INFLUENCE OF THERMOPHORESIS AND JOULE HEATING ON THE RADIATIVE FLOW OF JEFFREY FLUID WITH MIXED CONVECTION

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1 Brazilian Journal of Cheical Engineering ISSN Printed in Brazil Vol. 3 No. 4 pp October - Deceber 3 INFLUENCE OF THERMOPHORESIS AND JOULE HEATING ON THE RADIATIVE FLOW OF JEFFREY FLUID WITH MIXED CONVECTION S. A. Shehzad * A. Alsaedi and T. Hayat Departent of Matheatics Quaid-i-Aza University 453 Islaabad 44 Paistan. Departent of Matheatics Faculty of Science King Abdulaziz University P.O. Box 857 Jeddah 589 Saudi Arabia. E-ail: ali_qau7@yahoo.co (Subitted: June 9 ; Revised: Noveber 8 ; Accepted: Deceber ) Abstract - The ai of the present study is to address the agnetohydrodynaic (MHD) radiative flow of an incopressible Jeffrey fluid over a linearly stretched surface. Heat and ass transfer characteristics are accounted for in the presence of Joule heating and therophoretic effects. Series solutions by the hootopy analysis ethod are constructed for the velocity teperature and concentration fields. A convergence criterion for the series solutions is discussed. In addition the nuerical values of the sin friction coefficient local Nusselt and Sherwood nubers are first coputed and then analyzed. Keywords: Radiative flow; MHD Jeffrey fluid; Joule heating; Therophoresis. INTRODUCTION The analysis of non-newtonian fluids is significant because of several industrial and engineering applications. Such fluids are encountered in the process of anufacturing coated sheets foods drilling uds cosetic products dilute polyer solutions polyeric elts etc. Different odels of such fluids have beenare suggested. This is due to the versatility of fluid characteristics in nature (Qi and Xu 7; Kothandapani and Srinivas 8; Nadee and Abar ; Jail and Fetecau ab; Wang and Tan ; Hayat et al. ab; Jail et al. ; Nadee et al. ; Hayat et al. ). Although uch inforation is available on the boundary layer flow of viscous fluids such attepts for non- Newtonian fluids are liited. In fact the resulting differential equations in non-newtonian fluids are ore nonlinear than for a viscous fluid. To find the analytic/nuerical solution of such equations is not an easy tas. Investigation of heat transfer in the boundary layer over a stretching sheet has ey iportance in extrusion processes paper production glass drawing electronic chips crystal growing plastic anufacture etal spinning and cooling of etallic plates in a cooling bath etc. For the quality of the final product in such processes the cooling rate has great relevance (Bhattacharyya et al. ; Hayat et al. ; Rashidi and Pour ; Vyas and Rai ; Yao et al. ; Rashidi and Erfani ; Rashidi et al. ; Mainde and Aziz ; Sahoo and Poncet ). The igration of sall particles in the direction of decreasing theral gradient is called therophoresis (Hinds 98). This is used for particle collection. The velocity gained by the particle is called the therophoretic velocity and the force experienced by the suspended particle due to the teperature difference is called the therophoretic force (Tasai et al. 4). Therophoresis is a echanis for particles on a cold surface. It is quite significant in aerosol particle sapling deposition of *To who correspondence should be addressed

