral Sciences 84,

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12 S. Nadee and S. T. Hussain /JAFM, Vol. 9, No., pp ,