2 898 S. A. Shehzad A. Alsaedi and T. Hayat silicon thin fils radioactive particle deposition in nuclear reactor safety siulations scale foration on surfaces of heat exchangers icroelectronic anufacturing particulate aterial deposition on turbine blades odified cheical vapour deposition etc. Goren (977) studied the lainar flow of a viscous fluid with therophoresis in the presence of cold and hot plate conditions. The effect of therophoresis in MHD flow of a viscous fluid over an inclined plate was anlyzed by Asla et al. (8). Hayat and Qasi () investigated the radiative flow of Maxwell fluid in the presence of Joule heating and therophoresis effects. Hayat and Alsaedi () further exained the therophoresis effects in MHD flow of Oldroyd-B fluid over a linearly stretching surface. Heat and ass transfer effects in the therophoretic flow of viscous fluid over an inclined isotheral pereable surface have been explored by Noor et al. (). Motivated by the above entioned studies the ai here is to investigate the effects of therophoresis in MHD flow of a non-newtonian fluid over a stretching surface. Matheatical odelling is presented using constitutive equations of a Jeffrey fluid. Theral radiation and Joule heating effects are taen into account. The present study is structured in the following fashion. The atheatical forulation is copleted in the next section. Series solutions by the hootopy analysis ethod (HAM) (Liao 3; Yao 9; Rashidi et al. b; Hayat et al. c; Vosughi et al. ; Aziz et al. ; Hayat et al. d) are developed in the subsequent sections. Then convergence analysis and discussion are presented. Iportant results are suarized in the last section. X - Axis Y - Axis u = u w v = T = Tw C = C w. u B u y B Figure : Physical odel T T C C as y. The species concentration at the surface C w and abient concentration C are constants. Invoing the Rosseland approxiation [4] the resulting equations tae the following fors: u v + = x y () 3 3 u u u + v 3 u u ν u u + v = +λ x y y x y +λ y u u u u + () x y y x y σ B u + g [ βt( T T ) +βc( C C )] ρ 3 6 T T T σt T u + v = + x y ρc y 3 y p μ u σ B + + u ρc p y ρcp (3) FLOW FORMULATION C C C u + v = D ( V ). TC x y y y (4) We consider Cartesian coordinate syste in such a way that the x axis is along the stretching surface and the y axis is perpendicular to the x axis. Magnetohydrodynaic boundary layer flow of a Jeffrey fluid is considered. A unifor agnetic field B is applied parallel to the y axis (see Figure ). The induced agnetic field is neglected for sall agnetic Reynolds nuber. Heat and ass transfer characteristics are taen into account in the presence of theral radiation and therophoresis effects. The unifor teperature of the surface T w is larger than the abient fluid teperature T. Here ( uv ) are the velocity coponents along the x and y axes λ and λ are the ratio of relaxation/retardation ties and retardation tie respectively ν the dynaic viscosity ρ the density of the fluid σ the electrical conductivity g the gravitational acceleration β T and β c the theral expansion coefficients T the teperature c p the specific heat σ the Stefan-Boltzann constant the ean absorption coefficient D the diffusion coefficient and V T the therophoretic velocity. The associated boundary conditions are: Brazilian Journal of Cheical Engineering

3 Influence of Therophoresis and Joule Heating on the Radiative Flow of Jeffrey Fluid with Mixed Convection 899 u = u = ax v = T = T ( x) C = C ( x) at y = w w w u u T T C C y (5) φ + Sc( f φ φf ) Sc τ( φθ φθ ) = () f = f = θ= φ = at η = f f θ φ as η. (3) as y. where u w is the stretching velocity T w is the teperature at the wall C w is the concentration at the wall and T and C are the abient fluid teperature and concentration respectively. The ter V T in Eq. (4) can be defined as follows: ν T V = (6) T y T r in which is the therophoretic coefficient and T r is the reference teperature. A therophoretic paraeter τ is defined by the following relation: ( Tw T τ= ). T r The wall teperature and concentration fields are: (7) T = T + bx C = C + cx (8) w w where a b and c are the positive constants. Through the following transforations: a u = axf ( η ) v = avf( η) η= y v T T C C θη ( ) = φη ( ) = T T C C w w (9) Equation () is autoatically satisfied and Eqs. ()-(4) are reduced as follows: In the above expressions β =λa is the Deborah nuber M = σ B / ρc the Hartann nuber γ= Gr x the local buoyancy paraeter Rex 3 gβt( Tw T ) x / ν Grx = the local Grashof nuber uwx / ν βc( Cw C ) N = the constant diensionless concentration buoyancy paraeter Pr = the Prandtl β T( Tw T ) μc p 3 4σ T nuber R = the radiation paraeter uw ν Ec = the Ecert nuber and Sc = cp( Tw T ) D the Schidt nuber. The sin friction coefficient local Nusselt nuber and local Sherwood nuber are transfored to the following: Re () () Re ( ) / / x Cf = f +βf Nux x +λ / x = θ () and Sh Re = φ (). SERIES SOLUTIONS (4) The hootopic solutions for f θ and φ in a set of base functions: { η exp( nη) n } (5) f + ( +λ )( ff f ) +β( f ff ) ( +λ ) Mf + ( +λ ) γ( θ+ Nφ ) = 4 ( + R) θ + Pr[ fθ θ f ] + Pr Ecf 3 + M Pr Ecf = () () can be written to the following expressions: n n= = f ( η ) = a η exp( n η) (6) n n= = θ ( η ) = b η exp( n η) (7) Brazilian Journal of Cheical Engineering Vol. 3 No. 4 pp October - Deceber 3

4 9 S. A. Shehzad A. Alsaedi and T. Hayat φ ( η ) = c η exp( n η) (8) n n= = in which a n b n and c n are the coefficients. Initial guesses and the auxiliary linear operators are f ( η ) = ( exp( η) θ ( η) = exp( η) φ ( η ) = exp( η) (9) L = f f L = f f L = f f. () f θ The operators in Eq. (9) satisfy the following properties: η η η η = θ Lf ( C C e C e ) L ( C e C e ) () = L ( C e + C e ) φ φ η η 6 7 where C i ( i = 7) denote the arbitrary constants. The zeroth order deforation probles are expressible in the for ( q) L ( ˆ f f( η; q) f( η )) = q ( ˆ η θˆ η φˆ fn f f q q η q ) ( ; ) ( ; ) ( ; ) ( q) L ( ˆ θ θ( η; q) θ( η )) = q ( ˆ ˆ ˆ θnθ θηq f ηq φηq ) ( ; ) ( ; ) ( ; ) ( q) L ( ˆ φ φ( η; q) φ( η )) = q ( ˆ ˆ ˆ φnφ θηq f ηq φηq ) ( ; ) ( ; ) ( ; ) () (3) (4) ˆ ˆ (; ) (; ) ˆ ( ; ) ˆ f q = f q = f q = f ( ; q) = (5) θ ˆ(; q) = θˆ( ; q) = φ ˆ(; q) = and φˆ( ; q) =. Here q is the ebedding paraeter f θ and φ the non-zero auxiliary paraeters and the nonlinear operators N f N θ and N φ are given by: 3 ˆ ˆ ˆ ˆ ˆ ˆ f( η q) ˆ ( ) ( ) [ ( ) ( ; ) ( ; )] ( 3 ) f η q f η q N η θ η φ η = + +λ ( η ) f f q q q f q η η η ˆ 4 ( ) ˆ( ) ˆ f η q f η q ( η ) +β ˆ( η ) f q f q 4 ( + λ ) M η η η ( ˆ q Nˆ q ) + ( +λ ) γ( θ( η ) + φ( η; ) (6) 4 θη ˆ( q) fˆ( η q) fˆ( η q) N [ ˆ( ) ˆ( ; ) ˆ θ f η q θ η q φ( η ; q)] = + R + PrEc + M PrEc 3 η η η ˆ( ) ˆ ˆ f η q ˆ θ( η q) Pr θ( η q) + Pr f( η q) η η (7) ˆ ˆ ˆ ˆ ˆ ˆ φη ( q) ˆ φη ( q) f( η q) [ ( ) ( ; ) ( ; )] ( ) ˆ Nφ f η q θ η q φ η q = + Sc η φ( η ) f q q η η η ˆ ˆ ˆ θ( η q) φ( η q) θ( η q) Scτ φˆ( η q ). η η η (8) Brazilian Journal of Cheical Engineering

5 Influence of Therophoresis and Joule Heating on the Radiative Flow of Jeffrey Fluid with Mixed Convection 9 When q = and q = one has fˆ( η ;) = f ( η); fˆ( η ;) = f ( η) (9) θη ˆ( ;) =θ( η); θη ˆ( ;) =θη ( ) (3) φη ˆ( ;) =φ( η); φη ˆ( ;) =φη ( ). (3) Note that when q increases fro to then f ( η q ) θηq ( ) and φηq ( ) approach f ( η) to f ( η) θ ( η ) to θ( η ) and φ ( η ) to φη ( ). According to Taylor's series one has: f( η q) = f ( η ) + f ( η) q f = f( η; q) ( η ) =! η θη ( q) =θ( η ) + θ ( η) q = q= θ( η; q) θ( η ) =! η q= (3) (33) The general solutions for f θ and φ in ters of special solutions f θ and φ are η η 3 f ( η ) = f ( η ) + C + C e + C e (38) η η 4 5 θ( η ) =θ( η ) + Ce + Ce (39) η η 6 7 φ( η ) =φ( η ) + Ce + Ce. (4) CONVERGENCE ANALYSIS AND DISCUSSION The auxiliary paraeters f θ and φ play an iportant role in controlling and adjusting the convergence of the series solutions. To find a suitable value for each of these auxiliary paraeters the curves at the 9th order of approxiation are displayed. Figure indicates that the adissible values of f θ and φ are. f.. θ. and. φ.. Our series solutions converge in the whole region of η for = = =.7 f θ φ. φη ( q) =φ( η ) + φ ( η) q = φ( η; q) φ( η ) =.! η q= (34) The convergence of series ()-(4) is closely associated with f θ and φ. The values of f θ and φ are chosen such that the series ()-(4) converge at q =. Thus f( η ) = f ( η ) + f ( η) (35) = = θη ( ) =θ( η ) + θ ( η) φη ( ) =φ( η ) + φ ( η). = (36) (37) Figure : curves for the functions f θ and φ Figures 3-8 plot the effects of the various interesting paraeters on the velocity f ( η) teperature θη ( ) and concentration φη ( ). The fluid velocity and oentu boundary layer thicness increase with an increase in Deborah nuber β (see Figure 3). In Figure 4 the influence of the ratio of relaxation to the retardation ties is setched for the fluid velocity. It shows that the fluid velocity decreases upon increasing λ. Figure 5 shows that an increase in the Brazilian Journal of Cheical Engineering Vol. 3 No. 4 pp October - Deceber 3

6 9 S. A. Shehzad A. Alsaedi and T. Hayat local buoyancy paraeter γ yields an increase in the velocity. In fact the local buoyancy paraeter depends on the buoyancy force and an increase in buoyancy force gives rise to the fluid velocity increase. An increase in Hartann nuber M decreases the fluid velocity (see Figure 6). The Hartann nuber is a consequence of the Lorentz force. Obviously an increase in the Lorentz force opposes the flow and thus the fluid velocity decreases. Figure 7 shows that the velocity and oentu boundary layer thicness are decreasing functions of the Prandtl nuber Pr. Figure 8 displays the effects of the ratio of buoyancy paraeter N. The effects of the ratio of the buoyancy paraeter are siilar to that of γ in a qualitative way. The difference we noticed is that the fluid velocity increases ore rapidly in the case of increasing local buoyancy paraeter when copared with the ratio of the buoyancy paraeter. Figure 3: Effect of β on velocity f ( η) Figure 4: Effect of λ on velocity f ( η) Figure 5: Effect of γ on velocity f ( η) Figure 6: Effect of M on velocity f ( η) Figure 7: Effect of Pr on velocity f ( η) Figure 8: Effect of N on velocity f ( η) Brazilian Journal of Cheical Engineering

7 Influence of Therophoresis and Joule Heating on the Radiative Flow of Jeffrey Fluid with Mixed Convection 93 Effects of different paraeters on the teperature are seen in Figures 9-4. Figure 9 depicts the effects of the local buoyancy paraeter on the teperature field. The teperature field and theral boundary layer thicness are reduced with an increase in γ. Figure shows that the teperature increases when the Hartann nuber is increased. In fact large M corresponds to lower pereability and hence the teperature and the theral boundary layer thicness increase. Figures and illustrate the influences of Ecert nuber Ec and Prandtl nuber Pr on the teperature. It is observed fro these figures that Ec and Pr have quite opposite effects on the velocity and the associated theral boundary layer thicness. The teperature profile decreases ore quicly for Ec in coparison to Pr. The ratio of the buoyancy paraeter and the radiation paraeter correspond to a decrease and an increase in the teperature respectively (see Figures 3 and 4). Figure 9: Effect of γ on teperature θη ( ) Figure : Effect of M on teperature θη ( ) Figure : Effect of Ec on teperature θη ( ) Figure : Effect of Pr on teperature θη ( ) Figure 3: Effect of N on teperature θη ( ) Figure 4: Effect of R on teperature θη ( ) Brazilian Journal of Cheical Engineering Vol. 3 No. 4 pp October - Deceber 3

8 94 S. A. Shehzad A. Alsaedi and T. Hayat The effects of γ M N and R on concentration profile are seen through Figures 5-8. Figure 5 shows that the concentration and boundary layer thicness are decreasing functions of γ. It is found fro Figure 6 that the Hartann nuber increases the concentration profile and associated boundary layer thicness. The effects of N and R on φη ( ) are siilar. Increases in N and R decrease the concentration and related boundary layer thicness (see Figures 7 and 8). It is worth entioning here that the effects of M on the teperature are ore pronounced when copared with the concentration. This sae observation also holds for the influences of γ N and R on the teperature and concentration. Table shows the convergence of the series solutions through coputations of nuerical values. This table indicates that our series solutions converge fro the th-order of deforations for velocity and teperature and 5th-order of approxiation for the concentration. Table includes the nuerical values of the sin-friction coefficient for the different values of β λ τ M Sc Sc Ec and Pr when N =.3 and R =.4. The sin-friction coefficient increases with increasing β M Sc and Pr. It decreases when λ γ and Ec are increased. Interestingly the variation in the values of the sin-friction coefficient is very sall with increasing therophoretic paraeter τ. Table 3 consists of the values of the local Nusselt and Sherwood nubers. The values of the local Nusselt and Sherwood nubers increase slowly in coparison to the increase in the values of the sinfriction coefficient when β increases. The increase in the values of the local Nusselt nuber is very sall upon increasing τ but the increase in the values of the local Sherwood nuber is large. A coparative study of Tables and 3 shows that the values of the sin-friction coefficient and local Sherwood nuber are larger than the values of the local Nusselt nuber. Figure 5: Effect of γ on concentration φη ( ) Figure 6: Effect of M on concentration φη ( ) Figure 7: Effect of N on concentration φη ( ) Figure 8: Effect of R on concentration φη ( ) Brazilian Journal of Cheical Engineering

9 Influence of Therophoresis and Joule Heating on the Radiative Flow of Jeffrey Fluid with Mixed Convection 95 Table : Convergence of series solutions for different order of approxiations when β =. γ=.4 τ= M =.6 Sc =.7 λ = Ec =.5 Pr =. N =.3 R=.4 and = = =.7. f θ φ Order of approxiations f () θ () φ () Table : Nuerical values of sin-friction coefficient for different values of β λ τ M Sc Ec and Pr when N =.3 and R =.4. β λ γ τ M Sc Ec Pr ( f () f ()) +β +λ Brazilian Journal of Cheical Engineering Vol. 3 No. 4 pp October - Deceber 3

10 96 S. A. Shehzad A. Alsaedi and T. Hayat Table 3: Nuerical values of local Nusselt nuber -θ () and local Sherwood nuber φ ( ) for different values of β λ τ M Sc Ec and Pr when N=.3 and R=.4. β λ γ τ M Sc Ec Pr -θ () φ ( ) CLOSING REMARKS In this study we discussed the effects of theral radiation Joule heating and therophoresis in stretched flow of a Jeffrey fluid. The ain observations are as follows: Effects of β and λ on the fluid velocity are quite opposite. An increase in the ratio of the buoyancy paraeter N corresponds to an increase in f ( η). The fluid teperature rises with an increase in Ecert nuber Ec. Effects of the ratio of the buoyancy paraeter and radiation paraeter are qualitatively siilar. Variations in teperature are ore pronounced than those in concentration when M increases. The increase in the values of the sin-friction coefficient and the local Nusselt nuber is very sall in coparison to the increase in the local Sherwood nuber when τ increases. Nuerical values of the local Nusselt nuber increase and the values of the local Sherwood nuber decrease with increasing Prandtl nuber Pr. ACKNOWLEDGMENTS The research of Alsaedi was partially supported by the Deanship of Scientific Research (DSR) King Abdulaziz University Jeddah Saudi Arabia. We are also thanful to the reviewer for his/her useful suggestions. REFERENCES Asla M. S. Rahan M. M. and Sattar M. A. Effects of variable suction and therophoresis on Brazilian Journal of Cheical Engineering

11 Influence of Therophoresis and Joule Heating on the Radiative Flow of Jeffrey Fluid with Mixed Convection 97 steady MHD cobined free-forced convective heat and ass transfer flow over a sei-infinite pereable inclined plate in the presence of theral radiation. Int. J. Theral Sci. 47 p. 758 (8). Aziz R. C. Hashi I. and Abbasbandy S. Effects of therocapillarity and theral radiation on flow and heat transfer in a thin liquid fil on an unsteady stretching sheet. Math. Probles Eng. DOI:.55//73. Bhattacharyya K. Muhopadhyay S. and Laye G. C. Slip effects on an unsteady boundary layer stagnation-point flow and heat transfer towards a stretching sheet. Chin. Phy. Lett. 8 p. 947 (). Goren S. L. Therophoresis of aerosol particles in lainar boundary layer on flat plate. J. Colloid Interf. Sci. 6 p. 77 (977). Haitao Q. and Mingyu X. Unsteady flow of viscoelastic fluid with fractional Maxwell odel in a channel. Mech. Research Coun. 34 p. (7). Hayat T. and Alsaedi A. On theral radiation and Joule heating effects in MHD flow of an Oldroyd-B fluid with therophoresis. Arab. J. Sci. Eng. 36 p. 3 (). Hayat T. and Qasi M. Influence of theral radiation and Joule heating on MHD flow of a Maxwell fluid in the presence of therophoresis. Int. J. Heat Mass Transfer 53 p. 478 (). Hayat T. Qasi M. and Abbas Z. Radiation and ass transfer effects on the agnetohydrodynaic unsteady flow induced by a stretching sheet. Z Naturforsch A. 64 p. 3 (). Hayat T. Shehzad S. A. and Qasi M. Mixed convection flow of a icropolar fluid with radiation and cheical reaction. Int. J. Nu. Methods Fluids 67 p. 48 (). Hayat T. Shehzad S. A. Qasi M. and Obaidat S. Flow of a second grade fluid with convective boundary conditions. Theral Sci. 5 p. S53 (). Hayat T. Shehzad S. A. Qasi M. and Obaidat S. Radiative flow of Jeffery fluid in a porous ediu with power law heat flux and heat source. Nuclear Eng. Design 43 p. 5 (). Hayat T. Shehzad S. A. Qasi M. and Obaidat S. Steady flow of Maxwell fluid with convective boundary conditions. Z. Naturforsch 66a p. 47 (). Hayat T. Shehzad S. A. Qasi M. and Obaidat S. Theral radiation effects on the ixed convection stagnation-point flow in a Jeffery fluid. Z. Naturforsch 66a p. 66 (). Hinds W. C. Aerosol Technology: Properties Behavior and Measureent of Airborne Particles. John Wiley and Sons New Yor (98). Jail M. and Fetecau C. Helical flows of Maxwell fluid between coaxial cylinders with given shear stresses on the boundary. Nonlinear Analysis: Real World Appl. p. 43 (). Jail M. and Fetecau C. Soe exact solutions for rotating flows of a generalized Burgers' fluid in cylindrical doains. J. Non-Newtonian Fluid Mech. 65 p. 7 (). Jail M. Khan N. A. and Zafar A. A. Translational flows of an Oldroyd-B fluid with fractional derivatives. Coput. Math. Appl. 6 p. 54 (). Kothandapani M. and Srinivas S. Peristaltic transport of a Jeffrey fluid under the effect of agnetic field in an asyetric channel. Int. J. Non-Linear Mech 43 p. 95 (8). Liao S. J. Beyond Perturbation: Introduction to Hootopy Analysis Method. Chapan and Hall CRC Press Boca Raton (3). Mainde O. D. and Aziz A. Boundary layer flow of a nano fluid past a stretching sheet with convective boundary conditions. Int. J. Theral. Sci. 5 p. 36 (). Nadee S. and Abar N. S. Peristaltic flow of a Jeffrey fluid with variable viscosity in an asyetric channel. Zeitschrift fur Naturforschung A 64 p. 73 (). Nadee S. Zaheer S. and Fang T. Effects of theral radiation on the boundary layer flow of a Jeffery fluid over an exponentially stretching surface. Nuer. Algor. 57 p. 87 (). Noor N. F. M. Abbasbandy S. and Hashi I. Heat and ass transfer of therophoretic MHD flow over an inclined radiate isotheral pereable surface in the presence of heat source/sin. Int. J. Heat Mass Transfer 55 (). Rashidi M. M. and Erfani E. A new analytical study of MHD stagnation-point flow in porous edia with heat transfer. Coput. Fluids 4 p. 7 (). Rashidi M. M. and Pour S. A. M. Analytic approxiate solutions for unsteady boundary-layer flow and heat transfer due to a stretching sheet by hootopy analysis ethod. Nonlinear Analysis: Modelling and Control 5 p. 83 (). Rashidi M. M. Pour S. A. M. and Abbasbandy S. Analytic approxiate solutions for heat transfer of a icropolar fluid through a porous ediu with radiation. Coun. Nonlinear Sci. Nuer. Siulat. 6 p. 874 (). Rashidi M. M. Pour S. A. M. and Abbasbandy S. Brazilian Journal of Cheical Engineering Vol. 3 No. 4 pp October - Deceber 3

12 98 S. A. Shehzad A. Alsaedi and T. Hayat Analytic approxiate solutions for heat transfer of a icropolar fluid through a porous ediu with radiation. Coun. Nonlinear Sci. Nuer. Siulat. 6 p. 874 (). Sahoo B. and Poncet S. Flow and heat transfer of a third grade fluid past an exponentially stretching sheet with partial slip boundary condition. Int. J. Heat Mass Transfer 54 p. 5 (). Tasai C. J. Lin J. S. Shanar I. Aggarwal G. and Chen D. R. Therophoretic deposition of particles in lainar and turbulent tube flows. Aerosol Sci. Tech. 38 p. 3 (4). Vosughi H. Shivanian E. and Abbasbandy S. A new analytical technique to solve Volterra's integral equations. Math. Methods Appl. Sci. 34 p. 43 (). Vyas P. and Rai A. Radiative flow with variable theral conductivity over a non-isotheral stretching sheet in a porous ediu. Int. J. Contep. Math. Sciences 5 p. 685 (). Wang S. and Tan W. C. Stability analysis of soretdriven double-diffusive convection of Maxwell fluid in a porous ediu. Int. J. Heat Fluid Flow 3 p. 88 (). Yao B. Approxiate analytical solution to the Falner-San wedge flow with the pereable wall of unifor suction Coun. Nonlinear Sci. Nuer. Siulat. 4 p. 33 (9). Yao S. Fang T. and Zhang J. Heat transfer of a generalized stretching/shrining wall proble with convective boundary conditions. Co. Nonlinear Sci. Nu. Siu. 6 p. 75 (). Brazilian Journal of Cheical Engineering